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First Edition

Applied Statistics
Desktop Reference

Phil Crewson
Preface

Approach Used in this Desktop Reference

The Applied Statistics Desktop Reference Guide provides an overview of hypothesis

testing and the interpretation of output generated by statistical software. The Guide
will be most useful for those who have completed a course in statistics and need a
quick refresher or for students who would like to have a reference to support course
instruction. It is not intended to serve as a stand-alone statistical text.

The approach of the Guide is to connect classically taught statistics with statistical
software output. The Guide presents commonly used hypothesis tests and formulas,
provides an example of how to do the calculations, and relates the hypothesis test to
output generated by statistical software.

For many, the most valuable portions of the Guide will be the Glossary and the anno-
tated presentation of output from statistical software.

AcaStat Statistical Software

The software output examples provided in the text and the Glossary should be famil-
iar to anyone who has used one of the popular statistical software packages. Most of
the software output presented in the Guide is created with AcaStat statistical soft-
ware. AcaStat is an easy-to-use statistical software system that analyzes raw data
(electronic data not aggregated).

i
Some examples also use StatCalc. StatCalc analyzes summary data (multiple re-
cords reduced to counts, means, proportions). StatCalc is especially useful for verify-
ing hand calculations and creating z-scores and confidence intervals for means and
proportions.

The AcaStat and StatCalc applications are available at http://www.acastat.com. The

free version of AcaStat allows analysis of up to 50 observations which is sufficient to
explore a “Workbook” data file example provided with the software. Both AcaStat
and StatCalc are available for sale from the AcaStat Software website or the Mac
App Store.

Glossary

The Glossary contains over 200 definitions of statistical terms and concepts and in-
cludes a quick ‘Applied Stat’ review of 26 common statistical techniques with anno-
tated statistical output examples.

About the Author

Philip E. Crewson, Ph.D. taught undergraduate and graduate statistics at American

University for 15 years. He was co-editor of a monthly series in Radiology on biosta-
tistics for radiologists and has served as a Director of Research in government and
non-profit organizations and as a peer reviewer for academic journals. The content
of this Guide is an outline of academic course material used in undergraduate and
graduate university courses.

Copyright

Copyright © 2014 by Philip E. Crewson. All rights reserved under International and
Pan-American Copyright Conventions. By payment of the required fees, you have
been granted the non-exclusive, non-transferable right to access and read the text of
this e-book on-screen. No part of this text may be reproduced, transmitted, down-

ii
loaded, decompiled, reverse engineered, or stored in or introduced into any informa-
tion storage and retrieval system, in any form or by any means, whether electronic or
mechanical, now known or hereinafter invented, without the express written permis-
sion of the author.

iii
1
Research Design and Reporting

1.1!Research Design

The scientific model guides research design to ensure findings are quantifiable (meas-
ured in some fashion), verifiable (others can substantiate our findings), replicable (oth-
ers can repeat the study), and defensible (provides results that are credible to
others--this does not mean others have to agree with the results). For some the sci-
entific model may seem too complex to follow, but it is often used in everyday life
and should be evident in any research report, paper, or published manuscript. The
corollaries of common sense and proper paper format with the scientific model are
given below.

Scientific Model, Common Sense, and Paper Format Corollaries

Scientific Model Common Sense Paper Format

Research Question Why Introduction

Develop a theory Your answer Introduction
Identify variables How Method
Identify hypotheses Expectations Method
Test hypotheses Analyze data Results
Evaluate the results What it means Conclusion
Critical review What it doesn’t mean Conclusion

4
Overview of first four elements of the Scientific Model

The following discussion provides a very brief introduction to the first four elements
of the scientific model. Although these four elements are not the primary focus of
this text, they are the foundations for systematic research and data analysis and
should be carefully considered before, during, and after statistical analyses. The re-
maining elements that pertain to testing hypotheses and evaluating results are the pri-
mary focus for the remainder of the Desktop Reference Guide.

1. Research Question

The research question should be a clear statement about what the researcher intends
to investigate. It should be specified before research is conducted and openly stated
in reporting the results. One conventional approach is to put the research question in
writing in the introduction of a report starting with the phrase "The purpose of this
study is . . . .“ This approach forces the researcher to clearly identify the research ob-
jective, allow others the to benchmark how well the study design answers the pri-
mary goal of the research, and identify the key abstract concepts involved in the re-
search

Abstract concepts: The starting point for measurement. Abstract concepts are best
understood as general ideas in linguistic form that help us describe reality. They
range from the simple (hot, long, heavy, fast) to the more difficult (responsive, effec-
tive, fair). Abstract concepts should be evident in the research question and/or pur-
pose statement. An example of a research question is given below along with how it
might be reflected in a purpose statement.

Research question: Is the quality of public sector and private sector employees differ-
ent?

Purpose statement: The purpose of this study is to determine if the quality of public
and private sector employees is different.

5
2. Develop Theory

A theory is one or more propositions that suggest why an event occurs. It is the re-
searcher’s explanation for how the world works. These theoretical propositions pro-
vide a framework for analysis that are predisposed to determine “What is reality” not
“What should reality be.” A sound theory must have logical integrity and should con-
sider current knowledge on the event being explored. In other words, a thorough lit-
erature review before the design and implementation of a study is the hallmark of
good research.

3. Identify Variables

Variables are measurable abstract concepts that help describe relationships. This
measuring of abstract concepts is referred to as operationalization. In the previous
research question "Is the quality of public sector and private sector employees differ-
ent?” the key abstract concepts are employee quality and employment sector. To
measure "quality" a measurable representation of employee quality will need to be
found. Possible quality variables may be performance on a standardized intelligence
test, attendance at work, performance evaluations, etc. The variable for employment
sector seems to be fairly self-evident, but a good researcher must be very clear on
how they define and measure the concepts of public and private sector employment
(i.e., how is non-profit employment handled?).

Variables represent empirical indicators of an abstract concept. This does not mean
there is complete congruence between our measure and the abstract concept. Vari-
ables used in statistical analysis are unlikely to measure all aspects of an abstract
concept. Put simply, all variables have an error component.

Abstract concept = indicator + error

Because there is always error in measurement, multiple measures/indicators of one

abstract concept are felt to be better (more valid and more reliable) than one. As

6
shown below, one would expect that as more valid indicators of an abstract concept
are used, the effect of the error term would decline:

Abstract concept = indicator1 + indicator2 + indicator3 + error

The appropriate design and use of multiple indicators is beyond this text, but this ap-
proach is evident in many tools commonly used today such as psychological assess-
ments, employee satisfaction, socio-economic status, and work motivation meas-
ures.

4. Identify measurable hypotheses

A hypothesis is a formal statement that presents the expected relationship between

an independent and dependent variable. A dependent variable is a variable that con-
tains variations the researcher seeks to explain. An independent variable is a vari-
able that is thought to affect (cause) variations in the dependent variable. Although
causation is implied when statistically significant associations are identified between
independent and dependent variables, causation must not rely solely on the empirical
evidence. Proof (or causation) must always be an exercise in rational inference that
considers empirical evidence but also includes other factors such as study design,
prior research, and the basic elements required for causation (discussed later).

Levels of Data

There are four levels of variables. It is essential to be able to identify the levels of
data used in a research design. The level of data used in a research design directly im-
pacts which statistical methods are most appropriate for testing research hypothe-
ses. The four levels of data are listed below in order of their precision.

Nominal: Classifies objects by type or characteristic (sex, race, models of vehicles,

political jurisdictions).

Properties:

7
 • categories are mutually exclusive (an object or characteristic can only be con-
tained in one category of a variable)

• no logical order

Ordinal: Classifies objects by type or kind but also has some logical order (military
rank, letter grades).

Properties:

• categories are mutually exclusive

• logical order exists

• scaled according to amount of a particular characteristic they possess

Interval: Classifies by type, logical order, but also requires that differences between
levels of a category are equal (temperature in degrees Celsius, distance in kilometers,
age in years).

Properties:

• categories are mutually exclusive

• logical order exists

• scaled according to amount of a particular characteristic they possess

• differences between each level are equal

• no zero starting point

Ratio: Classification is similar to interval but ratio data has a true zero starting point
(total absence of the characteristic). For most purposes, the analyses presented in
this text assume interval/ratio are the same.

8
Association

Statistical techniques are used to explore connections between independent and de-
pendent variables. This connection between or among variables is often referred to
as association. Association is also known as covariation and can be defined as
measurable changes in one variable that occur concurrently with changes in another
variable. A positive association is represented by change in the same direction (in-
come rises with education level). Negative association is represented by concurrent
change in opposite directions (hours spent exercising and % body fat). Spurious as-
sociations are associations between two variables that can be better explained by a
third variable. As an example, if after taking cold medication for seven days the symp-
toms disappear, one might assume the medication cured the illness. Most of us, how-
ever, would probably agree that the change experienced in cold symptoms are proba-
bly better explained by the passage of time rather than pharmacological effect (i.e.,
the cold would resolve itself in seven days irregardless of whether the medication
was taken or not).

Causation

There is a difference between determining association and causation. Causation, of-

ten referred to as a relationship, cannot be proven with statistics. Statistical tech-
niques provide evidence that a relationship exists through the use of significance test-
ing and strength of association metrics. However, this evidence must be bolstered
by an intellectual exercise that includes the theoretical basis of the research and logi-
cal assertion. The following presents the basic elements necessary for claiming cau-
sation:

Required elements for causation

1. Association: Do the variables covary?

2. Precedence: Does the independent variable vary before the dependent variable?

9
3. Plausibility: Is the expected outcome consistent with theory/prior knowledge?

4. Nonspuriousness: Are no other variables more important?

Reliability and Validity

The accuracy of our measurements is affected by their reliability and validity. Reliabil-
ity is the extent to which the repeated use of a measure obtains the same values
when no change has occurred (can be evaluated empirically). Validity is the extent to
which the operationalized variable accurately represents the abstract concept it in-
tends to measure (cannot be confirmed empirically-it will always be in question). Reli-
ability negatively impacts all studies but is very much a part of any methodology/
operationalization of concepts. As an example, reliability can depend on who per-
forms the measurement and when, where, and how data are collected (from whom,
written, verbal, time of day, season, current public events).

There are several different conceptualizations of validity. Predictive validity refers to

the ability of an indicator to correctly predict (or correlate with) an outcome (e.g.,
GRE and performance in graduate school). Content validity is the extent to which the
indicator reflects the full domain of interest (e.g., past grades only reflect one aspect
of student quality). Construct validity (correlational validity) is the degree to which
one measure correlates with other measures of the same abstract concept (e.g., days
late or absent from work may correlate with performance ratings). Face validity evalu-
ates whether the indicator appears to measure the abstract concept (e.g., a person's
religious preference is unlikely to be a valid indicator of employee quality).

Study Design

There are two types of study designs; experimental and quasi-experimental.

Experimental: The experimental design uses a control group and applies treatment
to a second group. It provides the strongest evidence of causation through extensive

10
controls and random assignment to remove other differences between groups. Using
the evaluation of a job training program as an example, one could carefully select and
randomly assign two groups of unemployed welfare recipients. One group would be
provided job training and the other would not. If the two groups are similar in all
other relevant characteristics, one could assume any differences between the groups
employment one year later was caused by job training.

Whenever an experimental design is used, both internal and external validity can be-
come very important factors.

Internal validity: The extent to which accurate and unbiased association between the
IV and DVs were obtained in the study group.

External validity: The extent to which the association between the IV and DV is accu-
rate and unbiased in populations outside the study group.

Quasi-experimental: The quasi-experimental design does not have the level of con-
trols employed in an experimental design (most social science research). Although in-
ternal validity is lower than can be obtained with an experimental design, external va-
lidity is generally better (assuming a good random sample) and a well designed study
should allow for the use of statistical controls to compensate for extraneous variables.

Types of quasi-experimental design

Cross-sectional study: Data obtained at one point in time (most surveys).

Case study: An in-depth analysis of one entity, object, or event.

Panel study: Repeated cross-sectional studies over time with the same participants
(cohort study).

Trend study: Tracking indicator variables over a period of time (unemployment,

crime, dropout rates).

11
1.2 Reporting Results

The following provides an outline for presenting the results of systematic research.
Regardless of the specific format used, the key points to consider in reporting the re-
sults of research are:

1. Clearly state the research question up front.

2. Completely explain study assumptions and method of inquiry so that others may du-
plicate the study.

3. Objectively and accurately report the results of the analysis.

4. Present data in tables that are

4.1. Accurate

4.2. Complete, and

4.3. Titled and documented so that they could stand on their own without a report.

5. Correctly reference sources of information and related research.

6. Openly discuss weaknesses/biases of the research.

7. Develop a defensible conclusion based on the analysis, not personal opinion.

Paper Format

The following outline may be a useful guide in formatting a research report. It incorpo-
rates elements of the research design and steps for hypothesis testing (in italics). It
may be helpful to refer back to this outline after reviewing the sections on hypothesis
testing. The report should clearly identify each section so the reader does not get lost
or confused.

12
I. Introduction: The introduction section should include a definition of the central re-
search question (purpose of the paper), why it is of interest, and a review of the litera-
ture related to the subject and how it relates to the hypotheses.

Elements:

• Purpose statement

• Theory guiding study design

• Identify abstract concepts

II. Method: Describe the source of the data, sample characteristics, statistical tech-
nique(s) applied, level of significance necessary to reject the null hypotheses, and
how the abstract concepts were operationalized.

Elements:

• Independent variable(s) and level of measurement

• Dependent variable(s) and level of measurement

• Assumptions

1. Random sampling

2. Independent subgroups

3. Population normally distributed

• Hypotheses

1. Identify statistical technique(s)

2. State null hypothesis or alternative hypothesis

• Rejection criteria

13
 1. Indicate alpha (amount of error the researcher is willing to except)

2. Specify one or two-tailed tests

III. Results: Describe the results of the data analysis and the implications for the hy-
potheses. The results section should include such elements as univariate, bivariate,
and multivariate analyses; significance test statistics, the probability of error and re-
lated status of the hypotheses tests (were they rejected?).

Elements:

• Describe sample statistics

• Compute test statistics

• Decide results

IV. Conclusion: Summarize and evaluate the results. Put in plain words what the re-
search found concerning the central research question. Identify alternative variables,
discuss implications for further study, and identify the weaknesses of the research and
findings.

Elements:

• Interpretation (What do the results mean?)

• Weaknesses (What do the results not mean?)

V. References: Only include literature cited in the report.

VI. Appendix: Additional tables and information not included in the body of the re-
port.

14
Table Format

In the professional world, presentation is almost everything. Most consumers of re-

search do no want to review or wade through the output produced by statistical soft-
ware. As a result, most researchers develop the ability to create one-page summaries
of the research findings that contain one or more tables presenting summary data. An
example of survey responses to three questions by customer sex and age is given be-
low:

Survey of Customers

Customer Characteristics
Sex Age
Total (1) Female Male 18-29 30-39 40-49 50-59 60+
Sample Size -> 3686 1466 2081 1024 1197 769 493 98
% of Total (1) -> 100% 41% 59% 29% 33% 22% 14% 3%
Margin of Error -> 1.6% 2.6% 2.2% 3.1% 2.8% 3.5% 4.4% 9.9%

Staff Professional?
Yes 89.4% 89.6% 89.5% 89.5% 89.4% 89.5% 89.6% 90.0%
No 10.6% 10.4% 10.5% 10.5% 10.6% 10.5% 10.4% 10.0%

Treated Fairly?
Yes 83.1% 82.8% 83.8% 82.6% 83.0% 83.5% 84.3% 87.9%
No 16.9% 17.2% 16.2% 17.4% 17.0% 16.5% 15.7% 12.1%

Served Quickly?
Yes 71.7% 69.3% 74.1% 73.3% 69.3% 72.1% 74.2% 78.0%
No 28.3% 30.7% 25.9% 26.7% 30.7% 27.9% 25.8% 22.0%

(1) Total number of cases is based on responses to the question concerning being served
quickly. Non-responses to survey items cause the sample sizes to vary.

15
Critical Review Checklist

he following checklist may help when critically evaluating research prepared by others.
Is there anything else that should be added to the list?

1. Research question(s) clearly identified

2. Variables clearly identified

3. Operationalization of variables is valid and reliable

4. Hypotheses evident

5. Statistical techniques identified and appropriate

6. Significance level (alpha) stated a priori

7. Assumptions for statistical tests are met

8. Tables are clearly labeled and understandable

9. Text accurately describes the data in the tables

10. Statistical significance properly interpreted

11. Conclusions fit analytical results and original research question(s)

12. Inclusion of all relevant variables in the study design

13. Study design weaknesses are identified and addressed by researcher

14. Other?

15. _________________________________________

16. _________________________________________

16
2
Data File Basics

There are two general sources of data; primary and secondary data. Primary data is
designed and collected specifically to answer one or more research questions. Secon-
dary data was collected by others for purposes that may or may not match the re-
search questions being explored. An example of a primary data source would be an
employee survey the researcher designs and implements for an organization to evalu-
ate job satisfaction. An example of a secondary data source would be the use of cen-
sus data or other publicly available data such as the General Social Survey to explore
research questions that may not have been specifically envisioned when the data de-
sign was created.

Designing data files

The best way to envision a data file is to use the analogy of the common spreadsheet
software. In spreadsheets, there are columns and rows. For many data files, a
spreadsheet provides an easy means of organizing and entering data. In a rectangu-
lar data file, columns represent variables and rows represent observations. Variables
are commonly formatted as either numerical or string. A numerical variable is used
whenever the research plans to manipulate the data mathematically. Examples would
be age, income, temperature, and job satisfaction rating. A string variable is used
whenever the research plans to treat the data entries like words. Examples would be
names, cities, case identifiers, and race. Most variables that could be considered
string are coded as a numeric variable. As an example, data for the variable "sex"
might be coded 1 for male and 2 for female instead of using a string variable that
would require letters (e.g., "Male" and "Female"). This has two benefits. First, numeri-

17
cal entries are easier and quicker to enter. Second, manipulation of numerical data
with statistical software is generally much easier than manipulating string variables.

Data file format

There are several common data file formats. As a general rule, there are data files
that are considered system files and data files that are text files. System files are cre-
ated by and for specific software applications. Examples would be Microsoft Excel,
SAS, STATA, and SPSS. Text files contain data in ASCII format and are almost univer-
sal in that they can be imported into most statistical programs.

Text files

Text files can be either fixed or free formatted.

Fixed: In fixed formatted data, each variable will have a specific column location.
When importing fixed formatted data into a statistical package, these column locations
must be identified for the software program. The following is an example:

++++|++++|++++|++++|++++|

10123HoustonTX12Female1

Reading from left to right, the variables and their location in the data file are:

Variable Column location Data

Case 1-3 101
Age 4-5 23
City 6-12 Houston
State 13-14 TX
Education 15-16 12
Sex 17-22 Female
Marital status 23 1 1=single 2=married

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Free: In free formatted data, either a space or special value separates (delimits) each
variable. Common delimiters are tabs or commas. When importing free formatted
data into a statistical package, the software assumes that when a delimiter value is
found that it is the end of the previous variable and the next character will begin an-
other variable.

The following is an example of a comma separated value data file (know as a csv file):

101,23,Houston,TX,12,Female,1,

Reading from left to right, the variables are:

Case, Age, City, State, Education, Sex, Marital status

When reading either fixed or free data, statistical software counts the number of vari-
ables and assumes when it reaches the last variable that the next variable will be the
beginning of another observation (case).

Data dictionary

A data dictionary defines the variables contained in a data file (and sometimes the for-
mat of the data file). To properly define and document a data file, the researcher
should record information one the coding of each variable. The following is an exam-
ple of a description of variable coding for a three-question survey.

Employee Survey Response #:

Q1: How satisfied are you with the current pay system?
a. Very satisfied
b. Somewhat satisfied
c. Satisfied
d. Somewhat dissatisfied
e. Very dissatisfied
Q2: How many years have you been employed here?
Q3: Please fill in your department name:

19
To properly define and document a data file, record the following information:

An abbreviated name used by statistical software to represent

Variable name:
the variable (generally 8 characters or less)

Variable type: String, numerical, date

If a fixed data set, the column location and possibly row in data
Variable location:
file

Variable label: Longer description of the variable

If a numerical variable is coded to represent categories, the

Value label:
categories represented by the values must be identified

For the employee survey, the data dictionary for a comma separated data file will look
like the following:

Variable name: CASEID

Variable type: String
Variable location: First variable in csv file
Variable label: Response tracking number
Value labels: (not normally used for string variables)

Variable name: Q1
Variable type: Numerical (categorical)
Variable location: Second variable
Variable label: Satisfaction with pay system
Value labels: 1=Very satisfied
2=Somewhat satisfied
3=Satisfied
4=Somewhat dissatisfied
5=Very dissatisfied

20
Variable name: Q2
Variable type: Numerical
Variable location: Third variable
Variable label: Years employed
Value labels: None

Variable name: Q3
Variable type: String
Variable location: Fourth variable
Variable label: Department name
Value labels: None

If the data for four completed surveys are entered into a spreadsheet, it will look like
the following:

A B C D
1 CASEID Q1 Q2 Q3
2 1001 2 5 Admin
3 1002 5 10 MIS
4 1003 1 23 Accounting
5 1004 4 3 Legal

The data will look like the following if saved in a text file as comma separated (note:
the total number of commas for each record equals the total number of variables):

CASEID, Q1, Q2, Q3,

1001, 2, 5, Admin,
1002, 5, 10, MIS,
1003, 1, 23, Accounting,
1004, 4, 3, Legal,

21
Some statistical software will not read the first row in the data file as variable names.
In that case, the first row must be deleted before saving to avoid computation errors.
The data file without variable names will look like the following:

1001, 2, 5, Admin,
1002, 5, 10, MIS,
1003, 1, 23, Accounting,
1004, 4, 3, Legal,

This is a very efficient way to store and analyze large amounts of data, but it should
be apparent at this point that a data dictionary would be necessary to understand what
the aggregated data represent. Documenting data files is very important. Although
this was a simple example, many research data files have hundreds of variables and
thousands of observations that will be uninterpretable without careful documentation.

22
3
Descriptive Statistics

Descriptive statistics are used to describe a population or sample. Population de-

scriptive statistics are calculated using all possible observations in a population.
Sample descriptive statistics are calculated using a subset of observations in a popu-
lation. In the case of samples, the descriptive statistic only represents the sample,
not the underlying population they originated from.

When a descriptive statistic is presented to represent one variable, it is commonly re-

ferred to as a univariate statistic. Examples would be mean income of residents in a
city, percent of city residents with a high school education, etc. When the relation-
ship between two variables is presented, this is commonly referred to as a bivariate
statistic. Separately calculating mean income and percent with a high school educa-
tion for males and females would be an example of bivariate statistics.

3.1 Univariate Statistics – Counts and Percents

This section presents techniques for using counts and proportions to create univari-
ate descriptive statistics. Use the techniques presented in this section if the analysis
variable meets the following characteristics.

23
 Nominal Data Ordinal Data

Classifies objects by type or characteristic Classifies objects by type or kind but also
has some logical order
★ Categories are mutually exclusive
★ No logical order ★ Categories are mutually exclusive
★ Logical order exists
★ Scaled according to amount of a
particular characteristic they possess

Examples
sex, race, ethnicity income (low, moderate, high)
government agency job satisfaction (5 point scale)
religion course letter grade (A to F)
neighborhood (urban/suburb)
yes/no responses

Univariate descriptive statistics are used to describe one variable through the crea-
tion of a summary statistic. Common univariate statistics for nominal/ordinal data in-
clude ratios, rates, proportions, and percentages.

Ratios

A ratio measures the extent to which one category of a variable outnumbers another
category in the same variable.

Ratio Example: The community has 1370 Protestants and 930 Catholics.

L arg erFreq 1370

Ratio = Ratio =
SmallerFreq or 930
For every one Catholic there are 1.47 Protestants in the given population.

24
Rates

A rate measures the number of actual occurrences out of all possible occurrences
per an established period of time.

Rate Example: Create the death rate per 1,000 given there are 100 deaths a year in a
population of 10,000.

# PerYear 100
Rate = × RateMetric Rate = ×1, 000
TotalPopulation or 10, 000

The population death rate is 10 deaths per 1000 people.

Proportions

Proportions are a common method used to describe frequencies as they compare to

a total (relative frequencies). A proportion weights the frequency of occurrence
against the total possible. It is often reported as a percentage.

OccurenceFreq
Pr oportion =
TotalPossible

OccurenceFreq
Percent = ×100
TotalPossible

25
Software Output: An example using survey data.

Frequencies

Variable: Edu Years of education

Value Count Percent Cumulative
-----------------------------------------------------------------
08 1 2.00 2.00
09 2 [a] 4.00 [b] 6.00 [c]
10 2 4.00 10.00
11 2 4.00 14.00
12 21 42.00 56.00
13 2 4.00 60.00
14 4 8.00 68.00
15 3 6.00 74.00
16 6 12.00 86.00
17 3 6.00 92.00
18 1 2.00 94.00
19 2 4.00 98.00
20 1 2.00 100.00
-----------------------------------------------------------------
Total 50
Missing 0

Interpretation

[a] 2 out of 50 have 9 years of education

[b] 4.00% have 9 years of education (2/50 = .04; .04 * 100 = 4.00%)

[c] 6.00% have 9 years or less education (3/50 = .06; .06*100 = 6.00%)

26
3.2 Univariate Statistics – Mean, Median, and Mode

This section presents techniques for using interval/ratio data to create descriptive sta-
tistics. For the methods reviewed in this section, the analysis variable must by inter-
val or ratio level data.

Interval Data Ratio Data

Classifies objects by type or kind but also has Classifies objects by type or kind but also has
some logical order in consistent intervals. some logical order in consistent intervals.

★ Categories are mutually exclusive ★ Same as Interval but also has a zero
★ Logical order exists starting point
★ Scaled according to amount of a particular
characteristic they possess
★ Differences between each level are equal
★ No zero starting point

Examples of Interval/Ratio
Miles to work Income IQ
SAT score Education in years Temperature
Age Weight Height

The following reviews common univariate descriptive statistics used to represent cen-
tral tendency and univariate statistics that describe variation in the data.

Measures of Central Tendency

Mode: The mode is the most frequently occurring score. A distribution of scores can
be unimodal (one score occurred most frequently), bimodal (two scores tied for most
frequently occurring), or multimodal. In the table that follows the mode is 32. If there
were also two scores with the value of 60, the distribution would be bimodal (32 and
60).

27
Median: The median is the point on a rank ordered list of scores below which 50% of
the scores fall. It is especially useful as a measure of central tendency when there
are very extreme scores in the distribution, such as would be the case if we had
someone in the age distribution provided below who was 120. If the number of
scores is odd, the median is the score located in the position represented by (n+1)/2.
In the table below the median is located in the 4th position (7+1)/2 and would be re-
ported as a median of 42. If the number of scores are even, the median is the aver-
age of the two middle scores.

As an example, if the last score (65) is dropped from the following table, the median
would be represented by the average of the 3rd (6/2) and 4th score, or (32+42)/2=37.
Always remember to order the scores from low to high before determining the me-
dian.

Variable Age Also known as X

24

32 Mode

32

42 Median Xi Each score

55

60

65

7 Number of scores or cases (n)

310 Sum of scores

44.29 Mean

28
Mean:

X=
∑ X i

The mean is the sum of the scores ( ) divided by the number of scores (n) to com-
pute an arithmetic average of the scores in the distribution. The mean is the most of-
ten used measure of central tendency. It has two properties: 1) the sum of the devia-
tions of the individual scores (Xi) from the mean is zero, 2) the sum of squared devia-
tions from the mean is smaller than what can be obtained from any other value cre-
ated to represent the central tendency of the distribution. In the above table the mean
age is (310/7) or 44.29.

Weighted Mean:

When two or more means are combined to develop an aggregate or grand mean, the
influence of each mean must be weighted by the number of cases in its subgroup.

Example

Wrong Method:

29
Correct Method:

Measures of Variation

Range: The range is the difference between the highest and lowest score (high-low).
It describes the span of scores but cannot be compared to distributions with a differ-
ent number of observations. In the table below, the range is (65-24) or 41.

Variance: The variance represents the average of the squared deviations between the
individual scores and the mean. The larger the variance the more variability there is
among the scores in a given distribution. When comparing two samples with the
same unit of measurement (e.g., age), the variances are comparable even though the
sample sizes may be different. Generally, smaller samples have greater variability
among the scores than larger samples. The formula is almost the same for estimat-
ing population variance.

2
Σ( X i − µ )
σ2 =
Population Variance: N

S2 =
(
Σ Xi − X )
Sample Variance: n −1

30
Standard deviation: The standard deviation represents the square root of variance. It
provides a representation of the variation among scores that is directly comparable to
the raw scores.

2
Σ( X i − µ )
σ=
Population Standard Deviation: N

S=
(
Σ Xi − X )
 Sample Standard Deviation: n −1

 Example

Variable (Xi) Age (Xi-mean) (Xi-mean)* (Xi-mean)

24 -20.29 411.68
32 -12.29 151.04
32 -12.29 151.04 squared deviations
42 -2.29 5.24
55 10.71 114.70
60 15.71 246.80
65 20.71 428.90

n= 7 1509.43 sum of squares

mean = 44.29 251.57 sample variance
15.86 sample standard deviation

31
Software Example: Uses data from previous example.

Descriptive Statistics
Variable: Age
----------------------------------------------------------------------
Count 7 Pop Var 215.6327 [b]
Sum 310.0000 Sam Var 251.5714
Mean 44.2857 Pop Std 14.6844
Median 42.0000 Sam Std 15.8610
Min 24.0000 Std Error 5.9949
Max 65.0000 CV% 35.8152 [c]
Range 41.0000 95% CI (+/-) 14.6690 [d]
Skewness 0.0871 [a] One sample t-test (mu=0)|p<| 0.0003 [e]
----------------------------------------------------------------------

Missing Cases 0

Interpretation

[a] Skewness provides an indication of the how asymmetric the distribution is for a given sam-
ple. A negative value indicates a negative skew. Values greater than 1 or less than -1 indicate a
non-normal distribution.

[b] Population variance (Pop Var) and population standard deviation (Pop Std) will always be
less than sample variance and sample standard deviation, since the sum of squares is divided
by n instead of n-1.

[c] Coefficient of Variation (CV) is the ratio of the sample standard deviation to the sample
mean: (sample standard deviation/sample mean)*100 to calculate CV%. It is used as a measure
of relative variability, CV is not affected by the units of a variable.

[d] Add and subtract this value to the mean to create a 95% confidence interval.

[e] This represents the results of a one-sample t-test that compares the sample mean to a hy-
pothesized population value of 0 years of age. In this example, the sample mean age of 44 is sta-
tistically significantly different from zero.

32
3.3 Standardized Z-Scores

A standardized z-score represents the relative position of an individual score in a dis-

tribution as compared to the mean and the variation of scores in the distribution. A
negative z-score indicates the score is below the distribution mean. A positive z-
score indicates the score is above the distribution mean. Z-scores will form a distribu-
tion identical to the distribution of raw scores; the mean of z-scores will equal zero
and the variance and standard deviation of a z-distribution will always equal one.

To obtain a standardized score, subtract the mean from the individual score and di-
vide by the standard deviation.

Example: mean=44.29 and standard deviation=15.86

Variable (Xi) Age (Xi-mean) (Xi-mean)/s Z

Doug 24 -20.29 -20.29/15.86 -1.28

Mary 32 -12.29 -12.29/15.86 -0.77

Jenny 32 -12.29 -12.29/15.86 -0.77

Frank 42 -2.29 -2.29/15.86 -0.14

John 55 10.71 10.71/15.86 0.68

Beth 60 15.71 15.71/15.86 0.99

Ed 65 20.71 20.71/15.86 1.31 [a]

Interpretation

[a] Ed is 1.31 standard deviations above the mean age for those represented in the sample.

33
Software Output: Data from the previous example.

The following displays z-scores created by statistical software when conducting a de-
scriptive analysis. The statistical software calculates the mean and standard devia-
tion for the sample data and then calculates a z-score for each observation and out-
puts the result to the data file thereby creating a new variable (in this case Z-Age).

Obs Age Z-Age

1 24 -1.27897
2 32 -0.77459
3 32 -0.77459
4 42 -0.14411
5 55 0.67551
6 60 0.99075
7 65 1.30599

The following displays the descriptive statistics for the Z-Age variable. Notice that
the mean is zero and sample variance and sample standard deviation are one.

Descriptive Statistics
Variable: Z-Age
--------------------------------------------------------------------
Count 7 Pop Var 0.8571
Sum 0.0000 Sam Var 1.0000
Mean 0.0000 Pop Std 0.9258
Median -0.1441 Sam Std 1.0000
Min -1.2790 Std Error 0.3780
Max 1.3060 CV% ERR
Range 2.5850 95% CI (+/-) 0.9248
Skewness 0.0871 One sample t-test (mu=0) |p<| 1.0000
--------------------------------------------------------------------

Missing Cases 0

34
Z-scores may also be used to compare similar individuals in different groups. As an
example, to compare students with similar abilities taking two different classes with
the same level of course content but different instructors, the exam scores from the
two courses could be “normalized.” To more properly compare student A's perform-
ance in class A with Student B's performance in class B, the numerical test scores
could be adjusted by the variation (standard deviation) of test scores in each class
and the distance of each student's test score from the average (mean) for the class.
The student with the greater z-score performed relatively better, as measured by the
test score, than the student with the lower z-score.

Transformed Z-Scores

Z-scores have two major disadvantages for a lay audience. First, negative scores
have negative connotations. Second, z-scores and their associated decimal point
presentation can be difficult for others to interpret. These disadvantages can be miti-
gated by transforming the z-scores into another metric that removes decimals and
negative values. This is a common practice in standardized testing.

Procedure:

1. Transform raw scores in z-scores

2. Arbitrarily set a mean value that removes negative numbers

3. Arbitrarily set a standard deviation value

4. Calculate transformed z-score using the following formula:

35
where

= Arbitrary mean

= Arbitrary standard deviation

Example: transformed z-score

Fred took an IQ test and correctly answered 70 out of 100 points. The mean score
on this test is 75 with a standard deviation 5. The following transforms Fred’s z-score
(-1) to a scale where the mean is 100 and a standard deviation is 15.

Sample statistics:

Fred’s z-score:

Transformation: and

Fred’s transformed IQ test score is 85.

36
3.4 Bivariate Statistics – Counts and Percents

Bivariate descriptive statistics simultaneously analyze (compare) two variables to ex-

plore associations between the variables. When using a contingency table to explore
counts, the independent variable (if causation can be meaningfully defined) is repre-
sented by the columns and the dependent variable is represented by the rows. This
is a common practice that makes interpretation easier. The level of measurement can
be nominal or ordinal.

Interpreting Contingency Tables

(count) Female Male Row Total

Support tax 75 50 125
Do not support tax 25 50 75
Column Total 100 100 200

Using counts to create more meaningful percentage summary measures.

Column %: Of those who are female, 75% (75/100) support the tax.

Of those who are male, 50% (50/100) support the tax.

Row %: Of those supporting the tax, 60% (75/125) are female.

Of those not supporting the tax, 67% (50/75) are male.

Row Total %: Overall 62.5% (125/200) support the tax.

Overall 37.5% (75/200) do not support the tax

Column Total %: Overall 50% (100/200) are female and 50% (100/200) are male.

37
Software Output: An example using survey data.

Crosstabulation: EduCat (Rows) by Sex (Columns)

Column Variable Label: Respondent sex

Row Variable Label: Education Level

Count |
Row % |
Col % |
Total % | Male | Female | Total
-------------------------------------------
<12 yrs | 3 | 4 [a]| 7
| 42.86 | 57.14 [b]|
| 11.54 | 16.67 [c]| 14.00
| 6.00 | 8.00 [d]|
-------------------------------------------
HS Grad | 10 | 11 | 21
| 47.62 | 52.38 |
| 38.46 | 45.83 | 42.00
| 20.00 | 22.00 |
-------------------------------------------
College | 13 | 9 | 22
| 59.09 | 40.91 |
| 50.00 | 37.50 | 44.00
| 26.00 | 18.00 |
-------------------------------------------
| 26 | 24 | 50
Total | 52.00 | 48.00 | 100.00

Interpretation

[a] Count: There are 4 females who have less than 12 years of education.

[b] Row %: Of those who have less than a high school education (<12 yrs), 57.14% (4/7) are fe-
male.

[c] Column %: Of those who are female, 16.67% (4/24) have less than a high school education.

[d] Total %: 8% (4/50) of the sample is females with less than a high school education.

38
Summary Table

The following summary table is based on column percent from the crosstabulation
shown on the previous page. Counts are from the column marginals (Totals) for male
and female.

Education level for all subjects in the sample and by sex subgroups

Total Male Female

Observations (n) = 50 26 24

Education Level (column %)

Less than high school 14.0 11.5 16.7

High School 42.0 38.5 45.8

College 44.0 50.0 37.5

Interpretation

There are slightly more men in this sample than women. Using percentages to adjust for differ-
ences in sample size, men are more likely (50%) to have a college education than women
(37.5%).

39
3.5 Bivariate Statistics – Means

Bivariate descriptive statistics simultaneously analyze (compare) two variables to ex-

plore associations between the variables. At least one of the variables must be
interval/ratio data. For most descriptive statistics, the other variable will likely be
nominal or ordinal level data. Bivariate comparisons between two interval/ratio vari-
ables are discussed in the correlation chapter.

The following example provides separate income means for males and females.

Software Output: Income by Sex

Explore Descriptive Statistics by Subgroup

Analysis Variable: Respondent Income (approximated)

Category: All
--------------------------------------------------------------------
Count 1439 Pop Std Dev 19498.0888
Min 0.0000 Sample Std Dev 19504.8672
Max 75000.0000 Standard Error 514.1769
Sum 24060750.0000 95% CI (+/-) 1008.6397
Median 11250.0000 95% Lower Limit 15711.8259
Mean 16720.4656 95% Upper Limit 17729.1053
Range 75000.0000 Coeff of Variation 116.6527

40
By Subgroup Variable: SEX

Category: Male
--------------------------------------------------------------------
Count 620 Pop Std Dev 22407.0499
Min 0.0000 Sample Std Dev 22425.1420
Max 75000.0000 Standard Error 901.3426
Sum 14445000.0000 95% CI (+/-) 1770.0598
Median 18750.0000 95% Lower Limit 21528.3273
Mean 23298.3871 95% Upper Limit 25068.4469
Range 75000.0000 Coeff of Variation 96.2519

Category: Female
--------------------------------------------------------------------
Count 819 Pop Std Dev 15177.0254
Min 0.0000 Sample Std Dev 15186.2995
Max 75000.0000 Standard Error 530.9765
Sum 9615750.0000 95% CI (+/-) 1042.2369
Median 4500.0000 95% Lower Limit 10698.6056
Mean 11740.8425 95% Upper Limit 12783.0794
Range 75000.0000 Coeff of Variation 129.3459

Summary table created from the above output

Mean Income by sex

Total Male Female

Observations (n) 1439 620 819
Mean Income $16,721 $23,298 $11,741

41
4
Hypothesis Testing Basics

The chain of reasoning and systematic steps used in hypothesis testing are the back-
bone of every statistical test regardless of whether one writes out each step in a class-
room exercise or uses statistical software to conduct statistical tests on variables
stored in a data file.

4.1 Chain of reasoning for inferential statistics

★ Sample(s) must be randomly selected.

★ The sample estimate is compared to the underlying distribution of the same size
sampling distribution.

★ The probability that a sample estimate reflects the population parameter is deter-
mined from the sampling distribution.

The underlying logic of hypothesis testing is based on rejecting a statement of no dif-

ference or no association. This is called the null hypothesis. The null hypothesis is
only rejected when we have evidence beyond a reasonable doubt that a true differ-
ence or association exists in the population(s) from which we drew our random sam-
ple(s).

42
Reasonable doubt is based on probability sampling distributions. The benchmark for
reasonable doubt (defined by “alpha”) is established by the researcher. Alpha .05 is a
common benchmark for reasonable doubt. At alpha .05 we know from the sampling
distribution that a test statistic at or beyond this benchmark will only occur by ran-
dom chance five times out of 100 (5% probability). Since a test statistic that results
in an alpha of .05 could only occur by random chance 5% of the time, we assume
that the test statistic resulted because there are true differences between the popula-
tion parameters, not because we unwittingly drew a biased random sample.

The four possible outcomes in hypothesis testing

The following table summarizes the possible outcomes of hypothesis testing as they
relate to “truth” in the underlying population. It is important to remember that in hy-
pothesis testing we are using sample statistics to make predictions about the un-
known population values. The orange cells represent erroneous conclusions from hy-
pothesis testing. One error (Type I) is to reject the null hypothesis when there is no dif-
ference in the population (i.e., this is being wrong when we conclude significance).
The other error (Type II) is to not reject the null hypothesis when there is an actual dif-
ference. The blue cells represent correct decisions from hypothesis testing.

Hypothesis Testing In the Population

Outcomes There is no difference There is a difference
Decision From Sample Null Hyp. True Null Hyp. False

Type I error
Rejected Null Hyp. Correct Decision
(alpha)

Did not Reject Null Correct Decision Type II Error

43
When learning statistics we generally conduct statistical tests by hand. In these situa-
tions, we establish before the test is conducted what test statistic is needed (called
the critical value) to claim statistical significance. So, if we know for a given sampling
distribution that a test statistic of plus or minus 1.96 would only occur 5% of the time
randomly, any test statistic that is 1.96 or greater in absolute value would be statisti-
cally significant. In an analysis where a test statistic was exactly 1.96, there would be
a 5% chance of being wrong if statistical significance is claimed. If the test statistic
was 3.00, statistical significance could also be claimed but the probability of being
wrong would be much less (about .002 if using a 2-tailed test or two-tenths of one
percent; 0.2%). Both .05 and .002 are known as alpha; the probability of a Type I er-
ror.

When conducting statistical tests with computer software, the exact probability of a
Type I error is calculated. It is presented in several formats but is most commonly re-
ported as "p <" or "Sig." or "Signif." or "Significance." Using "p <" as an example, if
a priori the threshold for statistical significance is established at alpha .05, any test
statistic with significance at or less than .05 would be considered statistically signifi-
cant and the null hypothesis of no difference must be rejected. The following table
links p values with a constant alpha benchmark of .05 (note: alpha remains constant
while p-values are a direct result of the analysis on sample data):

P< Alpha Probability of Making a Type I Error Decision

.05 .05 5% chance difference is not significant Statistically significant

.10 .05 10% chance difference is not significant Not statistically significant

.01 .05 1% chance difference is not significant Statistically significant

44
4.2 The Normal Distribution

Although there are numerous sampling distributions used in hypothesis testing, the
normal distribution is the most common example of how data would appear if we cre-
ated a frequency histogram where the x axis represents the values of scores in a dis-
tribution and the y axis represents the frequency of scores for each value. Most
scores will be similar and therefore will group near the center of the distribution.
Some scores will have unusual values and will be located far from the center or apex
of the distribution. In hypothesis testing, we must decide whether the unusual val-
ues are simply different because of random sampling error or they are in the extreme
tails of the distribution because they are truly different from others. Sampling distribu-
tions have been developed that tell us exactly what the probability of this sampling er-
ror is when data originate from a random sample collected from a population that is
normally distributed.

Properties of a normal distribution

★ Forms a symmetric bell-shaped curve.

★ 50% of the scores lie above and 50% lie below the midpoint of the distribution.
★ The curve is asymptotic to the x axis.
★ The mean, median, and mode are located at the midpoint of the x axis.

45
Using theoretical sampling probability distributions

Sampling distributions approximate the probability that a particular value would occur
by chance alone. If the means were collected from an infinite number of repeated ran-
dom samples of the same sample size from the same population most means will be
very similar in value, in other words, they will group around the true population mean.
In a normal distribution, most means will collect about a central value or midpoint of a
sampling distribution. The frequency of means will decrease as the value of the ran-
dom sample mean increases its distance from the center of a normal sampling distri-
bution toward the tails. In a normal probability distribution, about 95% of the means
resulting from an infinite number of repeated random samples will fall between 1.96
standard errors above and below the midpoint of the distribution which represents
the true population mean and only 5% will fall beyond (2.5% in each tail of the distri-
bution).

The following are commonly used points on a distribution for deciding statistical sig-
nificance.

90% of scores +/- 1.65 standard errors

95% of scores +/- 1.96 standard errors

99% of scores +/- 2.58 standard errors

Standard error: The standard error is a mathematical adjustment to the sample stan-
dard deviation to account for the effect sample size has on the underlying sampling
distribution. It represents the standard deviation of the sampling distribution.

Alpha and the role of the distribution tails

The percentage of scores beyond a particular point along the x axis of a sampling dis-
tribution represent the percent of the time during an infinite number of repeated sam-
ples one would expect to have a score at or beyond that value on the x axis. This

46
value on the x axis is known as the critical value when used in hypothesis testing.
The midpoint represents the actual population value. Most scores will fall near the ac-
tual population value but will exhibit some variation due to sampling error. If a score
from a random sample falls 1.96 standard errors or farther above or below the mean
of the sampling distribution, we know from the probability distribution that there is
only a 5% or less chance of randomly selecting a set of scores that would produce a
sample mean that far from the true population mean. This area above and below
1.96 standard errors is the region of rejection.

When conducting significance testing, if we have a test statistic that is at least 1.96
standard errors above or below the mean of the sampling distribution, we assume we
have a statistically significant difference between our sample mean and the expected
mean for the population. Since we know a value that far from the population mean
will only occur randomly 5% or less of the time, we assume the difference is the re-
sult of a true difference between the sample and the population mean, and is not the
result of random sampling error. The 5% is also known as the probability of being
wrong when we conclude statistical significance.

1-tailed vs. 2-tailed statistical tests

A 2-tailed test is used when the researcher cannot determine a priori whether a differ-
ence between population parameters will be positive or negative. A 1-tailed test is
used when it is reasonable to expect a difference will be positive or negative.

47
4.3 Steps to Hypothesis Testing

Hypothesis testing is used to establish whether the differences exhibited by random

samples can be inferred to the populations from which the samples originated. Re-
gardless of whether statistical tests are conducted by hand or through statistical soft-
ware, there is an implicit understanding that systematic steps are being followed to
determine statistical significance. These general steps are 1) assumptions, 2) null
and alternative hypotheses, 3) rejection criteria, 4) computation of test statistics, and
5) decision regarding the null hypothesis.

I. General Assumptions

Population is normally distributed

Random sampling

Mutually exclusive comparison samples

Data characteristics match statistical technique

For interval / ratio data, use the following

t-tests, Pearson correlation, ANOVA, OLS regression

For nominal / ordinal data, use the following

Difference of proportions, chi square and related measures of association,

logistic regression

II. State the Hypothesis

Null Hypothesis (Ho): There is no difference between ___ and ___.

Alternative Hypothesis (Ha): There is a difference between __ and __.

48
 Note: The alternative hypothesis will indicate whether a 1-tailed or a 2-tailed test
is utilized to reject the null hypothesis.

Ha for 1-tail tested: The __ of __ is greater (or less) than the __ of __.

III. Set the Rejection Criteria

This determines how different the parameters and/or statistics must be before
the null hypothesis can be rejected. This "region of rejection" is based on alpha (
) -- the error associated with the confidence level. The point of rejection is
known as the critical value.

IV. Compute the Test Statistic

The collected data are converted into standardized scores for comparison with
the critical value.

V. Decide Results of Null Hypothesis

If the test statistic equals or exceeds the region of rejection bracketed by the criti-
cal value(s), the null hypothesis is rejected. In other words, the chance that the
difference exhibited between the sample statistics is due to sampling error is
remote--there is an actual difference in the population.

49
5
Confidence Intervals

For interval estimation, the data must be from a random sample. The following pre-
sents interval estimation techniques for proportions (nominal/ordinal data) and also
for means (interval/ratio data).

5.1 Interval Estimation for Proportions

Interval estimation (margin of error) uses sample data to determine a range (interval)
that, at an established level of confidence, is expected to contain the population pro-
portion.

Steps

Determine the confidence level (alpha is generally .05).

Use the z-distribution table to find the critical value for a 2-tailed test given the se-
lected confidence level (alpha).

Estimate the standard error of the proportion.

where

50
 p = sample proportion

q=1-p

Estimate the confidence interval.

CV = critical value

CI = p ± (CV)(Sp)

Interpret

Based on alpha .05, the researcher is 95% confident that the proportion in the popula-
tion from which the sample was obtained is between __ and __.

Note: Given the sample data and level of error, the confidence interval provides an es-
timated range of proportions that is most likely to contain the population proportion.
The term "most likely" is measured by alpha (i.e., in most cases there is a 5% chance
--alpha .05-- that the confidence interval does not contain the true population propor-
tion).

51
More About the Standard Error of the Proportion

The standard error of the proportion will vary as sample size and the proportion
changes. As the standard error increases, so will the margin of error.

  Sample Size (n)

Proportion (p) 100 300 500 1000 5000 10000
0.9 0.030 0.017 0.013 0.009 0.004 0.003
0.8 0.040 0.023 0.018 0.013 0.006 0.004
0.7 0.046 0.026 0.020 0.014 0.006 0.005
0.6 0.049 0.028 0.022 0.015 0.007 0.005
0.5 0.050 0.029 0.022 0.016 0.007 0.005
0.4 0.049 0.028 0.022 0.015 0.007 0.005
0.3 0.046 0.026 0.020 0.014 0.006 0.005
0.2 0.040 0.023 0.018 0.013 0.006 0.004
0.1 0.030 0.017 0.013 0.009 0.004 0.003

Effect of changes in the proportion

As a proportion approaches 0.5 the error will be at its greatest value for a given sam-
ple size. Proportions close to 0 or 1 will have the lowest error. The error above a pro-
portion of .5 is a mirror reflection of the error below a proportion of .5.

Effect of changes in sample size

As sample size increases the error of the proportion will decrease for a given propor-
tion. However, the reduction in error of the proportion as sample size increases is not
constant. Using a proportion of 0.9 as an example, increasing the sample size from
100 to 300 cut the standard error by about half (from .03 to .017). Increasing the sam-
ple size by another 200 only reduced the standard error by about one quarter (.017 to
.013).

52
Example: Interval Estimation for Proportions

Problem: A random sample of 500 employed adults found that 23% had traveled to a
foreign country. Based on these data, what is the estimate for the entire employed
adult population?

n=500, p = .23, q = .77

Use alpha .05 (i.e., the critical value is 1.96)

Estimate Sampling Error

Compute Interval

Interpret

Based on a 95% confident level, the actual proportion of all employed adults who
have traveled to a foreign country is between 19.3% and 26.7%.

53
Software output: Summary data from the previous example.

Margin of Error for Proportions

n = 500 Proportion = 0.23 Standard Error = 0.0188

Margin Limits
Confidence Error +/- Lower Upper
------------------------------------------------------------
90% 0.0311 0.1989 0.2611
95% 0.0369 0.1931 0.2669
99% 0.0486 0.1814 0.2786

Summary Table of Margin of Errors for each proportion: (Converted to %)

The promotion system is … Staff +/- Margin of Error 95% CI

Fair 50% 9.8% [b] 40.2% - 59.8%

Sometimes Fair 30% 8.9% 21.0% – 38.9%
Not Fair 20% [a] 7.8% 12.2% – 27.8%
Count (n) 100

Interpretation

[a] At the 95% confidence level, the percent of all employees in the population who believe
the promotion system is not fair is between 12.2% to 27.8%.

[b] The margin of error is different for each response category due to the change in propor-
tions. Proportions at 0.5 (50%) will have the highest level of error. Using one margin of error for
multiple comparisons

54
To avoid calculating a separate margin of error for each response category, it is com-
mon to calculate the most conservative standard error of a proportion (p=0.5) and
use this to represent the margin of error for all response options within a specific sub-
group. In the table below, a separate margin of error is calculated for the total sam-
ple, the male sample, and the female sample.

Summary Table: Using one confidence interval (CI) for all data in one subgroup

All Male Female

Total Count 1496 641 855
95% CI (+/-) 2.5% 3.9% 3.4%

Respondent's highest degree

< High school 18.6% 19.5% 18.0%
High school 52.1% 47.4% 55.7%
Junior college 6.0% [a] 5.8% 6.2% [b]
Bachelor 15.6% 16.8% 14.7%
Graduate 7.6% [c] 10.5% 5.4% [d]

Interpretation

[a] Based on the 95% confidence level, the percent of U.S. adults with a junior college educa-
tion level is between 3.5% and 8.5% (6.0% +/- 2.5%).

[b] Based on the 95% confidence level, the percent of Female U.S. adults with a junior college
education level is between 2.8% and 9.6% (6.2% +/- 3.4%).

[c] Based on the 95% confidence level, the percent of U.S. adults with a graduate education
level is between 5.1% and 10.1% (7.6% +/- 2.5%).

[d] Based on the 95% confidence level, the percent of Female U.S. adults with a graduate edu-
cation level is between 2.0% and 8.8% (5.4% +/- 3.4%).

55
Interval Estimation for the Difference Between Two Proportions

This approach uses sample data to determine a range (interval) that, at an estab-
lished level of confidence, will contain the difference between two population propor-
tions.

Steps

Determine the confidence level (generally alpha .05).

Use the z distribution table to find the critical value for a 2-tailed test (at alpha .05 the
critical value would equal 1.96).

Estimate Sampling Error

where and

Estimate the Interval

CI = (p1-p2) ± (CV)(Sp1-p2)

p1-p2 = difference between two sample proportions

CV = critical value

Interpret

Based on alpha .05, the researcher is 95% confident that the difference between the
proportions of the two subgroups in the population from which the sample was ob-
tained is between __ and __.

Note: Given the sample data and level of error, the confidence interval provides an es-
timated range of proportions that is most likely to contain the difference between the

56
population subgroups. The term "most likely" is measured by alpha or in most cases
there is a 5% chance (alpha .05) that the confidence interval does not contain the
true difference between the subgroups in the population.

5.2 Interval Estimation For Means

Interval estimation involves using sample data to determine a range (interval) that, at
an established level of confidence, is expected to contain the mean of the population.

Steps

1. Determine confidence level (df=n-1; alpha .05, 2-tailed)

2. Use either the z distribution (if n>120) or the t distribution (for all sizes of n).

3. Use the appropriate table to find the critical value for a 2-tailed test

4. Multiple hypotheses can be compared with the estimated interval for the popula-
tion to determine their significance. In other words, differing values of population
means can be compared with the interval estimation to determine if the hypothe-
sized population means fall within the region of rejection.

Estimation Formula

where

= sample mean

CV = critical value (consult z or t distribution table for df=n-1 and chosen alpha-- com-
monly .05)

57
Standard error of the mean

s
sx =
n

Note: assumes sample standard deviation was calculated using:

Example: Interval Estimation for Means

Problem: A random sample of 30 incoming college freshmen revealed the following

statistics: mean age 19.5 years; sample standard deviation 1.2. Based on a 5%
chance of error, estimate the range of possible mean ages for all incoming freshmen.

Estimation

Critical value (CV)

Df=n-1 or 29

Consult t-distribution for alpha .05, 2-tailed

CV=2.045

Standard error

sx = .219

Estimate Confidence Interval

58
CI 95 = 19.5 ± 2.045 (.219 )

CI 95 = 19.5 ±.448

Interpretation

Based on a 95% confidence level, the actual mean age of the all incoming freshmen
will be somewhere between 19 years (the lower limit) and 20 years (the upper limit) of
age.

Software Output: Summary data from previous example.

Margin of Error for Means

n = 30 Mean = 19.5 Standard Deviation = 1.2

Margin Limits
Confidence Error +/- Lower Upper
------------------------------------------------------------
90% 0.3723 19.1277 19.8723
95% 0.4481 19.0519 19.9481[a]
99% 0.6039 18.8961 20.1039

Interpretation

[a] Based on a 95% confidence level, the actual mean age of the all incoming freshmen will be
somewhere between 19 years (the lower limit) and 20 years (the upper limit) of age. Note that
both measures are rounded.

59
6
Z-Tests of Proportions and Chi-
Square
This section presents techniques for using counts and proportions to conduct hy-
pothesis testing. Use the techniques presented in this section if the analysis or de-
pendent variable meet the following characteristics.

Nominal Data Ordinal Data

Classifies objects by type or Classifies objects by type or
characteristic characteristic but also has some logical
order
1. Categories are mutually exclusive
2. No logical order 1. Categories are mutually exclusive
2. Logical order exists
3. Scaled according to amount of a
particular characteristic they possess

Examples of Nominal Data Examples of Ordinal Data

sex, race, ethnicity income (low, moderate, high)
government agency job satisfaction (5 point scale)
political party preference military rank (Lieutenant to General)
religion course letter grade (A to F)
neighborhood (urban/suburb) importance of welfare
census region (little, moderate, high)
yes/no responses
have children (yes/no)
voted (yes/no)

60
6.1 Z-test of Proportions

For hypothesis testing in this section both the independent and dependent variable
must be nominal or ordinal. In addition, the data are assumed to be from a random
sample. The relevant sampling distribution for this section is the standard normal (Z)
distribution. Tables of critical values are included in the Appendix.

Statistical Techniques Discussed in This Section

★ One-sample test of proportions

★ Two-sample test of proportions

Comparing a Population Proportion to a Sample Proportion (One-Sample Test)

A one-sample z-test of proportions is used to compare a proportion created by a ran-

dom sample to a proportion originating from or thought to represent the value for the
entire population. As an example, to make sure the random sample of 100 subjects is
not biased regarding a person’s sex, one approach would be to compare the propor-
tion of women in the sample to the known proportion of women in the underlying
population as reported in census data or by some other reliable source.

Example

Problem: Historical data indicates that about 10% of the agency's clients believe
they were given poor service. Now under new management for six months, a random
sample of 110 clients found that 15% believe they were given poor service.

Pu = .10 Ps = .15 n = 110

61
I. Assumptions

Independent random sampling

Nominal level data

Large sample size

II. State the Hypothesis

Ho: There is no statistically significant difference between the historical propor-

tion of clients reporting poor service and the current proportion of clients report-
ing poor service.

If 2-tailed test

Ha: There is a statistically significant difference between the historical proportion

of clients reporting poor service and the current proportion of clients reporting
poor service.

If 1-tailed test

Ha: The proportion of current clients reporting poor service is significantly greater
than the historical proportion of clients reporting poor service.

III. Set the Rejection Criteria

Use z-distribution table to estimate critical value

If 2-tailed test, Alpha .05, Zcv = 1.96

If 1-tailed test, Alpha .05, Zcv = 1.65

62
IV. Compute the Test Statistic

Estimate Standard Error

p = population proportion

q=1-p

n = sample size

Test Statistic

V. Decide Results of Null Hypothesis

If a 2-tailed test was used . . .

Since the test statistic of 1.724 did not meet or exceed the critical value of
1.96, there is insufficient evidence to conclude there is a statistically signifi-
cant difference between the historical proportion of clients reporting poor
service and the current proportion of clients reporting poor service.

If a 1-tailed test was used . . .

Since the test statistic of 1.724 exceeds the critical value of 1.65, conclude
the proportion of current clients reporting poor service is significantly greater
than the historical proportion of clients reporting poor service.

63
Software Output: Summary data from the previous example.

One-Sample Difference of Proportions!

Population Proportion! ! 0.10!

Sample Proportion! ! ! 0.15!
Sample Size (n)! ! 110!
Test statistic!! ! ! 1.748!
! ! ! ! p <!! ! 0.0805! (2tailed) [a]

Interpretation

[a] For a 2-tailed test, the p-value represents the probability of making a type 1 error (conclud-
ing there is statistical significance when there is none). Since there is about 8% chance of mak-
ing a type 1 error, which exceeds the 5% error limit established in the rejection criteria of the hy-
pothesis testing process (alpha), do not conclude there is statistical significance.

64
Comparing Proportions From Two Independent Samples (Two-Sample Test)

A two-sample z-test of proportions is used to compare two proportions created by

two random samples or two subgroups of one random sample.

Example

Problem: A survey was conducted of students from the Princeton public school sys-
tem to determine if the incidence of hungry children was consistent in two schools lo-
cated in lower-income areas. A random sample of 80 elementary students from
school A found that 23% did not have breakfast before coming to school. A random
sample of 180 elementary students from school B found that 7% did not have break-
fast before coming to school. Before putting more resources into school A, the
school superintendent wants to verify there is a statistically significant difference be-
tween the schools (i.e., the difference is beyond just random error created from two
small samples of students).

I. Assumptions

Independent random sampling

Nominal level data

Large sample size

II. State the Hypothesis

Ho: There is no statistically significant difference between the proportion of stu-

dents in school A not eating breakfast and the proportion of students in school B
not eating breakfast.

65
 Ha: There is a statistically significant difference between the proportion of stu-
dents in school A not eating breakfast and the proportion of students in school B
not eating breakfast.

III. Set the Rejection Criteria

Use z-distribution table to estimate critical value

Alpha.05, Zcv = 1.96

IV. Compute the Test Statistic

Estimate of Standard Error

where and

Test Statistic

V. Decide Results of the Null Hypothesis

Since the test statistic 3.721 exceeds the critical value of 1.96, conclude there is
a statistically significant difference between the proportion of students in school

66
 A not eating breakfast and the proportion of students in school B not eating
breakfast.

Software Output: Summary data from the previous example.

Two-Sample Difference of Proportions (2-tailed)

! !
! ! ! ! ! School A! School B
Sample Proportion! ! 0.23! ! 0.07
Sample Size (n)! ! 80! ! ! 180
! !
! Z-statistic! ! p < Significance
! ! ! ! 3.674! ! ! 0.0002 [a]

Interpretation

[a] For a 2-tailed test, the p-value represents the probability of making a type 1 error (conclud-
ing there is statistical significance when there is none). Since there is far less than a 5% chance
of making a type 1 error, conclude there is statistical significance.

67
Appropriate Sample Size

Sample size is an important issue when using statistical tests that rely on the stan-
dard normal distribution (z-distribution). As a rule of thumb, sample size is generally
considered large enough to use the z-distribution when n(p)>5 and p is less than q (1-
p). If p is greater than q (note: q = 1-p), then n(q) must be greater than 5. Otherwise
use the binomial distribution.

Example A

n=60, p=.10

n(p) 60(.10) = 6

Conclude sample size is sufficient

If n=40, p=.10 then 40(.10) = 4

Conclude sample size may not be sufficient

Example B

n=50, p=.80

q=1-p or q=.20

n(q) 60(.20) = 12

Conclude sample size is sufficient

68
6.2 Chi-Square

Chi-square is the most common technique used to compare two nominal/ordinal vari-
ables, especially when at least one of the variables has more than two categories or
the sample size is small. In most cases, chi-square is preferred over a difference of
proportions z-test. The relevant sampling distribution for this section is the chi-
square distribution.

Statistical Techniques Discussed in This Section

★ Goodness of Fit

★ Chi-Square Test of Independence

★ Measuring Association

Chi-Square Goodness-of-Fit Test

The chi-square goodness of fit test is used to compare frequencies (counts) among
multiple categories of nominal or ordinal level data for one-sample (univariate analy-
sis).

Problem: Evaluate variations in the proportion of defects produced from five assem-
bly lines. A random sample of 100 defective parts from the five assembly lines pro-
duced the following contingency table.

Line A Line B Line C Line D Line E

24 15 22 20 19

I. Assumptions

Independent random sampling

69
 Nominal or Ordinal level data

II. State the Hypothesis

Ho: There is no significant difference among the assembly lines in the observed
frequencies of defective parts.

Ha: There is a significant difference among the assembly lines in the observed fre-
quencies of defective parts.

III. Set the Rejection Criteria

Determine degrees of freedom (df) = k – 1 where k equals the number of catego-

ries df=5-1 or df=4

Establish the confidence level (.05, .01, etc.)

Use the chi-square distribution table to establish the critical value

At alpha .05 and 4 degrees of freedom, the critical value from the chi-square dis-
tribution is 9.488

IV. Compute the Test Statistic

where Fe = Frequency Expected

and . . .

n = sample size

k = number of categories or cells

Fo = observed frequency

70
 Line A Line B Line C Line D Line E
Fo 24 15 22 20 19
Fe (100/5=20) 20 20 20 20 20
Chi-Square .8 1.25 .2 0 .05 2.30

V. Decide Results of Null Hypothesis

Since the chi-square test statistic 2.30 does not meet or exceed the critical value
of 9.488, do not conclude there is a statistically significant difference among the
assembly lines in the observed frequencies of defective parts.

71
Chi-Square Test of Independence

The chi-square test of independence is used to compare frequencies (counts) of nomi-

nal or ordinal level data for two samples across two or more subgroups displayed in a
crosstabulation table. The chi-square test is more common and more flexible than z-
tests of proportions.

Problem: Evaluate the association between a person's sex and their attitudes toward
school spending on athletic programs. A random sample of adults in a school district
produced the following table (counts).

Female Male Row Total

Spend more money 15 25 40
Spend the same 5 15 20
Spend less money 35 10 45
Column Total 55 50 105

I. Assumptions

Independent random sampling

Nominal/Ordinal level data

No more than 20% of the cells have an expected frequency less than 5

No empty cells

II. State the Hypothesis

Ho: There is no association between a person's sex and their attitudes toward
spending on athletic programs.

Ha: There is an association between a person's sex and their attitudes toward
spending on athletic programs.

72
III. Set the Rejection Criteria

Determine degrees of freedom df=(# of rows - 1)(# of columns - 1)

df=(3 - 1)(2 - 1) or df=2

Establish the confidence level (.05, .01, etc.); Alpha = .05

Based on the chi-square distribution table, the critical value = 5.991

IV. Compute the Test Statistic

where

Fo= observed frequency

Frequency Observed Female Male Row Total

Spend more 15 25 40
Spend same 5 15 20
Spend less 35 10 45
Column Total 55 50 105

Fe= expected frequency for each cell

Fe= (frequency for the column)*(frequency for the row)/n

Frequency Expected Female Male Row Total

Spend more 55*40/105 = 20.952 50*40/105 = 19.048 40
Spend same 55*20/105 = 10.476 50*20/105 = 9.524 20
Spend less 55*45/105 = 23.571 50*45/105 = 21.429 45
Column Total 55 50 105

73
Chi-square Calculations Female Male
Spend more (15-20.952)2/20.952 (25-19.048)2/19.048
Spend same (5-10.476)2/10.476 (15-9.524)2/9.524
Spend less (35-23.571)2/23.571 (10-21.429)2/21.429

Chi-square Female Male

Spend more 1.691 1.860
Spend same 2.862 3.149
Spend less 5.542 6.096
21.200

V. Decide Results of Null Hypothesis

Since the chi-square test statistic 21.2 exceeds the critical value of 5.991, con-
clude there is a statistically significant association between a person's sex and
their attitudes toward spending on athletic programs. As is apparent in the contin-
gency table, males are more likely to support spending on athletic programs than
females.

Standardized Residuals

Standardized residuals are used to determine what categories (cells) were major con-
tributors to rejecting the null hypothesis. When the absolute value of the residual (R)
is greater than 2.00, the researcher can conclude it was a major influence on a signifi-
cant chi-square test statistic.

74
Example using the observed and expected frequencies in the previous example:

R Absolute R

Female Male Female Male

Spend More -1.300 1.364 1.300 1.364
Spend Same -1.692 1.774 1.692 1.774
Spend Less 2.354 -2.469 [a] 2.354 2.469

Interpretation

[a] Attitudes toward spending less for both females and males had a major contribution to the
chi-square result.

75
6.3!Coefficients for Measuring Association

The following are a few of several measures of association used with chi-square and
other contingency table analyses. When using the chi-square statistic, these coeffi-
cients can be helpful in interpreting the strength of the relationship between two vari-
ables once statistical significance has been established. The logic for using measures
of association is as follows:

Even though a chi-square test may show statistical significance between two vari-
ables, the relationship between those variables may not be substantively important.
Measures of association are available to help evaluate the relative strength of a statis-
tically significant relationship. In most cases, they are not used in interpreting the
data unless the chi-square statistic first shows there is statistical significance (i.e., it
doesn't make sense to say there is a strong relationship between two variables when
the statistical test shows this relationship is not statistically significant).

Nominal and Ordinal Variables

Phi

Phi is only used on 2x2 contingency tables. It is interpreted as a measure of the rela-
tive (strength) of an association between two variables ranging from 0 to 1.

Pearson's Contingency Coefficient (C)

Pearson’s contingency coefficient is interpreted as a measure of the relative (strength)

of an association between two variables. The coefficient will always be less than 1
and varies according to the number of rows and columns.

76
Cramer's V Coefficient (V)

Cramer’s V is useful for comparing multiple X2 test statistics and is comparable

across contingency tables of varying sizes. It is not affected by sample size and there-
fore is very useful in situations where a statistically significant chi-square may be the
result of large sample size instead of any substantive relationship between the vari-
ables. It is interpreted as a measure of the relative (strength) of an association be-
tween two variables. The coefficient ranges from 0 to 1 (perfect association). In prac-
tice, a Cramer's V of .10 may provide a good minimum threshold for suggesting there
may be a substantive relationship between two variables.

where q = smaller # of rows or columns

Describing Strength of Association

Characterizations

>.50 high association

.30 to .50 moderate association

.10 to .30 low association

.01 to .10 little if any association

77
Proportional Reduction of Error (PRE)

Lambda

Lambda is a proportional reduction in error (PRE) measure that ranges from 0 to 1.

Lambda indicates the extent to which the independent variable reduces the error as-
sociated with predicting the value of a dependent variable. Multiplied by 100, it repre-
sents the percent reduction in error.

Ordinal Variables Only

Gamma

Gamma is another PRE measure ranging from -1 to 1 that estimates the extent errors
are reduced in predicting the order of paired cases. Gamma ignores ties.

Kendall’s Tau b

Tau b is similar to Gamma but includes ties. It can range from -1 to 1 but since stan-
dardization is different from Gamma, it provides no clear explanation of PRE.

Inter-rater Agreement

Cohen’s Kappa

Cohen’s kappa measures agreement beyond chance. Although a negative value is

possible, it commonly ranges from 0 to 1 (perfect agreement). This measure requires
a balanced table where the number of rows is the same as the number of columns.
The diagonal cells represent agreement.

78
Software Output: Survey of U.S. adults.

Crosstabulation: GUNLAW (Rows) by SEX (Columns)

Column Variable Label: Respondent's Sex
Row Variable Label: Gun permits

Count |
Row % |
Col % |
Total % | Male | Female | Total
--------------------------------------------
Favor | 314| 497| 811
| 38.72| 61.28|
| 73.88| 88.91| 82.42
| 31.91| 50.51|
--------------------------------------------
Oppose | 111| 62| 173
| 64.16| 35.84|
| 26.12| 11.09| 17.58
| 11.28| 6.30|
--------------------------------------------
| 425| 559| 984
Total | 43.19| 56.81| 100.00

Chi-square Value DF p <

-----------------------------------------------------------
Pearson 37.622 1 0.0000 [a]
Likelihood Ratio 37.417 1 0.0000
Yate's Correction 36.592 1 0.0000 [b]

Measures of Association
-------------------------------------
Cramer's V .196 [c]
Pearson C .192
Lambda Symmetric .082 [d]
Lambda Dependent=Column .000
Lambda Dependent=Row .115 [e]

Note: 00.00% of the cells have an expected frequency <5

79
Interpretation

[a] The association between opinion on gun control and respondent sex is statistically signifi-
cant. This is the most common measure used for chi-square significance.

[b] When sample sizes are small, the continuous chi-square value tends to be too large. The
Yates continuity correction adjusts for this bias in 2x2 contingency tables. Regardless of sample
size, it is a preferred measure for chi-square tests on 2x2 tables.

[c] Weak association (both Cramer’s V and Pearson C).

[d] A symmetric lambda is used when identification of independent and dependent variables is
not useful.

[e] Knowing a person’s sex can reduce prediction error by 11.5%.

80
Summary Table

This table includes descriptive statistics for nominal level data using counts and per-
centages. It also includes inferential statistics using confidence intervals for propor-
tions, chi-square, and measures of association. Five calculations were required to cre-
ate the margin of errors. Four crosstabulations are presented in one table using col-
umn percent (Sex Education by Sex, Sex Education by Race, Gun Control by Sex,
Gun Control by Race).

Table 1: Attitudes toward sex education and gun permit policy issues by sex and race of respondent
(percent).
Sex Race
P <** P <**
Non-
Total Male Female (Cramer's V) White (Cramer's V)
White

Sample size > 75 36 39 63 12

* Margin of Error > 11.3% 16.3% 15.7% 12.4% 28.3%

Sex Education in School

.578 [b] .064
Favor 72% 75% 69% 76% 50%
Oppose 28% 25% 31% (.064) 24% 50% (.214)

Require Gun Permits

Favor 48% [a] 33% 62% .015 [c] 48% 50% .888
(.282) [d] (.017)
Oppose 52% 67% 38% 52% 50%

Source: 1993 General Social Survey of U.S. Adults

* Based on alpha .05 (95% confidence interval)
** Significance is based on the chi-square test of independence.

81
Interpretation

[a] Based on a 95% confidence level, the proportion of all U.S. adults who favor requiring gun
permits is 48% plus or minus 11.3% or between 37% to 59%.

[b] A statistically significant relationship between sex and attitudes toward sex education in
school is not evident (p > .05). Based on this random sample, there is a lack of convincing evi-
dence for a relationship in the underlying population.

[c] There is a statistically significant relationship between sex and attitudes toward gun per-
mits (p < .05). Women are significantly more likely (62%) to favor gun permits than men (33%).

[d] There is a weak to moderate association between sex and attitudes toward gun permits
based on a Cramer’s V of .282.

82
7
T-Tests of Means and One-Way
ANOVA
This section presents hypothesis testing techniques for interval/ratio data. The type
of analytical technique used will depend on the level of the independent variable.

Examples of Inferential Statistics using Interval/Ratio Data

Statistical Technique Dependent Variable Independent Variable

Confidence interval miles to work [c] None

t-test annual income [c] Sex [a]
ANOVA physicians per capita [c] census region [a]
ANOVA years education [c] Military rank [b]
Correlation, regression annual income [c] education [c]
Correlation, regression weight (kilograms) [c] height (centimeters) [c]

[a] Nominal [b] Ordinal [c] Interval/Ratio

This chapter focuses on t-tests and ANOVA. Later chapters will address correlation
and regression.

83
7.1 T-test of Means

For inferential statistics (confidence intervals and significance tests), the data are as-
sumed to be from a random sample. To conduct significance tests, the test statistic
is compared to a critical value from a sampling distribution. Tables of critical values
are included in the Appendix. The relevant sampling distributions for this section are
the t distribution and f distribution.

Statistical Techniques

★ Population Mean to a Sample Mean (One-Sample Test)

★ Independent Samples With Equal Variance (Two-Sample Test)

★ Independent Samples Without Equal Variance (Two-Sample Test)

Comparing a Population Mean to a Sample Mean (One-Sample Test)

A one-sample t-test is used to compare a mean from a random sample to the mean
(mu) of a population. It is especially useful to compare a mean from a random sam-
ple to an established data source such as census data to determine if a sample is un-
biased (i.e., representative of the underlying population). Examples include compar-
ing mean age and education levels of survey respondents to known values in the
population.

Problem: Compare the mean age of incoming students to the known mean age for all
previous incoming students. A random sample of 30 incoming college freshmen re-
vealed the following statistics: mean age 19.5 years, standard deviation 1 year. The
college database shows the mean age for previous incoming students was 18.

84
I. Assumptions

Interval/ratio level data

Random sampling

Normal distribution in population

II. State the Hypothesis

Ho: There is no significant difference between the mean age of past college stu-
dents and the mean age of current incoming college students.

Ha: There is a significant difference between the mean age of past college stu-
dents and the mean age of current incoming college students.

III. Set the Rejection Criteria

Significance level .05 alpha, 2-tailed test

Degrees of Freedom = n-1 or 29

Critical value from t-distribution = 2.045

IV. Compute the Test Statistic

Standard error of the sample mean

85
 Test statistic

V. Decide Results of Null Hypothesis

Given that the test statistic (8.197) exceeds the critical value (2.045), the null hy-
pothesis is rejected in favor of the alternative. There is a statistically significant
difference between the mean age of the current class of incoming students and
the mean age of freshman students from past years. In other words, this year's
freshman class is on average older than freshmen from prior years.

If the results had not been significant, the null hypothesis would not have been
rejected. This would be interpreted as the following: There is insufficient evi-
dence to conclude there is a statistically significant difference in the ages of cur-
rent and past freshman students.

Software Output: Summary data from previous example

One-Sample Difference of Means

Population Mean 18.0000

Sample Mean 19.5000
Std Deviation 1.0000
Sample Size (n) 30
Test statistic 8.2160
p < 0.0001 (2-tailed) [a]

Interpretation

[a] For a 2-tailed test, the p-value represents the probability of making a type 1 error (conclud-
ing there is statistical significance when there is none). Since there is less than a .01 % ( p <
.0001) chance of making a type 1 error, there is statistical significance.

86
Comparing Two Independent Sample Means (Two-Sample Test)

With Homogeneity of Variance

A two-sample t-test is used to compare two sample means. The independent vari-
able is nominal level data and the dependent variable is interval/ratio level data.

Problem: The number of years of education were collected from one random sample
of 38 police officers from City A and a second random sample of 30 police officers
from City B. The average years of education for the sample from City A is 15 years
with a standard deviation of 2 years. The average years of education for the sample
from City B is 14 years with a standard deviation of 2.5 years. Is there a statistically
significant difference between the education levels of police officers in City A and City
B?

I. Assumptions

Random sampling

Independent samples

Interval/ratio level data

Organize data

City A (labeled sample 1)

City B (labeled sample 2)

II. State Hypotheses

Ho: There is no statistically significant difference between the mean education

level of police officers working in City A and the mean education level of police
officers working in City B.

87
 For a 2-tailed hypothesis test

Ha: There is a statistically significant difference between the mean education

level of police officers working in City A and the mean education level of police
officers working in City B.

For a 1-tailed hypothesis test

Ha: The mean education level of police officers working in City A is significantly
greater than the mean education level of police officers working in City B.

III. Set the Rejection Criteria

Determine the degrees of freedom (df) = (n1+n2)-2 df = 38+30-2=66

Determine level of confidence -- alpha (1 or 2-tailed test)

Use the t-distribution table to determine the critical value

If using 2-tailed test, Alpha.05, tcv= 1.997

If using 1-tailed test, Alpha.05, tcv= 1.668

IV. Compute Test Statistic

Standard error

88
 Test Statistic

V. Decide Results

If using 2-tailed test . . .

There is no statistically significant difference between the mean years of educa-

tion for police officers in City A and mean years of education for police officers in
City B. The test statistic 1.835 does not meet or exceed the critical value of
1.997 for a 2-tailed test.

If using 1-tailed test . . .

Police officers in City A have significantly more years of education than police offi-
cers in City B. The test statistic 1.835 exceeds the critical value of 1.668 for a 1-
tailed test.

89
Software Output: Uses survey data to compare the mean incomes of males and fe-
males.

Two-Sample Difference of Means

Independent Variable: SEX

Dependent Variable: INCOME

Sample One: 0 Female

Sample Two: 1 Male

Sample 1 Sample 2
Female Male
-----------------------------------------------------------
Sample Mean 12279.4667 15266.4706
Std Deviation 4144.6323 3947.7352
Sample Size (n) 15 17

Homogeneity of Variance
------------------------------------------------------------
F-ratio 1.09 DF (14, 16) p < 0.4283 [a]

T-statistic DF p < (2-tailed)

------------------------------------------------------------
Equal Variance -2.0870 30 0.0455 [b]
Unequal Variance -2.0800 31.05 0.0459

Interpretation

[a] The F-ratio is not statistically significant. Therefore, use the equal variance test statistic.

[b] For a 2-tailed test, the p-value represents the probability of making a type 1 error (conclud-
ing there is statistical significance when there is none). Since there is about 4.6% (p < .0455)
chance of making a type 1 error, which does not exceed the 5% error limit (p=.05) established in
the rejection criteria of the hypothesis testing process (alpha), conclude there is statistical signifi-
cance.

90
Computing F-ratio

The F-ratio is used to determine whether the variances in two independent samples
are equal. If the F-ratio is not statistically significant, assume there is homogeneity of
variance and employ the standard t-test for the difference of means. If the F-ratio is
statistically significant, use an alternative t-test computation such as the Cochran
and Cox method.

Problem: Given the following summary statistics, are the variances equal?

Sample A =20 n=10

Sample B =30 n=30

Set the Rejection Criteria

Determine the degrees of freedom (df) for each sample

Numerator of the ratio is the sample with the larger variance (Sample B).

Numerator df = n - 1 df for numerator (Sample B) = 29

Denominator of the ratio is the sample with the smaller variance (Sample A).

Denominator df = n - 1 df for denominator (Sample A) = 9

Determine the level of confidence -- alpha

Consult F-Distribution table for df = (29,9), alpha.05

Fcv= 2.70

91
Compute the Test Statistic

where

= largest variance (Sample B)

= smallest variance (Sample A)

Decide Results

Compare the test statistic with the f critical value (Fcv) listed in the F distribution. If
the F-ratio equals or exceeds the critical value, the null hypothesis (Ho) (there is
no difference between the sample variances) is rejected. If there is a difference in the
sample variances, the comparison of two independent means should involve the use
of the Cochran and Cox method or one of several alternative techniques.

The test statistic (1.50) did not meet or exceed the critical value (2.70). Therefore,
there is no statistically significant difference between the variance exhibited in Sam-
ple A and the variance exhibited in Sample B. Assume homogeneity of variance for
tests of the difference between sample means.

92
Software Output: Comparing the mean incomes of high school graduates and high
school dropouts from survey data.

Two-Sample Difference of Means

Independent Variable: EduCat! Education Level

Dependent Variable: Income!Respondent individual income

Sample One: 1 <12 yrs

Sample Two: 2 HS Grad

Sample 1 Sample 2
<12 yrs HS Grade
------------------------------------------------------------
Sample Mean 23042.5000 31000.8235
Std Deviation 4450.5413 9442.4477
Sample Size (n) 6 17

Homogeneity of Variance
------------------------------------------------------------
F-ratio 5.08 DF (16, 5) p < 0.0408 [a]

T-statistic DF p < (2-tailed)

------------------------------------------------------------
Equal Variance -1.9660 21 0.0627 [b]
Unequal Variance -2.7220 21.67 0.0124 [c]

Interpretation
[a] The F-ratio is statistically significant. Use the unequal variance test statistic.
[b] If homogeneity of variance had not been considered, the research may have erroneously
failed to find statistical significance.

[c] Since there is only a 1.2% (p < .0124) chance of making a type 1 error, which does not ex-
ceed the 5% error limit (p=.05, alpha), conclude there is statistical significance.

93
7.2 One-Way Analysis of Variance (ANOVA)

One-way analysis of variance (ANOVA) is used to evaluate means from two or more
subgroups. A statistically significant ANOVA indicates there is more variation be-
tween subgroups than would be expected by chance. It does not identify which sub-
group pairs are significantly different from each other. ANOVA is used to evaluate mul-
tiple means of one independent variable to avoid conducting multiple t-tests (see mul-
tiple comparison problem in section 6.3).

Problem: The number of years of education were obtained from one random sample
of 38 police officers from City A, a second random sample of 30 police officers from
City B, and a third random sample of 45 police officers from City C. The average
years of education for the sample from City A is 15 years with a standard deviation of
2 years. The average years of education for the sample from City B is 14 years with a
standard deviation of 2.5 years. The average years of education for the sample from
City C is 16 years with a standard deviation of 1.2 years.

Is there a statistically significant difference between the education levels of police offi-
cers in City A, City B, and City C?

Data For Computations

City A City B City C

Mean (years) 15 14 16
S (Standard Deviation) 2 2.5 1.2
S2 (Variance) 4 6.25 1.44
n (number of cases) 38 30 45
Sum of Squares 152 187.5 64.8

94
I. Assumptions

Independent Random sampling

Interval/ratio level data

Population variances equal

Groups are normally distributed

II. State the Hypothesis

Ho: There is no statistically significant difference among the three cities in the
mean years of education for police officers.

Ha: There is a statistically significant difference among the three cities in the

mean years of education for police officers.

III. Set the Rejection Criteria

Determine the degrees of freedom for the F Distribution

Numerator Degrees of Freedom

df=k-1 where k=3 (number of independent samples/groups) df=2

Denominator Degrees of Freedom

df=n-k where n=113 (sum of all independent samples) df=110

Establish Critical Value

Determine the level of confidence -- alpha

At alpha.05, df=(2,110)

95
 Consult f-distribution, Fcv = 3.072

IV. Compute the Test Statistic

F=
∑ n ( X − X ) / (k −1)
k i g

# 2 & 2 2

%$∑ ( X −iX )1+ ∑ ( X − X ) + ∑ (

i X − X )
2
(' / ( n − k ) i 3

where

= each group mean

= grand mean for all the groups = ( sum of all scores)/N

= number in each group

Note: F = Mean Squares between groups

Mean Squares within groups

Estimate Grand Mean

Estimate F Statistic

F=
(
∑ nk Xi − X g ) / ( k −1)
#
) + ∑( X − X ) &(' / (n − k )
2 2 2
( ) (
%$∑ Xi − X 1 + ∑ Xi − X 2 i 3

96


V. Decide Results of Null Hypothesis

Compare the F statistic to the F critical value. If the F statistic equals or exceeds
the Fcv, the null hypothesis is rejected. This suggests that the population means
of the groups sampled are not equal -- there is a difference between the group
means.

Since the F-statistic (9.931) exceeds the F critical value (3.072), we reject the null
hypothesis and conclude there is a statistically significant difference between the
three cities in the mean years of education for police officers.

97
Software Output: Comparing amount of time spent watching TV per day by age
group.

Analysis of Variance (ANOVA): TVHOURS (Means) by AGECAT4 (Groups)

Independent Variable Label: Age categories

Dependent Variable Label: Hours watch TV per day

Group Summary Statistics

Group N Mean Std. Dev.

--------------------------------------------------------------
18-29 185 3.0108 2.2698
30-39 243 2.6008 2.0331
40-49 198 2.5859 1.7712
50+ 364 3.2665 2.6385

Analysis of Variance

Sum of Squares df Mean Squares

--------------------------------------------------------------
Between Groups 92.09 3 [a] 30.70 [b]
Within Groups 5093.45 986 [c] 5.17 [d]
Total 5185.54 989 [e]

F Statistic p <
--------------------------------
5.94 [f] 0.0005 [g]

Interpretation

[a] k-1, where k = 4 group means

[b] Between Groups Sum of Squares ÷ degrees of freedom

[c] n-k, where n = 990

[d] Within Groups Sum of Squares ÷ degrees of freedom

[e] n-1

98
[f] Between Groups Mean Squares ÷ Within Groups Mean Squares

[g] The F-Statistic is statistically significant. The p-value represents the probability of making
a type 1 error (concluding there is statistical significance when there is none). Since there is
about .05% (p < .0005) chance of making a type 1 error, conclude there is statistical signifi-
cance; ie, there is variation among the groups.

99
7.3 Multiple Comparison Problem
(Post hoc comparisons)

As statistical tests are repeatedly run between subgroups within the same variable,
the probability of making a type one error increases (i.e., if 100 t-tests were con-
ducted, by random chance 5 out of 100 may be statistically significant when if fact
they are not). Post hoc comparisons adjust for the problem of multiple comparisons
and provide information regarding which subgroup pairs have a statistically signifi-
cant difference. There are several alternative approaches to conducting post hoc
comparisons. Bonferroni is one of the most conservative (less likely to find statistical
significance).

Software Output: The following presents post hoc comparisons for the preceding
ANOVA output.

Bonferroni Post Hoc Comparison of Means

Group 1 Group 2 G1-G2 SE p<

-----------------------------------------------------------------
18-29 30-39 0.410 0.209 0.3005
18-29 40-49 0.425 0.207 0.2464
18-29 50+ -0.256 0.228 1.0000
30-39 18-29 -0.410 0.209 0.3005
30-39 40-49 0.015 0.184 1.0000
30-39 50+ -0.666 [a] 0.200 0.0056 [b]
40-49 18-29 -0.425 0.207 0.2464
40-49 30-39 -0.015 0.184 1.0000
40-49 50+ -0.681 0.209 0.0073 [c]
50+ 18-29 0.256 0.228 1.0000
50+ 30-39 0.666 0.200 0.0056
50+ 40-49 0.681 0.209 0.0073

100
Interpretation

[a] The difference between the mean of those 30-39 (G1) and the mean of those 50 or older
(G2) is a -0.666. A negative mean indicates those 50 or older have a greater mean number of
hours watching television than those 30-39.

[b] There is a statistically significant difference between the number of hours watching televi-
sion for those between the ages of 30 and 39 and the number of hours watching television for
those 50 and older.

[c] There is a statistically significant difference between the number of hours watching televi-
sion for those between the ages of 40 and 49 and the number of hours watching television for
those 50 and older.

101
Summary Report

This table uses descriptive statistics (means), t-tests, and ANOVA to compare city
manager characteristics. The independent variables are sex and city size (nominal/
ordinal). The dependent variables are age, education, and years as a manager
(interval/ratio).

Table 2: City manager characteristics (means) by sex and city size category.

Sex City Size Category

Total Male Female p<* Small Medium Large p < **

Sample size > 30 19 11 14 7 9

Manager Characteristics
Age 46.80 49.68 41.82 .028 [a] 41.71 46.86 54.67 .004 [b]
Years of Education 16.63 16.68 16.55 .838 16.14 17.71 16.56 .152
Years as Manager 16.08 18.40 12.09 .044 [c] 15.18 16.57 17.11 .861

Skill Preference Score

Negotiate-Analytical *** -0.15 -0.03 -0.37 .130 -0.39 -0.26 0.30 .003 [d]

Source: 1996 Survey of U.S. City Managers

* T-test of means, 2-tailed

** ANOVA
*** Based on two measures of skill importance (larger score indicates higher importance). The
preference score was created by subtracting the analytical score (management science) from
the negotiation score (political savvy). A positive preference score indicates a city manager finds
negotiation skills more important than analytical skills. A negative preference score indicates a
city manager finds analytical skills more important than negotiation skills.

102
Interpretation

[a] There is a significant difference between male and female city managers in mean age.
Male city managers (49.7 years) are significantly older than female city managers (41.8 years).

[b] There is a significant difference among small, medium, and large cities in the mean ages of
city managers. It appears that larger size cities are likely to have older city managers than
smaller cities. This conclusion should be confirmed with a post-host analysis.

[c] Males have significantly greater years experience (about 6 years more on average) as city
managers than females.

[d] There is a significant difference among small, medium, and large cities in the mean skill
preferences of city managers. It appears that large cities are likely to have city managers who
place more importance on negotiation skills than analytical skills. In contrast, small and medium
size cities prefer analytical skills over negotiation. This conclusion should be confirmed with a
post-host analysis.

103
8
Correlation

8.1 Pearson's Product Moment Correlation Coefficient

Correlation coefficients estimate strength and direction of association between two

interval/ratio level variables. The Pearson Correlation Coefficient presented here can
range from a -1.00 to 1.00. A positive coefficient indicates the values of variable A
vary in the same direction as variable B. A negative coefficient indicates the values of
variable A and variable B vary in opposite directions.

Example: The following data were collected to estimate the correlation between
years of formal education and income at age 35.

Susan Bill Bob Tracy Joan

Education (years) 12 14 16 18 12
Income ($1000) 25 27 32 44 26

Verify Conditions for using Pearson r

Interval/ratio data must be from paired observations.

A linear relationship should exist between the variables.

No extreme values in the data.

104
The following scattergram assists in evaluating linearity and also
helps to identify problems related to extreme values.

Y: Income

44.0| *
|
|
|
34.5|
| *
|
|
| * *
25.0| *
---------------|--------------|
12.0 15.0 18.0

X: Education

Compute Pearson's r

Education Income
  (Years) ($1000)
Name X Y XY X2 Y2
Susan 12 25 300 144 625
Bill 14 27 378 196 729
Bob 16 32 512 256 1024
Tracy 18 44 792 324 1936
Joan 12 26 312 144 676
Σ= 72 154 2294 1064 4990
n= 5

105
 n∑ XY − ∑ X ∑Y
rxy =
# 2& # 2&

%$ n ∑ X 2
− ( X )
∑ (' * %$n∑Y 2 − ( Y )
∑ ('

5 ( 2294) − 72 (154)
rxy =
"#5 (1064) − 72 2 $% * "#5 ( 4990 ) −154 2 $%

Interpret

A positive coefficient indicates the values of variable A vary in the same direction as
variable B. A negative coefficient indicates the values of variable A and variable B
vary in opposite directions.

Characterizations of Pearson r

.9 to 1 very high correlation

.7 to .9 high correlation

.5 to .7 moderate correlation

.3 to .5 low correlation

.0 to .3 little if any correlation

In this example, there is a very high positive correlation between the variation of edu-
cation and the variation of income. Individuals with higher levels of education earn
more than those with comparably lower levels of education.

106
Coefficient of Determination

The coefficient of determination represents the proportion of variation in the depend-

ent variable (Y) explained by the independent variable (X). In the case of Pearson r,
the coefficient of determination can be obtained by squaring the Pearson r coeffi-
cient.

Example: From previous example

Eighty-seven percent of the variance displayed in the income variable is associated

with the variance displayed in the education variable.

Hypothesis Testing for Pearson r

Problem: Determine statistical significance based on a Pearson r of .933 for annual

income and education obtained from a national random sample of 20 employed
adults.

I. Assumptions

Data originated from a random sample

Data are interval/ratio

Both variables are distributed normally

Linear relationship and homoscedasticity

II. State the Hypothesis

107
 Ho: There is no association between annual income and education for employed
adults.

Ha: There is an association between annual income and education for employed
adults.

III. Set the Rejection Criteria

Determine the degrees of freedom (df) df=n – 2 or 20-2=18

Determine the confidence level, alpha (1-tailed or 2-tailed)

Use the critical values from the t distribution at df=18

tcv @ .05 alpha (2-tailed) = 2.101

IV. Compute Test Statistic

V. Decide Results

Since the test statistic 11.022 exceeds the critical value 2.101, there is a statisti-
cally significant association in the national population between an employed
adult's education and their annual income.

108
Software Output: Example of one bivariate comparison with a scattergram. Compar-
ing individual income in U.S. dollars to years of education.

Pearson's Correlation

INCOME: [a]
42779.0| * *
| * * *
| * * * *
| * * *
28030.0| * *
| * * * * * *
| * *
| * *
| * *
12281.0| * *
---------------|--------------|
4.0 12.0 20.0
EDUC: [b]

Number of cases: 32
Missing: 0 [c]
Pearson Correlation: 0.751 [d]
p < (2-tailed signif.): 0.0000 [e]

Interpretation

[a] The Y axis of the scattergram. If theory suggests cause and effect, the Y axis is commonly
used for the dependent (response) variable.

[b] The X axis of the scattergram. If theory suggests cause and effect, the X axis is commonly
used for the independent variable.

[c] Since each observation (case) must have values for both income and education, any obser-
vations where one or both of these variables have no data will be removed from the analysis.

[d] Pearson correlation coefficient representing a high positive correlation between education
and income. Interpretation: As years of education increases so does personal income.

[e] There is a statistically significant association between income and education.

109
Software Output: Example of bivariate comparisons displayed in a correlation ma-
trix. Comparing individual income, education, and months of work experience.

Correlation: Pearson (R) Coefficients

-----------------------------------------------
Coeff | Correlation Matrix |
Cases |-----------------------------------|
p < | INCOME | EDUC | WORKEXP |
-----------------------------------------------
INCOME | 1.000 [a]| 0.751 [d]| -0.160 [f]|
| 32 [b]| 32 | 32 |
| . [c]| 0.000 [e]| 0.381 [g]|
-----------------------------------------------
EDUC | 0.751 | 1.000 | -0.520 |
| 32 | 32 | 32 |
| 0.000 | . | 0.002 [h]|
-----------------------------------------------
WORKEXP | -0.160 | -0.520 | 1.000 |
| 32 | 32 | 32 |
| 0.381 | 0.002 | . |
-----------------------------------------------
2-tailed significance tests
'.' p-value not computed

Interpretation

[a] The diagonal in the matrix represents Pearson correlations between the same variable
which will always be 1 since the variables are identical. The correlations above the diagonal are
a mirror reflection of these below the diagonal, so only interpret half of the matrix (in this case
three correlation coefficients).

[b] The number of paired observations for this comparison.

[c] A statistical test is not performed when comparing a variable to itself. Mathematically this
will always equal zero.

110
[d] Pearson correlation coefficient representing a high positive correlation between education
and income. Interpretation: As years of education increases so does personal income.

[e] There is a statistically significant association between income and education.

[f] Pearson correlation coefficient representing a very weak negative correlation between work
experience and income. Interpretation: As years of work experience increases personal income
decreases.

[g] There is not a statistically significant association between work experience and income.

[h] There is a statistically significant association between work experience and education.

111
8.2 Spearman Rho Coefficient

Spearman’s Rho is used to estimate strength and direction of association between

two ordinal level variables (paired observations that are ranked). The Spearman Rho
Coefficient presented here can range from a -1.00 to 1.00. A positive coefficient indi-
cates the values of variable A vary in the same direction as variable B. A negative co-
efficient indicates the values of variable A and variable B vary in opposite directions.

Verify the conditions are appropriate

Scores of two variables are ranks

Problem: Five college students' have the following rankings in math and philosophy
courses. Is there an association between student rankings in math and philosophy
courses?

Student Alice Jordan Dexter Betty Corina

Math class rank 1 2 3 4 5
Philosophy class rank 5 3 1 4 2

Compute Spearman Rho

where

n = number of paired ranks

d = difference between the paired ranks

112
Note: When two or more observations of one variable are the same, ranks are as-
signed by averaging positions occupied in their rank order.

Example:

Score 2 3 4 4 5 6 6 6 8
Rank 1 2 3.5 3.5 5 7 7 7 9

Math Rank Philosophy Rank X-Y (X-Y)2

X Y D d2
1 5 -4 16
2 3 -1 1
3 1 2 4
4 4 0 0
5 2 3 9
30

Interpret Coefficient

There is a moderate negative correlation between the math and philosophy course
rankings of students. Students who rank high as compared to other students in their
math course generally have lower philosophy course ranks and those with low math
rankings have higher philosophy course rankings than those with high math rankings.

Note: The formulas for Pearson r and Spearman rho are equivalent when there are no
tied ranks.

113
Hypothesis Testing for Spearman Rho

Significance testing for inference to the population.

Problem: Based on a Spearman Rho of .70 and a sample size of 20, is there an asso-
ciation between the music and physics rankings of students statistically significant?

I. Assumptions

Data originated from a random sample

Data are ordinal

Both variables are distributed normally

Linear relationship and homoscedasticity

Sample size is ≥ 10

II. State the Hypothesis

Null Hypothesis (Ho): There is no association between music and physics class
rankings for all students in the population.

Alternative Hypothesis (Ha): There is association between music and physics

class rankings for all students in the population.

III. Set the Rejection Criteria

Determine the degrees of freedom n – 2 or 20-2=18

Determine the confidence level, alpha (1-tailed or 2-tailed)

Use the critical values from the t distribution

114
 Note: The t distribution should only be used when the sample size is 10 or more.

tcv @ .05 alpha (2-tailed) = 2.101

IV. Compute the Test Statistic

Convert into

V. Decide Results of Null Hypothesis

Reject if the t-observed is equal to or greater than the critical value. The variables
are/are not related in the population. In other words, the association displayed
from the sample data can/cannot be inferred to the population.

Since the test statistic 4.159 exceeds the critical value 2.101, there is a statisti-
cally significant association in the national population between student music
and physics rankings. Students who rank high in musical ability will also likely
rank high in physics.

115
9
Simple OLS Regression

Linear ordinary least squares regression involves predicting the score for a dependent
variable (Y) based on the score of an independent variable (X). Data are tabulated for
two variables X and Y. If a linear relationship exists between the variables, it is appro-
priate to use linear regression to base predictions of the Y variable from values of the
X variable. See the multiple regression chapter for further discussion of assumptions.

9.1 Procedure

1. Determine the regression line with an equation of a straight line.

2. Use the equation to predict scores.

3. Determine the "standard error of the estimate" (Sxy) to evaluate the distribution
of scores around the predicted Y score.

Determine the regression line

Data are tabulated for two variables X and Y. Use Pearson’s r to help determine if
there is a linear relationship between the variables. If a significant linear relationship

116
exists between the variables, it is appropriate to use linear regression to base predic-
tions of the Y variable on the relationship developed from the original data.

Equation for a straight line

where

= predicted score

b = slope of a regression line (regression coefficient)

x = individual score from the X distribution

a = Y intercept (regression constant)

where

= mean of variable X

= mean of variable Y

Use the equation to predict scores

Insert the values for the intercept (a) and slope (b) of the completed equation for a
straight line and select an individual X score to predict a Y score.

117
Determine the Standard Error of the Estimate

The standard error of the estimate is used to provide a confidence interval around a
point estimate for predicted Y in the underlying population. In relative terms, larger
standard errors indicate less prediction accuracy for the population value than equa-
tions with smaller standard errors.

Confidence Interval for the Predicted Score

Sometimes referred to as the conditional mean of Y given a specific score of X.

Determine critical value (tcv)

Degrees of Freedom (df)= n-2

Select alpha (generally based on .05 for a 2-tailed test)

Obtain the critical value (tcv) from the t-distribution

118
Problem: The following data were collected to estimate the correlation between
years of formal education and income at age 35 and are the same data used in an ear-
lier example to estimate Pearson r.

Susan Bill Bob Tracy Joan

Education (years) 12 14 16 18 12
Income ($1000) 25 27 32 44 26

Education Income
  (Years) ($1000)
Name X Y XY X2
Susan 12 25 300 144
Bill 14 27 378 196
Bob 16 32 512 256
Tracy 18 44 792 324
Joan 12 26 312 144
Sum= 72 154 2294 1064
n= 5
Mean= 14.4 30.8

Determine Regression Line

Estimate the slope (b)

Estimate the y intercept

Given x (education) of 15 years, estimate predicted Y

119


or an estimated annual income of $32,485 for a person with 15 years of

education (this is the point estimate).

Determine the Standard Error of the Estimate

Standard error of the estimate of Y

2
" ^%
∑$# yi − y '&
se =
n−2

Determine the Confidence interval for predicted Y

where

Degrees of Freedom (df) = 5-2 or 3

Alpha .05, 2-tailed

Based on the t-distribution tcv = 3.182

120
Confidence interval for predicted Y

Given a 5% chance of error, the estimated income for a person with 15 years of edu-
cation will be $32,485 plus or minus $10,424 or somewhere between $22,060 and
$42,909 (the interval estimate).

Standard error of the slope estimate

To develop confidence intervals or test hypotheses, we need to estimate the standard

error of the slope estimate (sb).

2
" ^%
∑$# yi − y '& / (n − 2)
sb = 2
∑( x − X )
i

Based on the example data . . .

Calculating Confidence Interval for the slope (b)

121
Where CV= critical value (t-distribution, 2-tailed, .05 alpha, df=n-2) and 2.8 is the
point estimate for the slope.

or simplify to

Interpretation

The slope for education is between 0.8 and 4.8.

122
9.2 Hypothesis testing

The null hypothesis states that X is not related with Y (i.e., the slope is 0)

Ho: (slope coefficient is zero)

Ha: (slope coefficient is not zero)

Determine Critical Value

t-distribution, 2-tail, alpha .05, n-2=3

bcv = 3.182

2.8
t=
.625 t = 4.48

Decision

Reject the null hypothesis. The slope for education is statistically significant (i.e., t
test statistic of 4.48 is beyond the critical value of 3.182). As education in years in-
creases so will income.

123
9.3 Evaluating the power of the regression model

If we only had information on Y (Income), our best guess of an individual's income

would be the mean income. However, if we have a paired X variable (Education) that
is related to Y, we can use this additional variable to improve our ability to predict an
individual's income.

The independent variable's ability to model variations in Y can be evaluated by com-

paring the amount of deviation explained by our model using X to the total amount of
deviation in Y. This ratio is known as the Coefficient of Determination or R2 which rep-
resents the proportion of variation in Y explained by X. It can range from 0 to 1.

Components of Deviation (R2 ) (y=income; x=education)

The components of deviation for one observation are as follows:

yi − Y = the deviation of the Y observation from the mean of Y (Total Dev.)

^
yi −Y = deviation explained by X (Explained Dev.)

^
yi − yi = deviation not explained by X (Unexplained Dev.)

124
Example of Explained and Unexplained Deviations Using One Observation

= mean income $30.8k

yi = Tracy’s income $44k

xi
= Tracy’s education 18 years
^
y = Tracy’s predicted income is $40.9

Example of Deviations in a Two Dimensional Plot

125
The formula for estimating deviations for all observations is as follows:

TSS (Total Sum of Squares)

2
∑(Y −Y )
i
= the total deviation of Y

RSS (Regression Explained Sum of Squares)

2
"^ %
∑$#Y i −Y '&
= deviation explained by X

ESS (Error Sum of Squares)

2
" ^ %
∑$#Yi −Yi '&
= deviation not explained by X

2
"^ %
∑$#Y i −Y '&
R2 = 2

or
∑(Y −Y )
i

126
Example:

Impact of R2 on predictions:

A relatively high R2 is required to make accurate predictions (.90 or better). It is very

unlikely in social science that we will obtain R2 this high, thus we focus more on ex-
plaining relationships.

R2 is sample specific:

Two samples with the same variables, slope, and intercept could have different R2 be-
cause of the fit between the data and the regression line (different variation in Y; see
formula).

127
Software Output: Regressing individual income on education from survey data.

---------------------- Simple OLS Regression --------------------

Model Fit Statistics

Cases(n) R R Square Adj Rsq SE Est.

-----------------------------------------------------------------
32 0.751 [a] 0.564 [b] 0.550 2854.601 [c]

ANOVA Statistics

Sum of Sqs df Mean Sq F p <

-----------------------------------------------------------------
Regression 316481758.400 1 316481758.400 38.838 0.0000 [d]
Residual 244462446.475 30 8148748.216
Total 560944204.875 31

Coefficients

Variable b Coeff. std. error BetaWgt t p <

------------------------------------------------------------------
Constant 5077.512 [e] 1497.830 !3.390 ! 0.0020
EDUC 732.400 [f] 117.522 [g] 0.751 [h] 6.232 [i] 0.0000 [j]

Estimated Model

INCOME = 5077.513 + 732.400(EDUC)

Interpretation
[a] The correlation coefficient.
[b] Education explains 56% of the variation in income.
[c] Standard error of the estimate. Used for creating confidence intervals for predicted Y.
[d] The regression model is statistically significant.

128
[e] The Y intercept and mean value of income if education is equal to 0.
[f] The impact of a one-unit change in education on income. Interpretation: One additional
year of education will on average result in an increase in income of $732.
[g] Standard error of the slope coefficient. Used for creating confidence intervals for b and for
calculating a t-statistic for significance testing.
[h] Standardized regression partial slope coefficients. Used only in multiple regression models
(more than one independent variable), beta-weights in the same regression model can be com-
pared to one another to evaluate relative effect on the dependent variable. Beta-weights will indi-
cate the direction of the relationship (positive or negative). Regardless of whether negative or
positive, larger beta-weight absolute values indicate a stronger relationship.
[i] The t-test statistic. Calculated by dividing the b coefficient by the standard error of the
slope.
[j] Education has a statistically significant positive association with income.

129
10
Multiple OLS Regression

Multiple regression is an extension of the simple regression model that allows the in-
corporation of more than one independent variable into the explanation of Y. Multiple
regression helps clarify the relationship between each independent variable and Y by
holding constant the effects of the other independent variables in the model.

General equation:

Interpretation:

The value of Y is determined by a linear combination of the independent variables (Xk)

plus an error term (e).

We continue to use the least squares approach for fitting the data based on the for-
mula for the straight line; this as the least sum of the squared differences between ob-
served Y and predicted Y.

2
" ^ %
ESS = ∑$Yi −Y i '
# &

However, the fit cannot be visualized graphically on a two dimensional scattergram.

The line is best visualized on three or more dimensional planes.

130
Partial slope equation (Three variable example)

Process:

1. Assume there is some correlation between the independent variables.

2. Measure portion of X1 not explained by X2, which is a simple application of the bi-
variate model through the following process:

Use equation for a straight line to predict X1 given X2.

analogous to

Compute the prediction error where u represents the portion of X1 which X2 can-
not explain.

2. Repeat steps to measure portion of Y which is not explained by X2:

Use equation for a straight line to predict Y given X2.

analogous to

Compute the prediction error where v represents the portion of Y which X2 can-
not explain.

131
3. The above is incorporated into the formula for b1

^ ^
⎛ ⎞⎛ ⎞
⎜ X − X
∑ ⎝ 1 1 ⎠⎝
⎟⎜ Y − Y ⎟
⎠
b1 = 2
^
⎛ ⎞
X X
∑ ⎝ 1 1 ⎟⎠
⎜ −
or

This equation results in a measure of the slope for X1 that is independent of the linear
effect of X2 on X1 and Y. Put in more applied terms, a multiple regression model al-
lows us to evaluate the spuriousness of relationships. As an example. We found in
our bivariate model that education has a significant causal relationship with income.
Given the simplicity of this model, we don't know if the relationship might disappear if
we controlled for employee experience. In adding experience to our model, we face
four possible outcomes:

1. Education is significant but experience is not (evidence bivariate relationship be-

tween education and income is not spurious for experience).

2. Experience is significant but education is not (evidence bivariate relationship be-

tween education and income is spurious).

3. Both education and experience are significant (evidence bivariate relationship be-
tween education and income is not spurious).

4. Neither are significant (evidence of high correlation between education and expe-
rience).

132
Partial slope example

Data

Income Education Experience

($1000s) (Years) (Years)
Name Y X1 X2
Susan 25 12 4
Bill 27 14 5
Bob 32 16 2
Tracy 44 18 8
Joan 26 12 10

Model Fit Statistics

Cases(n) R R Square Adj Rsq SE Est.
--------------------------------------------------------------------------
5 0.974 0.948 0.896 2.538

ANOVA Statistics
Sum of Sqs df Mean Sq F p <
--------------------------------------------------------------------------
Regression 233.915 2 116.957 18.153 0.0522
Residual 12.885 2 6.443
Total 246.800

Coefficients
Variable b Coeff. std. error BetaWgt t p <
--------------------------------------------------------------------------
Constant -14.981 7.739 -1.936 0.1925
Education 2.900 0.490 0.962 5.925 0.0273
Experience 0.692 0.400 0.281 1.732 0.2255

Equation after regression analysis:

133
Interpreting the Parameter Estimates

Intercept (a0): The average value of income when all independent variables are
equal to "0" (-14.981 or -$14,981).

Education (b1): When we hold experience constant, a one unit change in education
results in an average change in income of 2.9k ($2,900).

Experience (b2): When we hold education constant, a one unit change in experience
results in an average change in income of .692k ($692).

Confidence Intervals for Slope Coefficients

This uses the same general procedure as bivariate (simple) regression model.

where tcv = critical value from the t distribution

n = sample size
k = number of independent variables

df = 2

alpha .05, 2-tailed with 2 degrees of freedom

tcv, = 4.303

134
The equation for the 95% confidence interval for the education slope is the following:

This is interpreted as there is a probability of .95 that the slope of education in the
population is between .800 and 5 ($800 and $5,000).

Significance Tests (2-tailed)

The significance test for the two slope coefficients uses the same degrees of free-
dom, sampling distribution, and critical value (tcv=4.303) as applied to the confidence
interval calculation.

Education: (significant; tb1 > tcv of 4.303)

Experience: (not significant; tb2 < tcv of 4.303)

Predicting Y

Based on the following equation . . .

and data for Jim who has 13 years of education and 10 years experience.

135
Jim's predicted average income is $29,639 (this is the predicted point estimate for
Jim).

Confidence Interval for Predicted Y

Again using the same degrees of freedom, sampling distribution, and critical value.

There is a probability of .95 that someone in the population with 13 years of educa-
tion and 10 years experience would earn between $18,720 and $40,560. Obviously
this model would not be very useful for predicting incomes in the population. This is
not surprising given the very small sample size (n=5).

Adjusted R2

Adjusted R2 is used to compensate for the addition of variables to the model. As

more independent variables are added to the regression model, unadjusted R2 will
generally increase (there will never be a decrease). This will occur even when the ad-
ditional variables do little to help explain the dependent variable. To compensate for
this, adjusted R2 is corrected for the number of independent variables in the model.
The result is an adjusted R2 than can go up or down depending on whether the addi-
tion of another variable adds or does not add to the explanatory power of the model.
Adjusted R2 will always be lower than unadjusted.

136
It has become standard practice to report the adjusted R2, especially when there are
multiple models presented with varying numbers of independent variables.

Adjusted R2 formula

2 ⎛ 2 k ⎞⎛ n − 1 ⎞
R = ⎜ R − ⎟⎜ ⎟
⎝ n − 1 ⎠⎝ n − k − 1 ⎠

2 ⎛ 2 ⎞⎛ 5 − 1 ⎞
R = ⎜ .948 − ⎟⎜ ⎟
⎝ 5 − 1 ⎠⎝ 5 − 2 − 1 ⎠

2
R = .896

Standardized Partial Slope Coefficients (beta weights)

Partial slope coefficients are not directly comparable to one another. This is because
each variable is likely based on a different metric (age, income, sex). Beta weights
(standardized slope coefficients) are used to compare the relative importance of re-
gression coefficients in a model. They are not useful for comparisons among models
from other samples.

⎛ s xi ⎞
β i = bi ⎜ ⎟
⎜ s ⎟
Formula: ⎝ y ⎠ or partial slope * (std dev. of Xi ÷ std dev. of Y)

Example:

Education partial slope coefficient (bi) = 1878.2

137
 sxi
Education standard deviation = 2.88

sy
Begin salary standard deviation = 7870.64

⎛ s xi ⎞ ⎛ 2.88 ⎞
β i = bi ⎜ ⎟ β i = 1878.2⎜ ⎟
⎜ s ⎟ 7870 .64 β i = .687
⎝ y ⎠ ⎝ ⎠

This is interpreted as the average standard deviation change in Y (salary) associated

with one standard deviation change in X (education), when all other independent vari-
ables are held constant. Or a one standard deviation change in education results in
an average change of .687 standard deviations in income.

F-statistic

The F-statistic represents a hypothesis test to determine if the independent variables

taken as a whole help explain the dependent variable. This is a test of the model, not
the partial regression coefficients.

I. Assumptions

See regression assumptions

II. State the Hypothesis

Ho: The regression coefficients taken together simultaneously are equal to zero.

Ha: The regression coefficients taken together simultaneously are equal to zero.

138
 H 0 ≠ β1 ≠ β2 ≠ βk ≠ 0

III. Set the Rejection Criteria

where

degrees of freedom for the numerator = k

degrees of freedom for the denominator = n-k-1

At alpha .05, DF = (2, 471)

F distribution critical value = 3.02

IV. Compute Test Statistic

Formula (A): This shows that the F-ratio is related to the explanatory power of
the entire regression equation

" %" n − k −1 %
'$ ( )
2
R
F =$ '
$ (1− R 2 ) '# k &
# &

Formula (B): This formula is more relevant for common regression output.

2
⎛ ^
∑ ⎜⎝ Yi − Y ⎞⎟⎠ / k
F= 2
^
⎛ ⎞
∑ ⎝ i ⎟⎠ /(n − k − 1)
⎜ Y − Y i

The mean sum of squares for what the model explains divided by the mean sum
of squares for what the model does not explain.

139
 Example:

Calculate F-test statistic -- Formula (A) and Formula (B)

" %" n − k −1 %
'$ ( )
2
R ⎛ .446 ⎞⎛ 471 ⎞
F =$ ' F = ⎜⎜ ⎟⎟⎜ ⎟
$ (1− R 2 ) '# k
# & &
⎝ (1 − .446 ) ⎠ ⎝ 2 ⎠ F = 189.59

or

2
⎛ ^
∑ ⎜⎝ Yi − Y ⎞⎟⎠ / k
F= 2
^
∑ ⎜⎝ Yi − Y i ⎞⎟⎠ /(n − k − 1)
⎛

V. Decision

Reject null hypothesis. The model of independent variables is significantly re-

lated to the dependent variable.

Note: It is possible to have independent variables that individually are not statisti-
cally significant but as a group do have a significant relationship with the depend-
ent variable. Lack of significance for each independent variable may be due to
high correlation with other independent variables in the model (multicolinearity).

Dummy Variables

It is possible to include nominal and ordinal level independent variables in a regres-

sion model by dichotomous coding of variables into integers 0 and 1. These are com-
monly called dummy variables. The interpretation of a dummy variable is similar to
other independent variables except as follows:

140
 The partial slope coefficient represents the on average change in
the dependent variable between the coded value of 1 for the inde-
pendent variable and the reference group of 0.

Creating Dummy Variables

Always use k-1 dummy variables in a regression model, where k is equal to the num-
ber of categories in the variable. In the following example, a dummy variable called
“Male” is created where 1=Male and 0=Female, the reference group. There is no
need to create a dummy variable for Female (k=2; or, K-1 = 1 dummy variable).

Example A: Creating a dummy variable from a two-category variable

Original Variable

Male Dummy: recoded into 0/1 variable (0=Female and 1=Male)

141
Example B: Creating a dummy variable from a three-category variable

For a three-category variable, two dummy variables are created to represent three
categories.

Original Variable

Clerical Dummy: recoded into 0/1 variable (0=Custodial and Manager; 1=Clerical)

Custodian Dummy: recoded into 0/1 variable (0=Clerical and Manager; 1=Custodian)

Managers are the reference group, represented in the model when the clerical and
custodian dummy variables are both equal to zero in the data.

142
Software Output: A multiple regression example regressing individual income on
education in years, work experience in months, and a dummy variable representing
sex (0=Female, 1=Male).

--------------------- Multiple OLS Regression -------------------

Explanatory Model
INCOME = Constant + EDUC + WORKEXP + SEX + e

Model Fit Statistics

Cases(n) R R Square Adj Rsq SE Est.
-----------------------------------------------------------------
32 0.865 0.748 0.721 [a] 2246.736 [b]

ANOVA Statistics
Sum of Sqs df Mean Sq F p <
-----------------------------------------------------------------
Regression 419605192.680 3 139868397.560 27.709 0.0000 [c]
Residual 141339012.195 28 5047821.864
Total 560944204.875 31

Coefficients
Variable b Coeff. std. error BetaWgt t p <
-----------------------------------------------------------------
Constant -338.10507 2027.91657 -0.167 0.8688
EDUC 870.14643 [d] 108.50630 0.89240 [e] 8.019 0.0000 [f]
WORKEXP 193.70166 [g] 82.88935 0.26208 2.337 0.0268
SEX 2821.21837 [h] 803.62244 0.33626 3.511 0.0015

Estimated Model
INCOME = -338.105 + 870.146(EDUC) + 193.702(WORKEXP)
+ 2821.218(SEX)

Interpretation

[a] Adjusted R2. Education, experience, and a person’s sex explain 72% of the variation in indi-
vidual income.

143
[b] Standard error of the estimate. Used for creating confidence intervals for predicted Y.

[c] The regression model is statistically significant.

[d] The impact of a one-unit change in education on income. One additional year of education
will on average result in an increase in income of $870 holding experience and sex constant.

[e] Standardized regression partial slope coefficients. Used only in multiple regression models
(more than one independent variable), beta-weights in the same regression model can be com-
pared to one another to evaluate relative effect on the dependent variable. Beta-weights will indi-
cate the direction of the relationship (positive or negative). Regardless of whether negative or
positive, larger beta-weight absolute values indicate a stronger relationship. In this model, edu-
cation has the greatest effect on income followed in order by sex and experience.

[h] Education has a statistically significant association with income. The probability of a type
one error is less than .0001 which far exceeds alpha of p = .05. Also, experience and sex are
statistically significant.

[i] One additional month of experience will on average result in an increase in income of $194
holding education and sex constant.

[j] Males will on average earn an income of $2,821 more than women holding education and
experience constant.

144
11
Regression Assumptions

Although ordinary least squares multiple regression is a very robust statistical tech-
nique, careful consideration should be given to the assumptions underlying obtaining
the Best Linear Unbiased Estimates (BLUE) from a regression model. In practice, it is
very difficult to completely satisfy all the assumptions but there are techniques to
help adjust the affects on the model. A few suggested corrections are provided in
the following brief review.

1. No measurement error

The independent (X) and dependent (Y) variables are accurately measured. Non-
random error (e.g., measuring the wrong income for a defined group of people) will se-
riously bias the model. Random error is the more common assumption violation. Ran-
dom error in an independent variable may lower R2 and the partial slope coefficients
can vary dramatically depending on the amount of error. Other independent variables
that do not have random measurement error will be biased if they are correlated with
another independent variable with measurement error. Error in measuring the depend-
ent variable many not bias the estimates if the error is random.

2. No specification error

The theoretical model is assumed to be a) linear, b) additive, and c) includes the cor-
rect variables.

Non-Linearity

145
Linear implies that the average change in the dependent variable associated with a
one unit change in an independent variable is constant regardless of the level of the
independent variable. If the partial slope for X is not constant for differing values of X,
X has a nonlinear relationship with Y and results in biased partial slopes. There are
two basic types of nonlinearity discussed in the following.

Not Linear but slope does not change direction.

In this example the variable’s slope is positive but the steepness of the curve in-
creases as the value of X increases.

Correction: A log-log model takes a nonlinear specification where the slope changes
as the value of X increases and makes it linear in terms of interpreting the parameter
estimates. It accomplishes this by transforming the dependent and all independent
variables by taking their log and replacing the original variables with the logged vari-
ables. The result is coefficients that are interpreted as a % change in Y given a 1%
change in X.

Example:

Note: All data must have positive values. A log of a negative value will equal 0 and
as a result biases the model. After modeling, the anti log of the coefficients can be
used to estimate y.

146
Not linear and the slope changes direction (pos or neg)

In this example the variable’s slope changes direction depending on the value of X.

Correction: A polynomial model may be used to correct for changes in the slope coef-
ficient sign (positive or negative): This is accomplished by adding additional variables
that are incremental powers of the independent variable to model bends in the slope.

Example:

Non-Additivity

Additive implies that the average change in the dependent variable associated with a
one unit change in an independent variable (X1) is constant regardless of the value of
another independent variable (X2) in the model. If this assumption is violated, we can
no longer interpret the slope by saying "holding other variables constant" since the
values of the other variables may possibly change the slope coefficient and therefore
its interpretation.

147
The figure above displays a non-additive relationship when (X1) is interval/ratio and
(X2) is a dummy variable. If the partial slope for (X1) is not constant for differing values
of (X2), (X1) and (X2) do not have an additive relationship with Y.

Correction: An interaction term may be added to the model using a dummy variable
where the slope of X1 is thought to depend on the value of a dummy variable X2. The
final model will look like the following:

Model:

where X1X2=the interaction between X1 and X2 or X1*X2

Interpretation:

b1 is interpreted as the slope for X1 when the dummy variable (X2) is 0

b1+ b2 is interpreted as the slope for X1 when the dummy variable (X2) is 1

Correction for two interval independent variables:

148
Model without interaction term

Model with interaction term

where b3 = X1*X2 (the interactive term)

Incorrect Independent Variables

Including the correct independent variables implies that an irrelevant variable has not
been included in the model and all theoretically important relevant variables are in-
cluded. Failing to include the correct variables in the model will bias the slope coeffi-
cients and may increase the likelihood of improperly finding statistical significance.
Including irrelevant variables will make it more difficult to find statistical significance.

Correction: Remove irrelevant variables and, if possible, include missing relevant vari-
ables.

3. Mean of errors equals zero

When the mean error (reflected in residuals) is not equal to zero, the y intercept may
be biased. Violation of this assumption will not affect the slope coefficients. The par-
tial slope coefficients will remain Best Unbiased Linear Estimates (BLUE).

4. Error term is normally distributed

The distribution of the error term closely reflects the distribution of the dependent vari-
able. If the dependent variable is not normally distributed the error term may not be
normally distributed. Violation of this assumption will not bias the partial slope coeffi-
cients but may affect significance tests.

149
Correction: Always correct other problems first and then re-evaluate the residuals

★ If the distribution of residuals is skewed to the right (higher values), try using the
natural log of the dependent variable

★ If the distribution of residuals is skewed to the left (lower values), try squaring the
dependent variable.

Always check the distribution of the residuals again after trying a correction.

5. Homoskedasticity

The variance of the error term is constant (homoskedastic) for all values of the inde-
pendent variables. Heteroskedasticity occurs when the variance of the error term
does not have constant variance. The parameter estimates for partial slopes and the
intercept are not biased if this assumption is violated; however, the standard errors
are biased and hence significance tests may not be valid.

Diagnosis Of Heteroskedasticity

150
Plot the regression residuals against the values of the independent variable(s). If there
appears an even pattern about a horizontal axis, heteroskedasticity is unlikely.

For small samples there may be some tapering at each end of the horizontal distribu-
tion.

151
If there is a cone or bow tie shaped pattern, heteroskedasticity is suspected.

Correction: If an excluded independent variable is suspected, including this variable

in the model may correct the problem. Otherwise, it may be necessary to use general-
ized least squares (GLS) or weighted least squares (WLS) models to create coeffi-
cients that are BLUE.

6. No autocorrelation

No autocorrelation assumes that the error terms are not correlated across observa-
tions. Violation of this assumption is likely to be a problem with time-series data
where the value of one observation is not completely independent of another observa-
tion. (Example: A simple two-year time series of the same individuals is likely to find
that a person's income in year 2 is correlated with their income in the prior year.) If
there is autocorrelation, the parameter estimates for the partial slopes and the inter-
cept are not biased but the standard errors are biased and hence significance tests
may not be valid.

Diagnosis

Suspect autocorrelation with any time series/longitudinal data

152
Use the Durbin-Watson (d) statistic

★ d=2 > no correlation between error terms

★ d=0 > perfect positive correlation between error terms

★ d=4 > perfect negative correlation between error terms

Correction: Use generalized least squares (GLS) or weighted least squares (WLS)
models to create coefficients that are BLUE.

7. No Multicolinearity

The assumption of no multicolinearity is an issue only for multiple regression models.

Multicolinearity occurs when one of the independent variables has a substantial linear
relationship with another independent variable in the equation. It occurs to some ex-
tent in any model and is more a matter of degree of colinearity rather than whether it
exists or not. Multicolinearity will result in variability in the partial slope coefficients
from one sample to the next or when models are changed slightly. In addition, stan-
dard errors are increased which reduces the likelihood of finding statistical signifi-
cance. The result of the above two situations is an unbiased estimator that is very in-
efficient.

Diagnosis

★ Failing to find any variables statistically significant yet the F-statistic shows the
model is significant.

★ Dramatic changes in coefficients as independent variables are added or deleted

from the model.

153
★ Examine covariation among the independent variables by calculating all possible
bivariate combinations of the Pearson correlation coefficient. Generally a high cor-
relation coefficient (say .80 or greater) suggests a problem. This is imperfect since
multicolinearity may not be reflected in a bivariate correlation matrix.

★ Regress each independent variable on the other independent variables. If any of

the R2s are near 1.0 (this indicates one independent variable is almost entirely ex-
plained by the other independent variables in the model), there is a high degree of
multicolinearity.

Correction:

★ Increase sample size to lower standard errors. This doesn't always work and is
normally not feasible since adding more cases is not a simple exercise in most
studies.

★ Combine two or more variables that are highly correlated into a single indicator of
an abstract concept.

★ Delete one of the variables that are highly correlated. This may result in a poorly
specified model.

★ Leave all the variables in model and rely on the joint hypothesis F-test to evaluate
the significance of the model. This is especially useful if multicolinearity is causing
most if not all of the independent variables to be not significant.

Summary

As noted at the beginning of this section, it is very difficult to completely satisfy all of
the assumptions required in regression modeling. The best starting point in design-
ing an adequate model is theory. Theory should drive what variables are included in

154
the regression model and it will also help inform the researcher on the possible inter-
actions among the independent variables. A model that does not perform as antici-
pated by theory suggests either the theory was wrong or that the regression esti-
mates and significance tests are being biased by one or more assumption violations.

The following page has a brief summary of the assumption violations, affect on the
model, and possible corrective action.

155
BLUE Check List: Diagnosing and correcting violations of regression assumptions

Assumption Violation Consequences Diagnosis Treatment

B Se R2

Random error in DV + - data errors; unreliable measure clean errors find better
measure

Random error in IV B - data errors; unreliable measure clean errors find better
measure

Nonlinear model B review theoretical relationships; transform variable(s)

(slope not constant for all break X1 into 2+ dummy (transformation will depend
X1 values) variables and evaluate on type of nonlinear
consistency of slopes relationship)

Nonadditive model B review theoretical relationships; create interaction term

break X2 into 2+ dummy (X1*X2)
(slope for X1 not variables and evaluate
constant for values of consistency of slopes
X2)

Irrelevant IV in model + coefficient is unlikely to be remove

statistically significant; removal
won't lower Adj R2

Relevant IV excluded B - - theory; adding should increase add to model

Adj R2

Residuals not normally B plot histogram of residuals positive skew > log DV
distributed negative skew > sqr DV

Homoskedasticity B plot residuals against each IV try including additional IV

(variance not constant ( funnel, cone, or bow tie shape use GLS or WLS models
for all values of Xi) indicative of heteroskedasticity)

Autocorrelation B always suspect when using use GLS or WLS models

(error terms correlated time series data; check Durbin-
across observations) Watson statistic

Multicolinearity Erratic + no IV significant; dramatic increase sample size;

coefficient change when IV are combine highly correlated
added or deleted; check IVs into a single measure;
correlation among IV; regress delete one of the correlated
each IV on other IV; R2 near 1.0 Ivs; use F-test to evaluate
suggests multicolinearity model signif.

B=Biased; ‘+’ =Increases; ‘-’ =Decreases; b=slope coefficient; Se=Standard Error; DV=Dependent Variable;
IV=Independent Variable

156
12
Logistic Regression

Logistic regression is used for multiple regression models where the dependent vari-
able is dichotomous (0 or 1 values) or ordinal if conducting multinomial logistic regres-
sion. By convention, the dependent variable should be coded 0 and 1 where 1 repre-
sents the value of greatest interest.

Assumptions:

See multiple regression for a more complete list of ordinary least squares (OLS) re-
gression assumptions. In addition to the key characteristic that the dependent vari-
able is discrete rather than continuous, the following assumptions for logistic regres-
sion differ from those for OLS regression:

1. Specification error - Linearity

Linearity is not required in logistic regression between the independent and depend-
ent variables. It does require that the logits of the independent and dependent vari-
ables are linear. If not linear, the model may not find statistical significance when it ac-
tually exists (Type II error).

2. Large Sample Size

Unlike OLS regression, large sample size is needed for logistic regression. If there is
difficulty in converging on a solution, a large number of iterations, or very large regres-
sion coefficients, there may be insufficient sample size.

157
Software Output: Evaluating the affect of age, education, sex, and race on whether
or not a person voted in the presidential election.

------------------------ Logistic Regression -------------------

Dependent Y: VOTE
Cases with Y=0: 417 28.86%! No, did not vote
Cases with Y=1: 1028 71.14%! Yes, voted
Total Cases: 1445

Explanatory Model
VOTE92 = Constant + AGE + EDUC + FEMALE + WHITE + e

Model Fit Statistics

Initial -2 Log Likelihood 1736.533
Model -2 Log Likelihood 1544.154 Iteration (4)
Cox & Snell Rsq 0.125 [a]
McFadden Rsq 0.111 [b]
Model Chi-Square 192.380 df 4 p< 0.0000 [c]

Coefficients
Variable b Coeff. SE Wald p < OR
------------------------------------------------------------------
Constant -4.1620 0.4184 98.9755 0.0000
AGE 0.0335 [d] 0.0040 69.1386 [e] 0.0000 [f] 1.0340 [g]
EDUC 0.2809 0.0244 132.4730 0.0000 1.3243
FEMALE -0.1249 0.1269 0.9686 0.3250 0.8826
WHITE 0.1000 0.1671 0.3581 0.5495 1.1052
------------------------------------------------------------------

Estimated Model
VOTE92 = -4.1620 + .0335(AGE) + .2809(EDUC) + -.1249(FEMALE)
+ .1000(WHITE)

Classification Table (.50 cutpoint)

Predicted Y
Observed Y 0 1 % Correct
+-----------+-----------+
0 | 103 | 314 | 24.70 [h]
+-----------+-----------+
1 | 64 | 964 | 93.77 [i]
+-----------+-----------+----------
Total 73.84 [j]

158
Interpretation

[a] This ranges from 0 to less than 1 and indicates the explanatory value of the model. The
closer the value is to 1 the greater the explanatory value. Analogous to R2 in OLS regression.

[b] Indicates explanatory value of the model. Ranges from 0 to 1 and adjusts for the number
of independent variables. Analogous to Adjusted R2 in OLS regression. In this model, the ex-
planatory value is relatively low.

[c] Tests the combined significance of the model in explaining the dependent (response) vari-
able. The significance test is based on the Chi-Square sampling distribution. Analogous to the
F-test in OLS regression. The independent variables in this model have a statistically significant
combined effect on voting behavior.

[d] A one unit increase in age (1 year) results in an average increase of .0335 in the log odds of
vote equaling 1 (voted in election).

[e] The test statistic for the null hypothesis that the b coefficient is equal to 0 (no effect). It is
calculated by (B/SE)2.

[f] Based on the Chi-Square sampling distribution, there is a statistically significant relation-
ship between age and voting behavior. The probability voting increases with age, holding educa-
tion, sex, and race constant.

[g] OR=Odds Ratio. The odds of a change in the dependent variable (vote) given a one unit
change in age is 1.03. Note: if the OR is >1, odds increase; if OR <1, odds decrease; if OR =1,
odds are unchanged by this variable. In this example, the odds of voting increases with a one
unit increase in age.

[h] Of those who did not vote, the model correctly predicted non-voting in 24.7% of the non-
voters included in the sample data.

[i] Of those who did vote, the model correctly predicted voting in 93.8% of the voters in-
cluded in the sample data.

[j] Of those who did or did not vote, the model correctly predicted 73.8% of the voting deci-
sions.

159
13
Tables

Critical Values for Z Distributions

Critical Values for T Distributions

Critical Values for Chi-square Distributions

Critical Values for F Distributions

160
Z Distribution Critical Values

Z Area Beyond Z Area Beyond Z Area Beyond

0.150 0.440 1.300 0.097 2.450 0.007
0.200 0.421 1.350 0.089 2.500 0.006
0.250 0.401 1.400 0.081 2.550 0.005
0.300 0.382 1.450 0.074 2.580 0.005
0.350 0.363 1.500 0.067 2.650 0.004
0.400 0.345 1.550 0.061 2.700 0.004
0.450 0.326 1.600 0.055 2.750 0.003
0.500 0.309 1.650 0.050 2.800 0.003
0.550 0.291 1.700 0.045 2.850 0.002
0.600 0.274 1.750 0.040 2.900 0.002
0.650 0.258 1.800 0.036 2.950 0.002
0.700 0.242 1.850 0.032 3.000 0.001
0.750 0.227 1.900 0.029 3.050 0.001
0.800 0.212 1.960 0.025 3.100 0.001
0.850 0.198 2.000 0.023 3.150 0.001
0.900 0.184 2.050 0.020 3.200 0.001
0.950 0.171 2.100 0.018 3.250 0.001
1.000 0.159 2.150 0.016 3.300 0.001
1.050 0.147 2.200 0.014 3.350 0.000
1.100 0.136 2.250 0.012 3.400 0.000
1.150 0.125 2.300 0.011 3.450 0.000
1.200 0.115 2.350 0.009 3.500 0.000

This is only a portion of a much larger table.

The area beyond Z is the proportion of the distribution beyond the critical value (region of rejec-
tion). Example: If a test at .05 alpha is conducted, the area beyond Z for a two-tailed test is .025
(Z-value 1.96). For a one-tailed test the area beyond Z would be .05 (Z-value 1.65).

161
T Distribution Critical Values (2-Tailed)

Alpha Alpha Alpha

DF 0.10 0.05 0.01 DF 0.10 0.05 0.01 DF 0.10 0.05 0.01
1 6.314 12.706 63.657 41 1.683 2.020 2.701 81 1.664 1.990 2.638
2 2.920 4.303 9.925 42 1.682 2.018 2.698 82 1.664 1.989 2.637
3 2.353 3.182 5.841 43 1.681 2.017 2.695 83 1.663 1.989 2.636
4 2.132 2.776 4.604 44 1.680 2.015 2.692 84 1.663 1.989 2.636
5 2.015 2.571 4.032 45 1.679 2.014 2.690 85 1.663 1.988 2.635
6 1.943 2.447 3.707 46 1.679 2.013 2.687 86 1.663 1.988 2.634
7 1.895 2.365 3.499 47 1.678 2.012 2.685 87 1.663 1.988 2.634
8 1.860 2.306 3.355 48 1.677 2.011 2.682 88 1.662 1.987 2.633
9 1.833 2.262 3.250 49 1.677 2.010 2.680 89 1.662 1.987 2.632
10 1.812 2.228 3.169 50 1.676 2.009 2.678 90 1.662 1.987 2.632
11 1.796 2.201 3.106 51 1.675 2.008 2.676 91 1.662 1.986 2.631
12 1.782 2.179 3.055 52 1.675 2.007 2.674 92 1.662 1.986 2.630
13 1.771 2.160 3.012 53 1.674 2.006 2.672 93 1.661 1.986 2.630
14 1.761 2.145 2.977 54 1.674 2.005 2.670 94 1.661 1.986 2.629
15 1.753 2.131 2.947 55 1.673 2.004 2.668 95 1.661 1.985 2.629
16 1.746 2.120 2.921 56 1.673 2.003 2.667 96 1.661 1.985 2.628
17 1.740 2.110 2.898 57 1.672 2.002 2.665 97 1.661 1.985 2.627
18 1.734 2.101 2.878 58 1.672 2.002 2.663 98 1.661 1.984 2.627
19 1.729 2.093 2.861 59 1.671 2.001 2.662 99 1.660 1.984 2.626
20 1.725 2.086 2.845 60 1.671 2.000 2.660 100 1.660 1.984 2.626
21 1.721 2.080 2.831 61 1.670 2.000 2.659 101 1.660 1.984 2.625
22 1.717 2.074 2.819 62 1.670 1.999 2.657 102 1.660 1.983 2.625
23 1.714 2.069 2.807 63 1.669 1.998 2.656 103 1.660 1.983 2.624
24 1.711 2.064 2.797 64 1.669 1.998 2.655 104 1.660 1.983 2.624
25 1.708 2.060 2.787 65 1.669 1.997 2.654 105 1.659 1.983 2.623
26 1.706 2.056 2.779 66 1.668 1.997 2.652 106 1.659 1.983 2.623
27 1.703 2.052 2.771 67 1.668 1.996 2.651 107 1.659 1.982 2.623
28 1.701 2.048 2.763 68 1.668 1.995 2.650 108 1.659 1.982 2.622
29 1.699 2.045 2.756 69 1.667 1.995 2.649 109 1.659 1.982 2.622
30 1.697 2.042 2.750 70 1.667 1.994 2.648 110 1.659 1.982 2.621
31 1.696 2.040 2.744 71 1.667 1.994 2.647 111 1.659 1.982 2.621
32 1.694 2.037 2.738 72 1.666 1.993 2.646 112 1.659 1.981 2.620
33 1.692 2.035 2.733 73 1.666 1.993 2.645 113 1.658 1.981 2.620
34 1.691 2.032 2.728 74 1.666 1.993 2.644 114 1.658 1.981 2.620
35 1.690 2.030 2.724 75 1.665 1.992 2.643 115 1.658 1.981 2.619
36 1.688 2.028 2.719 76 1.665 1.992 2.642 116 1.658 1.981 2.619
37 1.687 2.026 2.715 77 1.665 1.991 2.641 117 1.658 1.980 2.619
38 1.686 2.024 2.712 78 1.665 1.991 2.640 118 1.658 1.980 2.618
39 1.685 2.023 2.708 79 1.664 1.990 2.640 119 1.658 1.980 2.618
40 1.684 2.021 2.704 80 1.664 1.990 2.639 120 1.658 1.980 2.617
∞ 1.645 1.960 2.576
This is only a portion of a much larger table.

162
Chi-square Distribution Critical Values

Alpha
df .10 .05 .02 .01 .001
1 2.706 3.841 5.412 6.635 10.827
2 4.605 5.991 7.824 9.210 13.815
3 6.251 7.815 9.837 11.345 16.266
4 7.779 9.488 11.688 13.277 18.467
5 9.236 11.070 13.388 15.086 20.515
6 10.645 12.592 15.033 16.812 22.457
7 12.017 14.067 16.622 18.475 24.322
8 13.362 15.507 18.168 20.090 26.125
9 14.684 16.919 19.679 21.666 27.877
10 15.987 18.307 21.161 23.209 29.588
11 17.275 19.675 22.618 24.725 31.264
12 18.549 21.026 24.054 26.217 32.909
13 19.812 22.362 25.472 27.688 34.528
14 21.064 23.685 26.873 29.141 36.123
15 22.307 24.996 28.259 30.578 37.697
16 23.542 26.296 29.633 32.000 39.252
17 24.769 27.587 30.995 33.409 40.790
18 25.989 28.869 32.346 34.805 42.312
19 27.204 30.144 33.687 36.191 43.820
20 28.412 31.410 35.020 37.566 45.315
21 29.615 32.671 36.343 38.932 46.797
22 30.813 33.924 37.656 40.289 48.268
23 32.007 35.172 38.968 41.638 49.728
24 33.196 36.415 40.270 42.980 51.179
25 34.382 37.652 41.566 44.314 52.620
26 35.563 38.885 42.856 45.642 54.052
27 36.741 40.113 44.140 46.963 55.476
28 37.916 41.337 45.419 43.278 56.893
29 39.087 42.557 46.693 49.588 58.302
30 40.256 43.773 47.962 50.892 59.703

This is only a portion of a much larger table.

163
F Distribution Critical Values

Alpha .05
Denominator DF
 Numerator DF
DF 1 2 3 4 5 10 30 60 120 500
5 6.608 5.786 5.409 5.192 5.050 4.735 4.496 4.431 4.398 4.373
7 5.591 4.737 4.347 4.120 3.972 3.637 3.376 3.304 3.267 3.239
10 4.965 4.103 3.708 3.478 3.326 2.978 2.700 2.621 2.580 2.548
13 4.667 3.806 3.411 3.179 3.025 2.671 2.380 2.297 2.252 2.218
15 4.543 3.682 3.287 3.056 2.901 2.544 2.247 2.160 2.114 2.078
20 4.351 3.493 3.098 2.866 2.711 2.348 2.039 1.946 1.896 1.856
25 4.242 3.385 2.991 2.759 2.603 2.236 1.919 1.822 1.768 1.725
30 4.171 3.316 2.922 2.690 2.534 2.165 1.841 1.740 1.683 1.637
35 4.121 3.267 2.874 2.641 2.485 2.114 1.786 1.681 1.623 1.574
40 4.085 3.232 2.839 2.606 2.449 2.077 1.744 1.637 1.577 1.526
45 4.057 3.204 2.812 2.579 2.422 2.049 1.713 1.603 1.541 1.488
50 4.034 3.183 2.790 2.557 2.400 2.026 1.687 1.576 1.511 1.457
55 4.016 3.165 2.773 2.540 2.383 2.008 1.666 1.553 1.487 1.431
60 4.001 3.150 2.758 2.525 2.368 1.993 1.649 1.534 1.467 1.409
120 3.920 3.072 2.680 2.447 2.290 1.910 1.554 1.429 1.352 1.280
500 3.860 3.014 2.623 2.390 2.232 1.850 1.482 1.345 1.255 1.159
This is only a portion of a much larger table.

Alpha .01
Denominator DF
 Numerator DF
DF 1 2 3 4 5 10 30 60 120 500
5 16.258 13.274 12.060 11.392 10.967 10.051 9.379 9.202 9.112 9.042
7 12.246 9.547 8.451 7.847 7.460 6.620 5.992 5.824 5.737 5.671
10 10.044 7.559 6.552 5.994 5.636 4.849 4.247 4.082 3.996 3.930
13 9.074 6.701 5.739 5.205 4.862 4.100 3.507 3.341 3.255 3.187
15 8.683 6.359 5.417 4.893 4.556 3.805 3.214 3.047 2.959 2.891
20 8.096 5.849 4.938 4.431 4.103 3.368 2.778 2.608 2.517 2.445
25 7.770 5.568 4.675 4.177 3.855 3.129 2.538 2.364 2.270 2.194
30 7.562 5.390 4.510 4.018 3.699 2.979 2.386 2.208 2.111 2.032
35 7.419 5.268 4.396 3.908 3.592 2.876 2.281 2.099 2.000 1.918
40 7.314 5.179 4.313 3.828 3.514 2.801 2.203 2.019 1.917 1.833
45 7.234 5.110 4.249 3.767 3.454 2.743 2.144 1.958 1.853 1.767
50 7.171 5.057 4.199 3.720 3.408 2.698 2.098 1.909 1.803 1.713
55 7.119 5.013 4.159 3.681 3.370 2.662 2.060 1.869 1.761 1.670
60 7.077 4.977 4.126 3.649 3.339 2.632 2.028 1.836 1.726 1.633
120 6.851 4.787 3.949 3.480 3.174 2.472 1.860 1.656 1.533 1.421
500 6.686 4.648 3.821 3.357 3.054 2.356 1.735 1.517 1.377 1.232
This is only a portion of a much larger table.

164
14
Appendix

Basic Formulas

Glossary of Abbreviated Terms

Order of Mathematical Operations

165
Basic Formulas

Population Mean

Sample Mean

Population Variance

Sample Variance

Population Standard Deviation

Sample Standard Deviation

166
Z Score

s
sx =
Standard Error--Mean n

Standard Error--Proportion

167
Glossary of Abbreviated Terms

Mean of a sample

Y Any dependent variable

F The F ratio

Fe Expected frequency

Fo Observed frequency

Ho Null Hypothesis

Ha Alternate Hypothesis

N Number of cases in the population

n Number of cases in the sample

P Proportion

p < Probability of a Type I error

r Pearson's correlation coefficient

r2 Coefficient of determination

s Sample standard deviation

s2 Sample variance

X Any independent variable

Xi Any individual score

168
 Alpha-- probability of Type I error

µ Mean (mu) of a population

Population standard deviation

Population variance

"Summation of"

Chi-square statistic

Infinity

169
Order of Mathematical Operations

First

All operations in parentheses, brackets, or braces, are calculated from the inside out

Second

All operations with exponents

Third

Multiply and divide before adding and subtracting

170
A priori

Decisions that are made before the results of an inquiry are known. In hy-
pothesis testing, testable hypotheses and decisions regarding alpha and
whether a one-tailed or two-tailed test will be used should be established be-
fore collecting or analyzing the data.

Related Glossary Terms

Drag related terms here

Index Find Term

Abstract concept

The starting point for measurement. Abstract concepts are best understood
as general ideas in linguistic form that help us describe reality. They range
from the simple (hot, heavy, fast) to the more difficult (responsive, effective,
fair). Abstract concepts should be evident in the research question and/or
purpose statement.

An example of a research question is given below. In this example, the ab-

stract concepts are sector of employment (e.g., private, non-profit, public)
and employee quality.

Research Question: Is the quality of public sector and private sector employ-
ees different?

Related Glossary Terms

Research question

Index Find Term

Addition rule of probability

The probability of outcome A or outcome B occurring is equal to the sum of

their respective probabilities minus the probability of their joint occurrence.
In the following example, a joint occurrence is not possible.

Example

Out of a deck of 52 playing cards, what is the probability of picking an Ace

or a Jack? Assume a well shuffled deck and mutually exclusive event possi-
bilities (i.e., you cannot pull a Jack and an Ace at the same time).

Addition Rule: When two events, A and B, are mutually exclusive, the prob-
ability that A or B will occur is the sum of the probability of each event.

P(A or B) = P(A) + P(B)

or

P(Ace or Jack) = 4/52 + 4/52 = 8/52

There is about a 15% chance of pulling an Ace or Jack with one draw from
a deck of playing cards.

Related Glossary Terms

Multiplication rule of probability

Index Find Term

Alpha

The probability of a Type I error. Alpha represents the threshold for claiming
statistical significance.

When conducting statistical tests with computer software, the exact prob-
ability of a Type I error is calculated. It is presented in several formats but is
most commonly reported as "p <" or "Sig." or "Signif." or "Significance."
Using "p <" as an example, if a priori you established a threshold for statisti-
cal significance at alpha .05, any test statistic with significance at or less
than .05 would be considered statistically significant and you would be re-
quired to reject the null hypothesis of no difference.

The following table links p values with a constant alpha benchmark of .05.

P< Alpha Probability of Making a Type I Error Significant?

0.05 0.05 5% chance not statistically significant Yes

0.10 0.05 10% chance not statistically significant No

0.01 0.05 1% chance not statistically significant Yes

0.96 0.05 96% chance not statistically significant No

Related Glossary Terms

P-value, Type I error, Type II error

Index Find Term

Alternative hypothesis

This is a statement that asserts there is a difference between two population

parameters. It has the opposite assertion of a null hypothesis. An alterna-
tive hypothesis is also known as the research hypothesis.

Example

Alternate Hypothesis: There is a statistically significant difference between

the historical proportion of clients reporting poor service and the current pro-
portion of clients reporting poor service.

Null Hypothesis: There is no statistically significant difference between the

historical proportion of clients reporting poor service and the current propor-
tion of clients reporting poor service.

Related Glossary Terms

Hypothesis, Null hypothesis

Index Find Term

Antecedent variable

An independent variable that comes before other independent variables.

Related Glossary Terms

Dependent variable, Independent variable, Intervening variable

Index Find Term

Applied Stat: Bivariate Analysis-Counts

Simultaneously analyzing two variables to determine a relationship between

the variables.

________________

Software Output Example: Education level by individual’s sex.

Crosstabulation: EduCat (Rows) by Sex (Columns)

Column Variable Label: Respondent sex

Row Variable Label: Education Level

Count |
Row % |
Col % |
Total % | Male | Female | Total
-------------------------------------------
<12 yrs | 3 | 4 [a]| 7
| 42.86 | 57.14 [b]|
| 11.54 | 16.67 [c]| 14.00
| 6.00 | 8.00 [d]|
-------------------------------------------
HS Grad | 10 | 11 | 21
| 47.62 | 52.38 |
| 38.46 | 45.83 | 42.00
| 20.00 | 22.00 |
-------------------------------------------
College | 13 | 9 | 22
| 59.09 | 40.91 |
| 50.00 | 37.50 | 44.00
| 26.00 | 18.00 |
-------------------------------------------
| 26 | 24 | 50
Total | 52.00 | 48.00 | 100.00

Interpretation

[a] Count: There are 4 females who have less than 12 years of education.

[b] Row %: Of those who have less than a high school education (<12 yrs), 57.14%
(4/7) are female.

[c] Column %: Of those who are female, 16.67% (4/24) have less than a high
school education.

[d] Total %: 8% (4/50) of the sample is female with less than a high school educa-
tion.

Related Glossary Terms

Applied Stat: Counts-Chi-square, Bivariate table, Contingency table, Univariate analysis

Index Find Term

Applied Stat: Bivariate Analysis-Means

Simultaneously analyzing two variables to determine a relationship between

the variables. Examples include t-tests of means and Analysis of Variance.

________________

Software Output Example: Individual annual mean income by education

level.

Explore Means Procedure

Income by EduCat

EduCat Count Mean Median Std. Dev.

---------------------------------------------------------------
<12 yrs 7 23188.4286 24064.0000 4081.0739
HS Grade 21 30731.2381 28086.0000 8521.1830
College 22 40829.7273 35739.0000 21211.0346

Related Glossary Terms

Drag related terms here

Index Find Term

Applied Stat: Correlation

Used to explain the relationship between variables. Represents the notion

that one variable will vary systematically depending on the values of another
variable.

Correlation coefficients estimate strength and direction of association be-

tween two interval/ratio level variables. Used to create a summary measure
that reflects the covariation between two interval/ratio variables, the Pear-
son Correlation Coefficient presented here can range from a -1.00 to 1.00.
A positive coefficient indicates the values of variable A vary in the same di-
rection as variable B. A negative coefficient indicates the values of variable
A and variable B vary in opposite directions.

Assumptions

Interval/ratio data from paired observations.

A linear relationship should exist between the variables -- verified by plotting

the data on a scattergram.

No extreme values in the data.

Probability Distribution

T Distribution where . . .

Degrees of Freedom

df= n-2

Formula

n∑ XY − ∑ X ∑Y
rxy =
#
(∑ X ) &(' * #%$n∑Y − (∑Y ) &('
2 2

%$n∑ X −
2 2

________________

Software Output Example: Comparing individual income in U.S. dollars

to years of education.

Pearson's Correlation

INCOME: [a]

21779.0| * *
| * * *
| * * * *
| * * *
14030.0| * *
| * * * * * *
| * *
| * *
| * *
6281.0| * *
---------------|--------------|
4.0 12.0 20.0

EDUC: [b]

Number of cases: 32
Missing: 0 [c]
Pearson Correlation: 0.751 [d]
p < (2-tailed signif.): 0.0000 [e]

Interpretation

[a] The Y axis of the scattergram. If theory suggests cause and effect, the Y axis is
commonly used for the dependent (response) variable.

[b] The X axis of the scattergram. If theory suggests cause and effect, the X axis is
commonly used for the independent variable.

[c] Since each observation (case) must have values for both income and education,
any observations where one or both of these variables have no data will be removed
from the analysis.

[d] Pearson correlation coefficient representing a high positive correlation between

education and income. Interpretation: As years of education increases so does per-
sonal income.

[e] There is a statistically significant association between income and education.

Related Glossary Terms

Applied Stat: Correlation Matrix, Association, Covariation, Covary, Zero-order correlation

Index Find Term

Applied Stat: Correlation Matrix

Used to explain the relationship among multiple variables. Represents the

notion that one variable will vary systematically depending on the values of
another variable. The example below shows Pearson Product Moment Cor-
relations in matrix form. The cells above the diagonal are a mirror reflection
of the cells below the diagonal.

________________

Software Output Example: Multiple bivariate comparisons displayed in

a correlation matrix. Comparing individual income, education, and
months of work experience.

Correlation: Pearson (R) Coefficients

-----------------------------------------------
Coeff | Correlation Matrix |
Cases |-----------------------------------|
p < | INCOME | EDUC | WORKEXP |
-----------------------------------------------
INCOME | 1.000 [a]| 0.751 [d]| -0.160 [f]|
| 32 [b]| 32 | 32 |
| . [c]| 0.000 [e]| 0.381 [g]|
-----------------------------------------------
EDUC | 0.751 | 1.000 | -0.520 |
| 32 | 32 | 32 |
| 0.000 | . | 0.002 |
-----------------------------------------------
WORKEXP | -0.160 | -0.520 | 1.000 |
| 32 | 32 | 32 |
| 0.381 | 0.002 | . |
-----------------------------------------------
2-tailed significance tests
'.' p-value not computed

Interpretation

[a] The diagonal in the matrix represents Pearson correlations between the same
variable which will always be 1 since the variables are identical. The correlations
above the diagonal are a mirror reflection of these below the diagonal, so you only in-
terpret half of the matrix (in this case three correlation coefficients).

[b] The number of paired observations for this comparison.

[c] A statistical test is not performed when you are comparing a variable to itself.
Mathematically this will always equal zero.

[d] Pearson correlation coefficient representing a high positive correlation between

education and income. Interpretation: As years of education increases so does per-
sonal income.

[e] There is a statistically significant association between income and education.

[f] Pearson correlation coefficient representing a very weak negative correlation be-
tween work experience and income. Interpretation: As years of work experience in-
creases personal income decreases.

[g] There is not a statistically significant association between work experience and
income.

Related Glossary Terms

Applied Stat: Correlation

Index Find Term

Applied Stat: Counts-Chi-square

Chi-square compares the observed and expected counts of two or more

subgroups across two or more criteria. By convention, subgroups of the in-
dependent variable are placed in the columns and the subgroups of the de-
pendent variable are represented by the rows.

Assumptions

Independent random sampling

Nominal/Ordinal level data
No more than 20% of the cells have an expected frequency less than 5
No empty cells

Probability Distribution

Chi-square Distribution where . . .

Degrees of Freedom

df=(# of rows - 1)(# of columns - 1)

Formula

Chi-square Test Statistic

where

Fo= observed frequency

Fe= expected frequency for each cell

Fe= (frequency for the column)*(frequency for the row)/n

________________

Software Output Example: Data from survey of U.S. Adults.

Crosstabulation: GUNLAW (Rows) by SEX (Columns)

Column Variable Label: Respondent's Sex

Row Variable Label: Gun permits

Count |
Row % |
Col % |
Total % | Male | Female | Total
--------------------------------------------
Favor | 314| 497| 811
| 38.72| 61.28|
| 73.88| 88.91| 82.42
| 31.91| 50.51|
--------------------------------------------
Oppose | 111| 62| 173
| 64.16| 35.84|
| 26.12| 11.09| 17.58
| 11.28| 6.30|
--------------------------------------------
| 425| 559| 984
Total | 43.19| 56.81| 100.00

Chi-square Value DF p <

-----------------------------------------------------------
Pearson 37.622 1 0.0000 [a]
Likelihood Ratio 37.417 1 0.0000
Yate's Correction 36.592 1 0.0000 [b]

Measures of Association
-------------------------------------
Cramer's V .196 [c]
Pearson C .192
Lambda Symmetric .082 [d]
Lambda Dependent=Column .000
Lambda Dependent=Row .115 [e]

Note: 00.00% of the cells have an expected frequency <5

Interpretation

[a] Statistically significant (most common measure used for significance).

[b] When sample sizes are small, the continuous chi-square value tends to be too
large. The Yates continuity correction adjusts for this bias in 2x2 contingency tables.
Regardless of sample size, it is a preferred measure for chi-square tests on 2x2 ta-
bles.

[c] Weak association (both Cramer’s V and Pearson C).

[d] A symmetric lambda is used when identification of independent and dependent

variables is not useful.

[e] Knowing a person’s sex can reduce prediction error by 11.5%.

Related Glossary Terms

Applied Stat: Bivariate Analysis-Counts, Applied Stat: Counts-Cramer's V, Applied Stat:
Counts-Goodness of fit, Applied Stat: Counts-Lambda, Applied Stat: Counts-Yates continu-
ity correction, Applied Stat: Kappa, Bivariate table, Contingency table, Fisher's Exact Test,
Gamma, Nonparametric, Pearson's C, Phi, Proportional reduction in error (PRE)

Index Find Term

Applied Stat: Counts-Cramer's V

When using the chi-square statistic, Cramer's V coefficients can be helpful

in interpreting the strength of a relationship between two variables once sta-
tistical significance has been established. Cramer's V is also useful for com-
paring multiple X2 test statistics and is comparable across contingency ta-
bles of varying sizes. It is not affected by sample size and therefore is very
useful in situations where you suspect a statistically significant chi-square
was the result of large sample size instead of any substantive relationship
between the variables. It is interpreted as a measure of the relative (strength)
of an association between two variables from 0 (no association) to 1 (very
strong association). In practice, a Cramer's V of .10 may provide a good
minimum threshold for suggesting there is a substantive relationship be-
tween two variables.

Formula

where q = smaller # of rows or columns

________________

Software Output Example: Data from survey of U.S. Adults.

Crosstabulation: GUNLAW (Rows) by SEX (Columns)

Column Variable Label: Respondent's Sex

Row Variable Label: Gun permits

Count |
Row % |
Col % |
Total % | Male | Female | Total
--------------------------------------------
Favor | 314| 497| 811
| 38.72| 61.28|
| 73.88| 88.91| 82.42
| 31.91| 50.51|
--------------------------------------------
Oppose | 111| 62| 173
| 64.16| 35.84|
| 26.12| 11.09| 17.58
| 11.28| 6.30|
--------------------------------------------
| 425| 559| 984
Total | 43.19| 56.81| 100.00

Chi-square Value DF p <

-----------------------------------------------------------
Pearson 37.622 1 0.0000
Likelihood Ratio 37.417 1 0.0000
Yate's Correction 36.592 1 0.0000

Measures of Association
-------------------------------------
Cramer's V .196 [a]
Pearson C .192
Lambda Symmetric .082
Lambda Dependent=Column .000
Lambda Dependent=Row .115

Note: 00.00% of the cells have an expected frequency <5

Interpretation

[a] Weak association (both Cramer’s V and Pearson C).

Related Glossary Terms

Applied Stat: Counts-Chi-square, Association

Index Find Term

Applied Stat: Counts-Goodness of fit

Uses chi-square to test the significance of the distribution of observations in

a set of categories in one variable.

Also used in logistic regression to estimate the overall performance of the re-
gression model.

Assumptions

Independent random sampling

Nominal or Ordinal level data

Probability Distribution

Chi-square Distribution where . . .

Degrees of Freedom

(df) = k – 1 where k equals the number of categories

Formula

where Fe = Frequency Expected

and . . .

n = sample size
k = number of categories or cells
Fo = observed frequency

________________

Software Output Example: Education Level of a small sample of U.S.

Adults.

Frequencies
Variable: EduCat Education Level

Value Count Percent Cumulative

-----------------------------------------------------------
<12 yrs 7 14.00 14.00
HS Grade 21 42.00 56.00
College 22 44.00 100.00
-----------------------------------------------------------
Total 50
Missing 0
Value Count 3

Goodness-of-Fit Value DF p <

-----------------------------------------------------------
One-Sample Chi-square 8.440 2 0.0147 [a]

Note: Frequency Expected = 16.67

Interpretation

[a] There is a statistically significant difference among the education categories.

Related Glossary Terms

Applied Stat: Counts-Chi-square

Index Find Term

Applied Stat: Counts-Lambda

Lambda is a proportional reduction in error (PRE) measure. It represents the

proportion by which you reduce the error in predicting the dependent vari-
able by incorporating values from the independent variable. The value of
lambda can range from 0 (no improvement in prediction) to 1 (knowing one
variable means you can predict with out error). A symmetric lambda is used
when identification of independent/dependent variables is not useful.

________________

Software Output Example: Data from survey of U.S. Adults.

Crosstabulation: GUNLAW (Rows) by SEX (Columns)

Column Variable Label: Respondent's Sex

Row Variable Label: Gun permits

Count |
Row % |
Col % |
Total % | Male | Female | Total
--------------------------------------------
Favor | 314| 497| 811
| 38.72| 61.28|
| 73.88| 88.91| 82.42
| 31.91| 50.51|
--------------------------------------------
Oppose | 111| 62| 173
| 64.16| 35.84|
| 26.12| 11.09| 17.58
| 11.28| 6.30|
--------------------------------------------
| 425| 559| 984
Total | 43.19| 56.81| 100.00

Chi-square Value DF p <

-----------------------------------------------------------
Pearson 37.622 1 0.0000
Likelihood Ratio 37.417 1 0.0000
Yate's Correction 36.592 1 0.0000

Measures of Association
-------------------------------------
Cramer's V .196
Pearson C .192
Lambda Symmetric .082 [a]
Lambda Dependent=Column .000
Lambda Dependent=Row .115 [b]

Note: 00.00% of the cells have an expected frequency <5

Interpretation

[a] A symmetric lambda is used when identification of independent and dependent

variables is not useful.

[b] Knowing a person’s sex can reduce prediction error by 11.5%.

Related Glossary Terms

Applied Stat: Counts-Chi-square, Association, Measurement scale

Index Find Term

Applied Stat: Counts-Yates continuity correction

When sample sizes are small, the continuous chi-square tends to be too
large. The continuity correction adjusts for this bias in 2x2 contingency ta-
bles. Regardless of sample size, it is a preferred measure for chi-square
tests on 2x2 tables. If a cell has zero observations, Fisher's exact test is
more appropriate for chi-square significance tests.

________________

Software Output Example: Data from survey of U.S. Adults.

Crosstabulation: GUNLAW (Rows) by SEX (Columns)

Column Variable Label: Respondent's Sex

Row Variable Label: Gun permits

Count |
Row % |
Col % |
Total % | Male | Female | Total
--------------------------------------------
Favor | 314| 497| 811
| 38.72| 61.28|
| 73.88| 88.91| 82.42
| 31.91| 50.51|
--------------------------------------------
Oppose | 111| 62| 173
| 64.16| 35.84|
| 26.12| 11.09| 17.58
| 11.28| 6.30|
--------------------------------------------
| 425| 559| 984
Total | 43.19| 56.81| 100.00

Chi-square Value DF p <

-----------------------------------------------------------
Pearson 37.622 1 0.0000
Likelihood Ratio 37.417 1 0.0000
Yate's Correction 36.592 1 0.0000 [a]

Note: 00.00% of the cells have an expected frequency <5

Interpretation

[a] When sample sizes are small, the continuous chi-square value tends to be too
large. The Yates continuity correction adjusts for this bias in 2x2 contingency tables.
Regardless of sample size, it is a preferred measure for chi-square tests on 2x2 ta-
bles.

Related Glossary Terms

Applied Stat: Counts-Chi-square

Index Find Term

Applied Stat: Frequencies

Used to represent the relative distribution of values in one variable. The cate-
gories can be any level of measurement and are presented as counts, per-
cents, and cumulative percents.

________________

Software Output Example: Education level.

Frequencies
Variable: Edu Years of education
Value Count Percent Cumulative
--------------------------------------------------------------
08 1 2.00 2.00
09 2 [a] 4.00 [b] 6.00 [c]
10 2 4.00 10.00
11 2 4.00 14.00
12 21 42.00 56.00
13 2 4.00 60.00
14 4 8.00 68.00
15 3 6.00 74.00
16 6 12.00 86.00
17 3 6.00 92.00
18 1 2.00 94.00
19 2 4.00 98.00
20 1 2.00 100.00
--------------------------------------------------------------
Total 50
Missing 0

Interpretation

[a] 2 out of 50 have 9 years of education

[b] 4.00% have 9 years of education (2/50=.04; .04 * 100=4.00%)

[c] 6.00% have 9 years or less education (3/50=.06; .06*100 = 6.00%)

Related Glossary Terms

Cumulative frequency, Cumulative percentage, Frequency distribution, Relative frequency

Index Find Term

Applied Stat: Kappa

Cohen's kappa is a common technique for estimating paired interrater agree-

ment for nominal and ordinal level data. Kappa is a coefficient that repre-
sents agreement obtained between two readers beyond what would be ex-
pected by chance alone . A value of 1.0 represents perfect agreement. A
value of 0.0 represents no agreement.

Assumptions

Elements being rated (images, diagnoses, clinical indications, etc.) are inde-
pendent of each other.

One rater’s classifications are made independently of the other rater’s classi-
fications.

The same two raters provide the classifications used to determine kappa.

The rating categories are independent of each other.

Formula

p − pe
kˆ = 0
1 − pe

where

p0 = Proportion of cases where there is agreement between two raters

p e = Proportion of cases where raters would agree by chance

________________

Software Output Example: Breast Imaging by Two Radiologists.

Crosstabulation: Radiologist A by Radiologist B

Column Variable Label: Radiologist A

Row Variable Label: Radiologist B

Count |
Row % |
Col % |
Total % |Not Dise |Diseased | Total
--------------------------------------------
Not Diseased | 41| 11| 52
| 78.85| 21.15|
| 82.00| 22.00| 52.00
| 41.00| 11.00|
--------------------------------------------
Diseased | 9| 39| 48
| 18.75| 81.25|
| 18.00| 78.00| 48.00
| 9.00| 39.00|
--------------------------------------------
| 50| 50| 100
Total | 50.00| 50.00| 100.00

Agreement Statistics
------------------------------------------
Proportion Agree 0.800 [a]
Kappa Statistic 0.600 [b]
Kappa Standard Error 0.100
Kappa Significance (p<) 0.000 [c]

Interpretation

[a] The radiologist interpretations agree in 80% of the patients (note they are review-
ing imaging of the same patients independently).

[b] Kappa indicates a moderate level of agreement.

[c] The agreement between the two radiologists is statistically significant.

Related Glossary Terms

Applied Stat: Counts-Chi-square, Association

Index Find Term

Applied Stat: Mean-1 sample

Compares a mean obtained from a random sample to a population mean. It

is generally used to determine if there is a difference between what was ob-
tained in a random sample and an established benchmark that either origi-
nated from all members of a comparable population or is assumed to repre-
sent the population.

Assumptions

Interval/ratio level data

Random sampling

Normal distribution in population

Probability Distribution

T- Distribution

Degrees of Freedom = n-1

Formula

X −µ
t=
Sx

Where

Standard error of the sample mean

s
sx =
n

________________

Software Output Example: Starting income after completing two years

of college.

One-Sample Difference of Means

--------------------------------------------------

Population Mean 38000.0000 [a]

Sample Mean 35000.0000 [b]
Std Deviation 6000.0000 [c]
Sample Size (n) 30
Test statistic -2.7386
p < 0.0104 (2-tailed) [d]

Interpretation

[a] Represents the historical mean starting income of all graduates of two year tech-
nical colleges in the United States (in constant dollars).

[b] Represents the mean starting income of a random sample of 30 graduates from
this year’s technical colleges in the United States.

[c] The standard deviation for the sample of 30 graduates (you generally use the
sample standard deviation unless the population standard deviation is known).

[d] There is a statistically significant difference between the starting income of this
year’s graduating class and the historical average starting pay. This year’s graduates
have a mean starting pay that is significantly less than the mean pay for prior gradu-
ates.

Note: The above example uses StatCalc output.

Related Glossary Terms

Applied Stat: Mean-2 sample

Index Find Term

Applied Stat: Mean-2 sample

Compares two means from two random samples; such as mean incomes of
males and females, Republicans and Democrats, Urban and Rural residents.
If there are more than two comparable groups, ANOVA is a more appropri-
ate technique.

Assumptions

Random sampling

Independent samples

Interval/ratio level data

Homogeneity of variance

Probability Distribution

T distribution

Degrees of Freedom = (n1+n2)-2

Formula

X1 − X 2
t=
S x −x
1 2

Where the Standard Error

(n1 − 1) s12 + (n2 − 1) s 22 n1 + n2

S x −x =
1 2
n1 + n2 − 2 n1n2

________________

Software Output Example: Comparing the mean income of males and

females.

Two-Sample Difference of Means

Independent Variable: SEX

Dependent Variable: INCOME

Sample One: 0 Female

Sample Two: 1 Male

Sample 1 Sample 2
Female Male
-----------------------------------------------------------
Sample Mean 12279.4667 15266.4706
Std Deviation 4144.6323 3947.7352
Sample Size (n) 15 17

Homogeneity of Variance
------------------------------------------------------------
F-ratio 1.09 DF (14, 16) p < 0.4283 [a]

T-statistic DF p < (2-tailed)

------------------------------------------------------------
Equal Variance -2.0870 30 0.0455 [b]
Unequal Variance -2.0800 31.05 0.0459

Interpretation

[a] The F-ratio is not statistically significant. Therefore, use the equal variance test
statistic.

[b] For a 2-tailed test, the p-value represents the probability of making a type 1 error
(concluding there is statistical significance when there is none). Since there is about
4.6% (p < .0455) chance of making a type 1 error, which does not exceed the 5% er-
ror limit (p=.05) established in the rejection criteria of the hypothesis testing process
(alpha), you conclude there is statistical significance.

Related Glossary Terms

Applied Stat: Mean-1 sample

Index Find Term

Applied Stat: Mean-Analysis of Variance

Used to evaluate means from two or more subgroups. A statistically signifi-

cant ANOVA indicates there is more variation between subgroups than
would be expected by chance. It does not identify which subgroup pairs
are significantly different from each other. Used to evaluate multiple means
of one independent variable to avoid conducting multiple t-tests.

Assumptions

Independent Random sampling

Interval/ratio level data
Population variances equal
Groups are normally distributed

Probability Distribution

F-Distribution where . . .

Numerator Degrees of Freedom

df=k-1 where k=number of independent samples/groups

Denominator Degrees of Freedom

df=n-k where n=sum of all observations from the independent samples/

groups

Formula

F-Test Statistic

F=
∑ n ( X − X ) / (k −1)
k i g

#
) ∑( X − X ) + ∑( X − X ) &(' / (n − k )
2 2 2

%$∑ ( X
i − X1 + i 2 i 3

where

= each group mean

= grand mean for all the groups = ( sum of all scores)/N

= number in each group

Note: F= Mean Squares between groups

Mean Squares within groups

________________

Software Output Example: Comparing amount of time spent watching

TV per day by four age groups (n=990).

Analysis of Variance (ANOVA): TVHOURS (Means) by AGECAT4 (Groups)

Independent Variable Label: Age categories

Dependent Variable Label: Hours watch TV per day

Analysis of Variance

Sum of Squares df Mean Squares

--------------------------------------------------------------
Between Groups 92.09 3 [a] 30.70 [b]
Within Groups 5093.45 986 [c] 5.17 [d]
Total 5185.54 989 [e]

F Statistic p <
---------------------------------
5.94 [f] 0.0005 [g]

Interpretation
[a] k-1, where k = 4 group means

[b] Between Groups Sum of Squares ÷ degrees of freedom

[c] n-k, where n = 990

[d] Within Groups Sum of Squares ÷ degrees of freedom

[e] n-1

[f] Between Groups Mean Squares ÷ Within Groups Mean Squares

[g] The F-Statistic is statistically significant. The p-value represents the probability
of making a type 1 error (concluding there is statistical significance when there is
none). Since there is about .05% (p < .0005) chance of making a type 1 error, you
conclude there is statistical significance; ie, there is variation among the groups.

Related Glossary Terms

Applied Stat: Mean-Bonferroni post hoc

Index Find Term

Applied Stat: Mean-Bonferroni post hoc

The Bonferroni post hoc test adjusts alpha to compensate for multiple com-
parisons. In this situation, one may falsely conclude a significant effect
where there is none. In particular, use of a .05 cut-off value for significance
theoretically guarantees that if there were 20 pairwise comparisons, there
will by chance alone appear to be one with significance at the .05 level. The
Bonferroni adjustment is often offered as an additional test for ANOVA mod-
els to guide post ANOVA exploration and interpretation regarding which
paired mean comparisons are likely to have statistically significant differ-
ences. There are other corrections for this problem. Some are useful for un-
ordered groups, such as patient height versus sex, while others are applied
to ordered groups (to evaluate a trend), such as patient height versus sex
when stratified by age. Consequently, when multiple statistical tests are con-
ducted between the same variables, the significance cut-off value is often
adjusted to represent a more conservative estimate of statistical signifi-
cance. There is a debate about which post hoc method to use, and some
hold the view that this correction is overused. The Bonferroni correction ad-
justs the threshold for significance, which is equal to the desired p- value
(e.g., .05, .01) divided by the number of paired outcome variables being ex-
amined. One limitation of the Bonferroni correction is that by reducing the
level of significance associated with each test, we have reduced the power
of the test, thereby increasing the chance of incorrectly retaining the null hy-
pothesis.

________________

Software Output Example: Comparing amount of time spent watching

TV per day by age group.

Analysis of Variance (ANOVA): TVHOURS (Means) by AGECAT4 (Groups)

Independent Variable Label: Age categories

Dependent Variable Label: Hours watch TV per day

Analysis of Variance

Sum of Squares df Mean Squares

--------------------------------------------------------------
Between Groups 92.09 3 30.70
Within Groups 5093.45 986 5.17
Total 5185.54 989

F Statistic p <
-----------------------------
5.94 0.0005

Bonferroni Post Hoc Comparison of Means

Group 1 Group 2 G1-G2 SE p<

------------------------------------------------------------
18-29 30-39 0.410 0.209 0.3005
18-29 40-49 0.425 0.207 0.2464
18-29 50+ -0.256 0.228 1.0000
30-39 18-29 -0.410 0.209 0.3005
30-39 40-49 0.015 0.184 1.0000
30-39 50+ -0.666 [a] 0.200 0.0056 [b]
40-49 18-29 -0.425 0.207 0.2464
40-49 30-39 -0.015 0.184 1.0000
40-49 50+ -0.681 0.209 0.0073 [c]
50+ 18-29 0.256 0.228 1.0000
50+ 30-39 0.666 0.200 0.0056
50+ 40-49 0.681 0.209 0.0073

Interpretation

[a] The difference between the mean of those 30-39 (G1) and the mean of those 50
or older (G2) is a -0.666. A negative mean indicates those 50 or older have a greater
mean number of hours watching television than those 30-39.

[b] There is a statistically significant difference between the number of hours watch-
ing television for those between the ages of 30 and 39 and the number of hours
watching television for those 50 and older.

[c] There is a statistically significant difference between the number of hours watch-
ing television for those between the ages of 40 and 49 and the number of hours
watching television for those 50 and older.

Related Glossary Terms

Applied Stat: Mean-Analysis of Variance, Post hoc test

Index Find Term

Applied Stat: Mean-Confidence Interval

Interval estimation involves using sample data to determine a range (interval)

that, at an established level of confidence, is expected to contain the mean
of the population.

Steps

1. Determine confidence level (df=n-1; alpha .05, 2-tailed)

2. Use either z distribution (if n>120) or t distribution (for all sizes of n).

3. Use the appropriate table to find the critical value for a 2-tailed test

4. Multiple hypotheses can be compared with the estimated interval for

the population to determine their significance. In other words, differing val-
ues of population means can be compared with the interval estimation to de-
termine if the hypothesized population means fall within the region of rejec-
tion.

Estimation Formula

where

= sample mean

CV = critical value (consult distribution table for df=n-1 and chosen alpha--
commonly .05)

= Standard error of the mean

Note: assumes sample standard deviation was calculated using:

Example: Interval Estimation for Means

Problem: A random sample of 30 incoming college freshmen revealed the

following statistics: mean age 19.5 years; sample standard deviation 1.2.
Based on a 5% chance of error, estimate the range of possible mean ages
for all incoming freshmen.

Estimation

Critical value (CV)

Df=n-1 or 29

Consult t-distribution for alpha .05, 2-tailed

CV=2.045

Standard error

sx = .219

Estimate Confidence Interval

CI 95 = 19.5 ± 2.045 (.219 )

CI 95 = 19.5 ±.448

________________

Software Output Example: Summary data from the previous example

Margin of Error for Means

__________________________________________________________

n = 30 Mean = 19.5000 Standard Deviation = 1.2000

Margin Limits
Confidence Error +/- Lower Upper
----------------------------------------------------------
90% 0.3723 19.1277 19.8723
95% 0.4481 19.0519 19.9481
99% 0.6039 18.8961 20.1039

Note: The above example uses StatCalc output.

Related Glossary Terms

Confidence interval, Confidence limits

Index Find Term

Applied Stat: Percentage

The number of cases in a category of a variable divided by the number of

cases in all categories of the same variable. The quantity is then multiplied
by 100 to convert the proportion into a percentage.

________________

Software Output Example: Education level.

Frequencies
Variable: Edu Years of education
Value Count Percent Cumulative
--------------------------------------------------------------
08 1 2.00 2.00
09 2 [a] 4.00 [b] 6.00 [c]
10 2 4.00 10.00
11 2 4.00 14.00
12 21 42.00 56.00
13 2 4.00 60.00
14 4 8.00 68.00
15 3 6.00 74.00
16 6 12.00 86.00
17 3 6.00 92.00
18 1 2.00 94.00
19 2 4.00 98.00
20 1 2.00 100.00
--------------------------------------------------------------
Total 50
Missing 0

Interpretation

[a] Count: 2 out of 50 have 9 years of education

[b] Percentage: 4% have 9 years of education (2/50=.04; .04 * 100=4.00%)

[c] Cumulative Percentage: 6% have 9 years or less education (3/50=.06; .06*100 =

6.00%)

Related Glossary Terms

Applied Stat: Proportion, Relative frequency

Index Find Term

Applied Stat: Proportion

A frequency proportion represents a subfrequency divided by the total fre-

quency. This weights the count in the subgroup by the total. When multi-
plied by 100, a proportion becomes a percentage.

Related Glossary Terms

Applied Stat: Percentage, Relative frequency

Index Find Term

Applied Stat: Proportion-1 sample

Compares a proportion obtained from a random sample to a population pro-

portion. It is generally used to determine if there is a difference between
what was obtained in a random sample and an established benchmark that
either originated from all members of a comparable population or is as-
sumed to represent the population.

Assumptions

Independent random sampling

Nominal level data

Large sample size

Probability Distribution

Use z-distribution table

Formula

ps = sample proportion

pu
= population proportion

Where

Standard Error of the proportion

qu = 1− pu

n = sample size

________________

Software Output Example: Comparing historical proportion of clients

reporting poor service and the current proportion of clients reporting
poor service.

One-Sample Difference of Proportions!

Population Proportion! ! 0.10!

Sample Proportion! ! ! 0.15!
Sample Size (n)! ! 110!
Test statistic!! ! ! 1.748!
! ! ! ! p <!! ! 0.0805! (2tailed) [a]

Interpretation

[a] For a 2-tailed test, the p-value represents the probability of making a type 1 error
(concluding there is statistical significance when there is none). Since there is about
8% chance of making a type 1 error, which exceeds the 5% error limit established in
the rejection criteria of the hypothesis testing process (alpha), you would not con-
clude there is statistical significance.

Note: The above example uses StatCalc output.

Related Glossary Terms

Applied Stat: Proportion-2 sample
Applied Stat: Proportion-2 sample

Compares two proportions from two random samples; such as males and
females, Republicans and Democrats, Urban and Rural residents. If there
are more than two comparable groups, chi-square is a more appropriate
technique.

Assumptions

Independent random sampling

Nominal level data

Large sample size

Probability Distribution

Use z-distribution

Formula

p1 = Proportion from sample one

p2 = Proportion from sample two

Standard Error of the difference in proportions

^ ^

and
q = 1− p

________________

Software Output Example: Comparing the difference between the pro-

portion of students in school A not eating breakfast before coming to
school and the proportion of students in school B not eating breakfast
before coming to school.

Two-Sample Difference of Proportions (2-tailed)

! !
! ! ! ! ! School A! School B
Sample Proportion! ! 0.23! ! 0.07
Sample Size (n)! ! 80! ! ! 180
! !
! Z-statistic! ! p < Significance
! ! ! ! 3.674! ! ! 0.0002 [a]

Interpretation

[a] For a 2-tailed test, the p-value represents the probability of making a type 1 error
(concluding there is statistical significance when there is none). Since there is far less
than a 5% chance of making a type 1 error, you would conclude there is statistical sig-
nificance between the schools in the proportion of children who need nutrition sup-
port at school.

Note: The above example uses StatCalc output.

Related Glossary Terms

Applied Stat: Proportion-1 sample

Index Find Term

Applied Stat: Proportion-Confidence Interval

Interval estimation (margin of error) uses sample data to determine a range

(interval) that, at an established level of confidence, is expected to contain
the population proportion.

Steps

Determine the confidence level (alpha is generally .05)

Use the z-distribution table to find the critical value for a 2-tailed test given
the selected confidence level (alpha)

Estimate the standard error of the proportion

where

p = sample proportion

q=1-p

Estimate the confidence interval

CV = critical value

CI = p ± (CV)(Sp)

Interpret

Based on alpha .05, you are 95% confident that the proportion in the popula-
tion from which the sample was obtained is between __ and __.

Note: Given the sample data and level of error, the confidence interval pro-
vides an estimated range of proportions that is most likely to contain the
population proportion. The term "most likely" is measured by alpha (i.e., in
most cases there is a 5% chance --alpha .05-- that the confidence interval
does not contain the true population proportion).

More About the Standard Error of the Proportion

The standard error of the proportion will vary as sample size and the propor-
tion changes. As the standard error increases, so will the margin of error.

 
Sample Size (n)
Proportion (p) 100 300 500 1000 5000 10000
0.9 0.030 0.017 0.013 0.009 0.004 0.003
0.8 0.040 0.023 0.018 0.013 0.006 0.004
0.7 0.046 0.026 0.020 0.014 0.006 0.005
0.6 0.049 0.028 0.022 0.015 0.007 0.005
0.5 0.050 0.029 0.022 0.016 0.007 0.005
0.4 0.049 0.028 0.022 0.015 0.007 0.005
0.3 0.046 0.026 0.020 0.014 0.006 0.005
0.2 0.040 0.023 0.018 0.013 0.006 0.004
0.1 0.030 0.017 0.013 0.009 0.004 0.003

Effect of changes in the proportion

As a proportion approaches .5 the error will be at its greatest value for a

given sample size. Proportions close to 0 or 1 will have the lowest error.
The error above a proportion of .5 is a mirror reflection of the error below a
proportion of .5.

Effect of changes in sample size

As sample size increases the error of the proportion will decrease for a given
proportion. However, the reduction in error of the proportion as sample size
increases is not constant. As an example, at a proportion of 0.9, increasing
the sample size from 100 to 300 cut the standard error by about half (from
.03 to .017). Increasing the sample size by another 200 only reduced the
standard error by about one quarter (.017 to .013).

Example: Interval Estimation for Proportions

Problem: A random sample of 500 employed adults found that 23% had
traveled to a foreign country. Based on these data, what is your estimate for
the entire employed adult population?

n=500, p = .23 q = .77

Use alpha .05 (i.e., the critical value is 1.96)

Estimate Sampling Error

Compute Interval

Interpret

You are 95% confident that the actual proportion of all employed adults who
have traveled to a foreign country is between 19.3% and 26.7%.

________________

Software Output Example: Summary data from the previous example

Margin of Error for Proportions

n = 500 Proportion = 0.23 Standard Error = 0.0188

Margin Limits
Confidence Error +/- Lower Upper
------------------------------------------------------------
90% 0.0311 0.1989 0.2611
95% 0.0369 0.1931 0.2669
99% 0.0486 0.1814 0.2786

Note: The above example uses StatCalc output.

Related Glossary Terms

Confidence interval, Confidence limits

Index Find Term

Applied Stat: Rate

The number of occurrences of a trait divided by the number of possible oc-

currences per an established unit of time.

Rate Example: Create death rate per 1000 given there are 100 deaths a
year in a population of 10,000.

Related Glossary Terms

Applied Stat: Ratio

Index Find Term

Applied Stat: Ratio

The number of cases in one category divided by the number of cases in an-
other category.

Ratio = f1/f2 or frequency in one group (larger group) divided by the fre-
quency in another group.

Ratio Example: Your community has 1370 Protestants and 930 Catholics

1370/930 = 1.47 or for every one Catholic there are 1.47 Protestants in the
given population.

Related Glossary Terms

Applied Stat: Rate

Index Find Term

Applied Stat: Regression-Logistic

Used for multiple regression modeling when the dependent variable is nomi-
nal level data. Binomial logistic is used when the dependent variable is a di-
chotomy (0,1). Multinomial logistic regression is used when there are more
than two nominal categories.

Assumptions

See multiple regression for a more complete list of ordinary least squares
(OLS) regression assumptions. In addition to the key characteristic that the
dependent variable is discrete rather than continuous, the following assump-
tions for logistic regression differ from those for OLS regression:

1. Specification error - Linearity

Linearity is not required in logistic regression between the independent and

dependent variables. It does require that the logits of the independent and
dependent variables are linear. If not linear, the model may not find statisti-
cal significance when it actually exists (Type II error).

2. Large Sample Size

Unlike OLS regression, large sample size is needed for logistic regression. If
there is difficulty in converging on a solution, a large number of iterations, or
very large regression coefficients, there may be insufficient sample size.

________________

Software Output Example: Evaluating the affect of age, education,

sex, and race on whether or not a person voted in the presidential elec-
tion.

------------------------ Logistic Regression -------------------

Dependent Y: VOTE92

Cases with Y=0: 417 28.86%! No, did not vote

Cases with Y=1: 1028 71.14%! Yes, voted
Total Cases: 1445

Explanatory Model

VOTE92 = Constant + AGE + EDUC + FEMALE + WHITE + e

Model Fit Statistics

Initial -2 Log Likelihood 1736.533

Model -2 Log Likelihood 1544.154 Iteration (4)
Cox & Snell Rsq 0.125 [a]
McFadden Rsq 0.111 [b]
Model Chi-Square 192.380 df 4 p< 0.0000 [c]

Coefficients
Variable b Coeff. SE Wald p < OR
----------------------------------------------------------------
Constant -4.1620 0.4184 98.9755 0.0000
AGE 0.0335 [d] 0.0040 69.1386 [e] 0.0000 [f] 1.0340 [g]
EDUC 0.2809 0.0244 132.4730 0.0000 1.3243
FEMALE -0.1249 0.1269 0.9686 0.3250 0.8826
WHITE 0.1000 0.1671 0.3581 0.5495 1.1052
----------------------------------------------------------------

Estimated Model
VOTE92 = -4.1620 + .0335(AGE) + .2809(EDUC) + -.1249(FEMALE)
+ .1000(WHITE)

Classification Table (.50 cutpoint)

Predicted Y
Observed Y 0 1 % Correct
+-----------+-----------+
0 | 103 | 314 | 24.70 [h]
+-----------+-----------+
1 | 64 | 964 | 93.77 [i]
+-----------+-----------+----------
Total 73.84 [j]

Interpretation

[a] This ranges from 0 to less than 1 and indicates the explanatory value of the
model. The closer the value is to 1 the greater the explanatory value. Analogous to
R2 in OLS regression.

[b] Indicates explanatory value of the model. Ranges from 0 to 1 and adjusts for the
number of independent variables. Analogous to Adjusted R2 in OLS regression. In
this model, the explanatory value is relatively low.

[c] Tests the combined significance of the model in explaining the dependent (re-
sponse) variable. The significance test is based on the Chi-Square sampling distribu-
tion. Analogous to the F-test in OLS regression. The independent variables in this
model have a statistically significant combined effect on voting behavior.

[d] A one unit increase in age (1 year) results in an average increase of .0335 in the
log odds of vote equaling 1 (voted in election).

[e] The test statistic for the null hypothesis that the b coefficient is equal to 0 (no ef-
fect). It is calculated by (B/SE)2.

[f] Based on the Chi-Square sampling distribution, there is a statistically significant

relationship between age and voting behavior. The probability that one will vote in-
creases with age, holding education, sex, and race constant.

[g] OR=Odds Ratio. The odds of a change in the dependent variable (vote) given a
one unit change in age is 1.03. Note: if the OR is >1, odds increase; if OR <1, odds
decrease; if OR =1, odds are unchanged by this variable. In this example, the odds of
voting increases with a one unit increase in age.

[h] Of those who did not vote, the model correctly predicted non-voting in 24.7% of
the non-voters included in the sample data.

[i] Of those who did vote, the model correctly predicted voting in 93.8% of the vot-
ers included in the sample data.

[j] Of those who did or did not vote, the model correctly predicted 73.8% of the vot-
ing decisions.

Related Glossary Terms

Drag related terms here

Index Find Term

Applied Stat: Regression-OLS Multiple

An extension of the simple (bivariate) regression model that allows us to in-

corporate more than one independent variable into our explanation of Y. It
helps clarify the relationship between each independent variable and Y by
holding constant the effects of the other independent variables.

Assumptions

1. No measurement error

2. No specification error

3. Mean of errors equals zero

4. Error term is normally distributed

5. Homoskedasticity

6. No autocorrelation

7. No Multicolinearity

________________

Software Output Example: A multiple regression example regressing

individual income on education in years, work experience in months,
and a dummy variable representing sex of an individual (0=Female,
1=Male).

------------------- Multiple OLS Regression -----------------

Explanatory Model

INCOME = Constant + EDUC + WORKEXP + SEX + e

Model Fit Statistics

Cases(n) R R Square Adj Rsq SE Est.

---------------------------------------------------------
32 0.865 0.748 0.721 [a] 2246.736 [b]

ANOVA Statistics

Sum of Sqs df Mean Sq F p <

--------------------------------------------------------------
Regression 419605192.680 3 139868397.560 27.709 0.0000 [c]
Residual 141339012.195 28 5047821.864
Total 560944204.875 31

Coefficients

Variable b Coeff. std. error BetaWgt t p <

--------------------------------------------------------------
Constant -338.105 2027.916 -0.167 0.8688
EDUC 870.146 [d] 108.506 0.892 [e] 8.019 0.0000 [f]
WORKEXP 193.701 [g] 82.889 0.262 2.337 0.0268
SEX 2821.218 [h] 803.622 0.336 3.511 0.0015

Estimated Model

INCOME = -338.105 + 870.146(EDUC) + 193.702(WORKEXP)

+ 2821.218(SEX)

Interpretation

[a] Adjusted R2. Education, experience, and a person’s sex explain 72% of the
variation in individual income.

[b] Standard error of the estimate. Used for creating confidence intervals for pre-
dicted Y.

[c] The regression model is statistically significant.

[d] The impact of a one-unit change in education on income. One additional year of
education will on average result in an increase in income of $870 holding experience
and sex constant.

[e] Standardized regression partial slope coefficients. Used only in multiple regres-
sion models (more than one independent variable), beta-weights in the same regres-
sion model can be compared to one another to evaluate relative effect on the depend-
ent variable. Beta-weights will indicate the direction of the relationship (positive or
negative). Regardless of whether negative or positive, larger beta-weight absolute val-
ues indicate a stronger relationship. In this model, education has the greatest effect
on income followed in order by sex and experience.

[h] Education has a statistically significant association with income. The probability
of a type one error is less than .0001 which far exceeds alpha of p = .05. Also, experi-
ence and sex are statistically significant.

[i] One additional month of experience will on average result in an increase in in-
come of $194 holding education and sex constant.

[j] Males will on average earn an income of $2,821 more than women holding edu-
cation and experience constant.

Related Glossary Terms

Applied Stat: Regression-OLS Simple, Bias, Multicolinearity, Multiple correlation coefficient,
Partial slope, Principle of least squares, Slope (b), Standardized partial slopes

Index Find Term

Applied Stat: Regression-OLS Simple

Simple (bivariate) regression involves evaluating the linear relationship be-

tween one independent variable (X) and one dependent variable (Y) by fitting
a line to a scatter of data points. If there is a relationship between X and Y,
knowing the value of X will help explain and predict variations in Y.

Assumptions

See multiple regression for a complete list of assumptions. Basic theoretical

assumptions for simple regression are the following (note that in practice it
is difficult to find a perfect regression model that fully satisfies all the as-
sumptions):

1. Both the independent (X) and the dependent (Y) variables are interval or
ratio data.

2. There is a linear relationship between X and Y (confirm with scattergram).

3. Errors in prediction Y are normally distributed.

4. Errors in prediction Y are all independent of each other.

5. The distribution of the errors in prediction of the value of Y is constant re-

gardless of the value of X.

Formula

Equation for a straight line

where

= predicted score

b = slope of a regression line (regression coefficient)

x = individual score from the X distribution

a = Y intercept (regression constant)

where

= mean of variable X

= mean of variable Y

________________

Software Output Example: Regressing the effect of education on indi-

vidual income.

---------------------- Simple OLS Regression --------------------

Model Fit Statistics

Cases(n) R R Square Adj Rsq SE Est.

-----------------------------------------------------------------
32 0.751 [a] 0.564 [b] 0.550 2854.601 [c]

ANOVA Statistics

Sum of Sqs df Mean Sq F p <

-----------------------------------------------------------------
Regression 316481758.400 1 316481758.400 38.838 0.0000 [d]
Residual 244462446.475 30 8148748.216
Total 560944204.875 31

Coefficients

Variable b Coeff. std. error BetaWgt t p <

------------------------------------------------------------------
Constant 5077.512 [e] 1497.830 !3.390 ! 0.0020
EDUC 732.400 [f] 117.522 [g] 0.751 [h] 6.232 [i] 0.0000 [j]

Estimated Model

INCOME = 5077.513 + 732.400(EDUC)

Interpretation
[a] The correlation coefficient.
[b] Education explains 56% of the variation in income.
[c] Standard error of the estimate. Used for creating confidence intervals for pre-
dicted Y.
[d] The regression model is statistically significant.
[e] The Y intercept and mean value of income if education is equal to 0.
[f] The impact of a one-unit change in education on income. Interpretation: One ad-
ditional year of education will on average result in an increase in income of $732.
[g] Standard error of the slope coefficient. Used for creating confidence intervals
for b and for calculating a t-statistic for significance testing.
[h] Standardized regression partial slope coefficients. Used only in multiple regres-
sion models (more than one independent variable), beta-weights in the same regres-
sion model can be compared to one another to evaluate relative effect on the depend-
ent variable. Beta-weights will indicate the direction of the relationship (positive or
negative). Regardless of whether negative or positive, larger beta-weight absolute val-
ues indicate a stronger relationship.
[i] The t-test statistic. Calculated by dividing the b coefficient by the standard error
of the slope.
[j] Education has a statistically significant positive association with income.

Related Glossary Terms

Applied Stat: Regression-OLS Multiple, Principle of least squares, Regression line, Slope
(b), Specification error

Index Find Term

Association

Changes in one variable are accompanied by changes in another variable.

Finding associations is one of the major objectives of statistical analysis.
The question answered by association is whether knowledge of one set of
data allows us to infer or predict the characteristics of another set of data.
Descriptive associations include those relationships identified between two
sets of observations (e.g., sex and income) in nature. No conclusions about
causality can be made based solely on data that shows statistical associa-
tion between variables.

Related Glossary Terms

Applied Stat: Correlation, Applied Stat: Counts-Cramer's V, Applied Stat: Counts-Lambda,
Applied Stat: Kappa, Causation, Covariation, Covary, Gamma, Negative association, Non-
orthogonal, Pearson's C, Phi, Positive association, Proportional reduction in error (PRE),
Spearman's rho, Spurious association

Index Find Term

Autocorrelation

The error terms are correlated across observations in a multiple regression

model. Violation of this assumption is likely to be a problem with time-
series data where the value of one observation is not completely independ-
ent of another observation. (Example: A simple two-year time series of the
same individuals is likely to find that a person's income in year 2 is corre-
lated with their income in the prior year.) If there is autocorrelation, the pa-
rameter estimates for partial slopes and the intercept are not biased but the
standard errors are biased and hence significance tests may not be valid.

Diagnosis

Suspect autocorrelation with any time series/longitudinal data

Use the Durbin-Watson (d) statistic.

d=2 > no correlation between error terms

d=0 > perfect positive correlation between error terms

d=4 > perfect negative correlation between error terms

Correction:

Use generalized least squares (GLS) or weighted least squares (WLS) mod-
els.

Related Glossary Terms

Regression model

Index Find Term

Bar chart

A graphic display used for discrete variables. Each category is represented

by a bar of equal width. The length of each bar represents the number or
percentage of observations in that category.

Example of a Horizontal Column Chart

Related Glossary Terms

Column Chart, Frequency polygon, Pie chart

Index Find Term

Beta-weight

Standardized regression partial slope coefficients. Beta-weights indicate the

amount of change in the standardized scores of Y (dependent variable) for a
one unit change in the standardized scores of each independent variable,
controlling for the effects of all independent variables included in the regres-
sion model. Beta-weights in the same regression model can be compared
to one another to evaluate relative effect on the dependent variable. Beta-
weights will indicate the direction of the relationship (positive or negative).
Regardless of whether negative or positive, larger beta-weight absolute val-
ues indicate a stronger relationship.

Partial slope coefficients are not directly comparable to one another. This is
because each variable is often based on a different metric. Beta weights
(standardized slope coefficients) are used to compare the relative impor-
tance of regression coefficients in a model. They are not useful for compari-
sons among models from other samples.

Formula:

⎛ s xi ⎞
β i = bi ⎜ ⎟
⎜ s ⎟
⎝ y ⎠ or partial slope * (std dev. of Xi ÷ std dev. of Y)

Example:

Education partial slope coefficient (bi) = 1878.2

sxi
Education standard deviation = 2.88

sy
Begin salary standard deviation = 7870.64

⎛ s xi ⎞ ⎛ 2.88 ⎞
β i = bi ⎜ ⎟ β i = 1878.2⎜ ⎟
⎜ s ⎟ β i = .687
⎝ y ⎠ ⎝ 7870.64 ⎠

This is interpreted as the average standard deviation change in Y (salary) as-

sociated with one standard deviation change in X (education), when all other
independent variables are held constant. Or a one standard deviation
change in education results in an average change of .687 standard devia-
tions in income.

Related Glossary Terms

Regression model

Index Find Term

Bias

A misrepresentation of what is being measured. Bias can be the result of

many factors to include measurement device or technique. Estimates of
population parameters are sometimes referred to as either biased or unbi-
ased. A statistic is biased if, in the long run, it consistently over or underesti-
mates the parameter it is estimating.

Related Glossary Terms

Applied Stat: Regression-OLS Multiple, Interpretation bias, Regression model

Index Find Term

Biased estimate

A sample statistic that tends to over or underestimate the true population

value.

Related Glossary Terms

Regression model

Index Find Term

Binomial variable

A variable that has only two values or attributes. Sex is a common example
(male or female).

Related Glossary Terms

Drag related terms here

Index Find Term

Bivariate normal distributions

This assumption specifies that two comparison groups must be normally dis-
tributed.

Related Glossary Terms

Drag related terms here

Index Find Term

Bivariate table

A table that displays the joint frequency distributions of two variables. Gen-
erally referred to as a contingency table or crosstabulation. By convention,
the independent variable is usually represented in the columns and the de-
pendent variable is represented in the rows.

Example:

Crosstabulation: Manager (Rows) by Sex (Columns)

Column Variable Label: Respondent sex

Row Variable Label: In Management Position?

Count |
Row % |
Col % |
Total % |Male |Female | Total
--------------------------------------------
No | 21| 15| 36
| 58.33| 41.67|
| 84.00| 71.43| 78.26
| 45.65| 32.61|
--------------------------------------------
Yes | 4| 6| 10
| 40.00| 60.00|
| 16.00| 28.57| 21.74
| 8.70| 13.04|
--------------------------------------------
| 25| 21| 46
Total | 54.35| 45.65| 100.00

Related Glossary Terms

Applied Stat: Bivariate Analysis-Counts, Applied Stat: Counts-Chi-square

Index Find Term

Causation

Causation represents a direct stimulus-response link between two variables.

Since causation is a logical assertion and cannot be proven with statistics,
statistical techniques are best used to explore (not prove) connections be-
tween independent and dependent variables. The following logical conclu-
sions are a minimum requirement for concluding causation.

Association: Do the variables covary empirically? Strong associations are

more likely to be causal than weak associations.

Precedence: Does the independent variable vary before the effect exhibited
in the dependent variable?

Nonspuriousness: Can the empirical correlation between two variables be

explained away by the influence of a third variable?

Plausibility: Is the expected outcome plausible and consistent with theory,

prior knowledge, and other studies?

Related Glossary Terms

Association

Index Find Term

Central limit theorem

As sample size increases, the sampling distribution of means approximates

a normal distribution and is usually close to normal at a sample size of 30.

________________

Software Output Example: Skewed income distribution for a popula-

tion of 1,614 families.

Descriptive Statistics
Variable: REALINC FAMILY INCOME IN CONSTANT $
--------------------------------------------------------------------
Count 1614 Pop Var 297866464.1298
Sum 37950010.0000 Sam Var 298051130.2576
Mean 23513.0173 Pop Std 17258.8083
Median 18375.0000 Sam Std 17264.1574
Min 245.0000 Std Error 429.7280
Max 68600.0000 CV% 73.4238
Range 68355.0000 95% CI (+/-) 842.8885
Skewness 0.7979 t-test(mu=0) |p<| 0.0001
Kurtosis 2.8317
--------------------------------------------------------------------

Horizontal Bar Chart

--------------------------------------------------------------------
Bar Interval Count
---------------------- ------
245.00 to 7840.00 309 XXXXXXXXXXXXXXXXXXXXXXXXXXX
7840.01 to 15435.00 325 XXXXXXXXXXXXXXXXXXXXXXXXXXXX
15435.01 to 23030.00 341 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
23030.01 to 30625.00 118 XXXXXXXXXX
30625.01 to 38220.00 153 XXXXXXXXXXXXX
38220.01 to 45815.00 151 XXXXXXXXXXXXX
45815.01 to 53410.00 104 XXXXXXXXX
53410.01 to 61005.00 68 XXXXX
61005.00 to 68600.00 45 XXX
--------------------------------------------------------------------

Software Output Example: Simulation of taking a random sample of 30

observations from the population 1,000 times and calculating a mean
from each random sample.

-----------------------------------------------------
| Results of Random Sampling |
| |
| Variable Randomized: REALINC |
| Size of Each Sample: 30 |
| Number of Random Samples: 1000 |
| |
| This is the mean of means produced from 1000 |
| repeated random samples. |
-----------------------------------------------------

Descriptive Statistics
Variable: Mean
---------------------------------------------------------------------
Count 1000 Pop Var 10491553.2947
Sum 19247808.4165 Sam Var 10502055.3500
Mean 19247.8084 Pop Std 3239.0667
Median 19230.4583 Sam Std 3240.6875
Min 7231.5833 Std Error 102.4795
Max 30788.3333 CV% 16.8367
Range 23556.7500 95% CI (+/-) 201.0998
Skewness 0.0753 t-test(mu=0) |p<| 0.0001
Kurtosis 3.1711
---------------------------------------------------------------------

Horizontal Bar Chart

---------------------------------------------------------------------
Bar Interval Count
---------------------- ------
7231.58 to 9849.00 3 |
9849.01 to 12466.42 7 |
12466.43 to 15083.83 96 XXXXXXXX
15083.84 to 17701.25 211 XXXXXXXXXXXXXXXXXXX
17701.26 to 20318.67 329 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
20318.68 to 22936.08 229 XXXXXXXXXXXXXXXXXXXX
22936.09 to 25553.50 100 XXXXXXXXX
25553.51 to 28170.92 22 XX
28170.92 to 30788.33 3 |
----------------------------------------------------------------------

Related Glossary Terms

Independent random sample, Theoretical frequency distribution

Index Find Term

Central tendency

A single summary value that suggests a representative measure of a set of

observations. Measures of central tendency include mean, median, and
mode.

Related Glossary Terms

Mean, Median, Mode

Index Find Term

Codebook

A document that tells the location and meaning of variables and values in a
data file.

Related Glossary Terms

Data Dictionary

Index Find Term

Coefficient of determination

Represents the proportion of variation in the dependent variable (Y) ex-

plained by the independent variable (X). In the case of Pearson r, the coeffi-
cient of determination can be obtained by squaring the Pearson r coeffi-
cient. See also the coefficient of multiple determination.

Related Glossary Terms

Coefficient of multiple determination (Rsq)

Index Find Term

Coefficient of multiple determination (Rsq)

A statistic that represents the proportion of total variation explained in the

dependent variable by all independent variables included in the model. Multi-
ply by 100 to produce a percentage of the explained variation.

Related Glossary Terms

Coefficient of determination

Index Find Term

Coefficient of variation (CV)

The Coefficient of Variation (CV) is the ratio of the sample standard deviation
to the sample mean: (sample standard deviation/sample mean)*100 to calcu-
ate CV%. Used as a measure of relative variability, CV is not affected by the
units of a variable. CV is useful for comparing the variation in two series of
data which are measured in two different units. Examples include a compari-
son between variation in height and variation in weight or comparing differ-
ent experiments involving the same units of measure but conducted by dif-
ferent persons.

Related Glossary Terms

Drag related terms here

Index Find Term

Column Chart

A graphic display used for discrete variables. Each category is represented

by a bar of equal width. The height of each bar represents the number or
percentage of observations in that category.

Example of a Vertical Column Chart

Related Glossary Terms

Bar chart

Index Find Term

Concept

An idea that represents a set of objects. Concepts are sometimes referred

to as abstract concepts in that they are idealizations that may not be meas-
ured or observed directly.

Related Glossary Terms

Drag related terms here

Index Find Term

Conceptual model

This is a construct system that connects our view of the world with data. It
connects theory, hypotheses, and data with our vision of reality.

Related Glossary Terms

Theory

Index Find Term

Confidence interval

An interval of values within which we can state with a degree of confidence

that the true population parameter falls. Used in conjunction with the point
estimate. Also known as the interval estimate.

Related Glossary Terms

Applied Stat: Mean-Confidence Interval, Applied Stat: Proportion-Confidence Interval, Confi-
dence level, Confidence limits, Interval estimation, Margin of error, Point estimation

Index Find Term

Confidence level

An alternative representation of alpha. It represents the probability that an

interval estimate will contain the true population value.

Related Glossary Terms

Confidence interval, Constant

Index Find Term

Confidence limits

The boundary values that contain the interval estimate. An interval esti-
mate is an interval of values within which we can state with a degree of confi-
dence that the true population parameter falls. Used in conjunction with the
point estimate. Also known as the confidence interval and is often ex-
pressed as a margin of error.

Related Glossary Terms

Applied Stat: Mean-Confidence Interval, Applied Stat: Proportion-Confidence Interval, Confi-
dence interval

Index Find Term

Constant

When an object assumes only one value it is a constant. If an object is not a

constant, it is a variable. We sometimes refer to constants as controls. As an
example, a study of only female patients would have the constant female.

Related Glossary Terms

Confidence level

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Construct validity

The degree to which one measure correlates with other measures of the
same abstract concept.

Related Glossary Terms

Content validity, Criterion-related validity, External validity, Face validity, Internal validity, Pre-
dictive validity, Validity

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Content validity

The extent to which the indicator reflects the full domain of interest.

Related Glossary Terms

Construct validity, Criterion-related validity, External validity, Face validity, Internal validity,
Predictive validity, Validity

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Contingency table

Used to represent the relationship between two variables. The variables are
usually nominal or ordinal level and the relationship is described by counts
and row and column percent. Also referred to as a crosstabulation or cros-
stab.

Related Glossary Terms

Applied Stat: Bivariate Analysis-Counts, Applied Stat: Counts-Chi-square, Partial tables

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Continuous variable

A measure that can take on any value within a given interval or set of inter-
vals. An infinite number of possible values such as distance in kilometer and
loan interest rates.

Related Glossary Terms

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Control variable

A variable that is held to one value to help clarify the relationship between
other variables. As an example, sex may be controlled to investigate the re-
lationship between education level and income (i.e., a separate analysis for
males and females).

Related Glossary Terms

Partial tables, Statistical control

Index Find Term

Covariation

Changes in one variable are accompanied by changes in another variable.

Also known as association. This is one of the major objectives of statistical
analysis. The question answered by association is whether knowledge of
one set of data allows us to infer or predict the characteristics of another set
of data. Descriptive associations include those relationships identified be-
tween two sets of observations (e.g., sex and income) in nature. Sometimes
called correlational, no conclusions about causality can be made based
solely on the data. Experimental relationships are experiment based which
involves the ability of researchers to manipulate the levels of one variable to
observe changes in another variable.

Related Glossary Terms

Applied Stat: Correlation, Association

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Covary

Changes in one variable are accompanied by changes in another variable.

Also known as association.

Related Glossary Terms

Applied Stat: Correlation, Association

Index Find Term

Criterion-related validity

The degree to which one measure is similar to another measure. An exam-

ple would be high school GPA and graduate rate in college.

Related Glossary Terms

Construct validity, Content validity, Predictive validity, Validity

Index Find Term

Critical value

The point on the x-axis of a sampling distribution that is equal to alpha. It is

interpreted as standard error. As an example, a critical value of 1.96 is inter-
preted as 1.96 standard errors above the mean of the sampling distribution.

Related Glossary Terms

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Cross-sectional survey

Observations collected a one point in time.

Related Glossary Terms

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Cumulative frequency

A cumulative frequency conveys how many observations are contained

above or below a value of a variable. Generally described as a cumulative
frequency distribution that reports the number of observations that occur up
to a given value. Most statistical software provides this by default when you
run a frequency procedure.

Related Glossary Terms

Applied Stat: Frequencies, Cumulative percentage, Deciles, Frequency distribution

Index Find Term

Cumulative percentage

The percentage of cases within an interval and all preceding intervals. Gen-
erally displayed in a frequency distribution table.

Related Glossary Terms

Applied Stat: Frequencies, Cumulative frequency, Deciles

Index Find Term

Data

Observations of our environment that are used to answer questions such as

how much, how many, how long, how often, how fast, where, and what
kind. These observations are often represented by numbers. The impor-
tance of numbers is that they have less ambiguity than words as to their
meaning. Data are generally referred to in the plural.

Related Glossary Terms

Raw score, Secondary analysis

Index Find Term

Data Dictionary

A document that tells the meaning of variables and values in a data file.

________________

Software Output Example: Data from survey of U.S. Adults.

FILE DATA DICTIONARY

_________________________________________________________________

File Name: Workbook.dcs

File Size: 1,877 Bytes
Created: 2007-03-15 17:56:08
Created by: AcaStat Software

Variable Name: ! ! Idnum

Variable Label: !! Respondent Number
Value Labels: ! ! None
Missing Values: !! None

Variable Name: ! ! Age

Variable Label: !! Respondent age in years
Value Labels: ! ! None
Missing Values: !! None

Variable Name: ! ! Edu

Variable Label: !! Years of education
Value Labels: ! ! None
Missing Values: !! None

Variable Name: ! ! Sex

Variable Label: !! Respondent sex
Value Labels:
1 = Male
2 = Female
Missing Values: !! None

Variable Name: ! ! Manager

Variable Label: !! In Management Position?
Value Labels:
1 = No
2 = Yes
Missing Values: !! 9

Variable Name: ! ! JobSat

Variable Label: !! Satisfied with job
Value Labels:
0 = No
1 = Yes
Missing Values: !! None

Variable Name: ! ! Income

Variable Label: !! Respondent individual income
Value Labels: ! ! None
Missing Values: !! None

Variable Name: ! ! EduCat

Variable Label: !! Education Level
Value Labels:
1 = <12 yrs
2 = HS Grade
3 = College
Missing Values: !! None

Related Glossary Terms

Codebook

Index Find Term

Data reduction

Summarizing data into generalized measures based on numerous values of

raw data. Statistical analysis reduces large data sets that describe original
(raw) data into measures that lose some detail but provide simplicity in inter-
pretation.

Related Glossary Terms

Drag related terms here

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Deciles

The points that divide a distribution of scores into 10ths.

Related Glossary Terms

Cumulative frequency, Cumulative percentage

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Deduction

Expectations are based on general principles instead of observations.

Related Glossary Terms

Theory

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Degrees of freedom

The number of observations in the data that are free to vary after the sample
statistics have been calculated. Once the second to last calculation for a sta-
tistic has been completed, the final result is automatically determined with-
out calculation. This last value is not free to vary, so generally one is sub-
tracted from the count of observations (n-1).

Related Glossary Terms

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Dependent variable

A measure not under the control of the researcher that reflects responses
caused by variations in another measure (the independent variable). Also
called the response variable.

Related Glossary Terms

Antecedent variable, Intervening variable, Response variable

Index Find Term

Descriptive analysis

A direct and exhaustive measure of a complete group. A descriptive analysis

can be a description of an entire population or a sample. It becomes an in-
ferential analysis if you choose to use sample descriptors to describe a
larger underlying population. To successfully describe the underlying popula-
tion, you must base your measures on a random sample of that population.

Related Glossary Terms

Descriptive statistic

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Descriptive statistic

A statistic that classifies and summarizes numerical data from a sample.

Related Glossary Terms

Descriptive analysis, Interval data

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Dichotomous variable

This is a variable that is classified by two values. These variables are also
referred to as binary variables or dummy variables. Dummy variables are
qualitative variables that measure presence or absence of a characteristic.
As an example, a dummy variable representing the characteristic "male"
could be represented as 0=female (non-male) and 1=male.

Related Glossary Terms

Discrete variable, Dummy variable

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Discrete variable

A variable that is limited to a finite number of values. Such as religion or

number of parks in a city (you can't have 1.5 parks).

Related Glossary Terms

Dichotomous variable

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Dispersion

The extent to which observations differ (vary) from one another. Also known
as variation. Common summary measures of dispersion are range, vari-
ance, and standard deviation.

Related Glossary Terms

Variance

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Dispersion measures

Statistics that indicate the variation in a distribution of scores. Examples in-

clude variance, standard deviation, range.

Related Glossary Terms

Drag related terms here

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Dummy variable

A variable that is classified by two values. Dummy variables are qualitative

variables that measure presence or absence of a characteristic. As an exam-
ple, a dummy variable representing the characteristic "male" could be repre-
sented as 0=female (non-male) and 1=male. These variables are also re-
ferred to as binary variables or dichotomous variables.

Related Glossary Terms

Dichotomous variable

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Ecological fallacy

Improperly drawing conclusions about individuals based entirely on the ob-

servation of groups.

Related Glossary Terms

Errors in human inquiry

Index Find Term

Efficiency

The extent to which multiple sample outcomes are clustered around the
mean of the sampling distribution. The efficiency of a statistic is the de-
gree to which the statistic is stable from sample to sample. If one statistic
has a smaller standard error than another, then the statistic with smaller stan-
dard error is the more efficient statistic.

Related Glossary Terms

Drag related terms here

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Elaboration

A multivariate approach that expands the investigation of bivariate relation-

ships by controlling for other variables. This approach produces a separate
(partial) bivariate table (contingency or crosstabulation) for each value of a
third variable.

Related Glossary Terms

Drag related terms here

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Element

A term used in sampling to represent the unit of analysis. Examples include

people, cities, states, etc.

Related Glossary Terms

Drag related terms here

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Errors in human inquiry

Inaccurate observation: we tend to be casual observers. Science requires a

more deliberate and conscious activity

Overgeneralization: we use a few events to develop general patterns to sim-

plify our understanding. Science uses large samples (numerous events) and
replication.

Selective observation: we tend to pay attention to events/observations that

support our understanding and exclude those that conflict. Science uses all
observations and the peer review process discourages subjective opinion.

Illogical reasoning: individuals are comfortable with using exceptions to

prove a pattern/rule. Science has rules for causation and peer review en-
courages honesty.

Premature closure of Inquiry: we are information satisficers who are comfort-

able basing patterns and explanations on erroneous observations. Science
can suffer from the same ailment but one of the goals of science is to evalu-
ate certain knowledge (obvious explanations of reality such as the world is
flat, the universe revolves around the earth, "we have tried that before and it
didn't work").

Related Glossary Terms

Ecological fallacy

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Experimental design

Experimental research design uses a control group and applies a treatment

to a second group. This design provides the strongest evidence of causa-
tion through extensive controls and random assignment to remove other dif-
ferences between groups.

Related Glossary Terms

Quasi-experimental

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External validity

The extent to which the association between the independent and depend-
ent variable is accurate and unbiased in populations outside the study
group.

Related Glossary Terms

Construct validity, Content validity, Face validity, Hawthorne effect, Quasi-experimental, Va-
lidity

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Face validity

The extent to which an indicator appears to measure the abstract concept.

Related Glossary Terms

Construct validity, Content validity, External validity, Validity

Index Find Term

Fisher's Exact Test

Examines the relationship in a 2x2 contingency tables. It is most commonly

used in place of chi square when n is small (generally 30 or less). It should
be used in place of chi square when the expected frequency of any cell is
less than 1 or 20% of expected frequencies are less than 5.

________________

Software Output Example: Data from survey of U.S. Adults.

Crosstabulation: Manager (Rows) by Sex (Columns)

Column Variable Label: Respondent sex

Row Variable Label: In Management Position?

Count |
Row % |
Col % |
Total % |Male |Female | Total
--------------------------------------------
No | 21| 15| 36
| 58.33| 41.67|
| 84.00| 71.43| 78.26
| 45.65| 32.61|
--------------------------------------------
Yes | 4| 6| 10
| 40.00| 60.00|
| 16.00| 28.57| 21.74
| 8.70| 13.04|
--------------------------------------------
| 25| 21| 46
Total | 54.35| 45.65| 100.00

Chi-square Value DF p <

---------------------------------------------------------
Pearson 1.060 1 0.3032
Likelihood Ratio 1.059 1 0.3034
Yate's Correction 0.450 1 0.5023
Fisher's Exact (2-tailed) 0.5017 [a]

Interpretation

[a] Not statistically significant.

Related Glossary Terms

Applied Stat: Counts-Chi-square

Index Find Term

Formative evaluation

A method of judging the value of a program while the program activities are
forming or happening. Formative evaluation focuses on the process (Bhola
1990).

Example: Collecting continuous feedback from participants in a program in

order to revise the program as needed.

Related Glossary Terms

Summative evaluation

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Frequency distribution

A count of how frequently each value of a variable occurs. Counts can be

displayed in table or graphical form such as a histogram.

Related Glossary Terms

Applied Stat: Frequencies, Cumulative frequency, Frequency polygon, Histogram, Kurtosis,
Normal distribution, Relative frequency

Index Find Term

Frequency polygon

A graphical presentation of a frequency distribution. The y axis (vertical) rep-

resents the counts and the x axis (horizontal) represents the values of a vari-
able. The difference between a polygon and histogram is that a line graph is
used to represent each value's count rather than a bar.

Related Glossary Terms

Bar chart, Frequency distribution, Histogram, Normal distribution, Pie chart

Index Find Term

Gamma

A measure of association used in bivariate tables of two ordinal level vari-

ables. Gamma is a proportional reduction in error technique that can range
from -1 (negative association) to 1 (positive association). The closer the
gamma coefficient is to an absolute value of one, the stronger the associa-
tion.

Related Glossary Terms

Applied Stat: Counts-Chi-square, Association

Index Find Term

Hawthorne effect

A commonly cited research error. A series of experiments in the 1920s at

the Hawthorne Western Electric Company found that the increases in pro-
duction researchers attributed to the manipulation of lighting in a control
group was actually the result of the attention they gave to the study group,
not the intervention (lighting).

Related Glossary Terms

External validity, Internal validity

Index Find Term

Heteroskedasticity

Occurs when the variance of the error term does not have constant variation
across multiple values of an independent variable.

Related Glossary Terms

Regression model

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Histogram

A graphical presentation of frequency distribution. The y axis (vertical) repre-

sents the counts and the x axis (horizontal) represents the values of a vari-
able. Unlike a column or bar chart, columns or bars in a histogram are con-
tinuous intervals and flow from one to the other. A histogram will not have a
gap between the bars or columns except to differentiate one interval from
another.

Related Glossary Terms

Frequency distribution, Frequency polygon, Normal distribution

Index Find Term

Homogeneity of variance

The assumption of homogeneity of variance is that the variance within each

of the populations is equal. This is an assumption of t-tests and analysis of
variance (ANOVA).

Related Glossary Terms

Drag related terms here

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Homoscedasticity

The variance of the Y scores in a correlation are uniform for the values of the
X scores. In other words, the Y scores are equally spread above and below
the regression line.

Related Glossary Terms

Regression model

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Hypothesis

A formal statement that presents the expected association between an inde-

pendent and dependent variable. A dependent variable is a variable that con-
tains variations for which we seek an explanation. An independent variable
is a variable that is thought to affect (cause) variations in the dependent vari-
able.

Related Glossary Terms

Alternative hypothesis

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Independent random sample

Random samples collected so that the selection of a particular case for one
sample has no effect on the probability that any particular case will be se-
lected for the other samples.

Related Glossary Terms

Central limit theorem

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Independent variable

A measure that can take on different values which are subject to manipula-
tion by the researcher.

Related Glossary Terms

Antecedent variable, Intervening variable

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Induction

General principles are based on observations.

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Inference

A procedure for stating the degree of confidence that the measures we have
of our environment are accurate. Inferential measures are based on less
than the entire population and represent an estimate of the true but un-
known value of a population characteristic. Inference depends on the con-
cept of random sampling.

Related Glossary Terms

Inferential statistic

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Inferential statistic

Statistics that use characteristics of a random sample to include sampling

error to predict the true values in the larger population. Any time one uses
sample data to make estimates of a population one is employing an inferen-
tial statistic. Inferential statistics are treated as estimates of population pa-
rameters.

Related Glossary Terms

Inference

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Instrumental variable

For regression modeling, an instrumental variable for a specific independent

variable Xi is another variable that a) is correlated with Xi, b) has no effect on
the dependent variable other than indirectly through Xi, and c) is not corre-
lated with the error term in the model. The basic premise is to use instru-
mental variables to purge the independent variable of random error by incor-
porating a two-stage regression analysis.

Related Glossary Terms

Regression model

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Interaction

Occurs when the statistical significance and/or direction of a bivariate rela-

tionship varies depending on a particular category of a controlling variable.

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Intercept (a)

Represents the point where the regression line intercepts the Y axis and indi-
cates the average value of Y when X is equal to zero (0).

Related Glossary Terms

Regression model

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Interdependent

Changes in one variable are accompanied by changes in another variable.

This is one of the major objectives of statistical analysis. The question an-
swered by association is whether knowledge of one set of data allows us to
infer or predict the characteristics of another set of data.

Related Glossary Terms

Drag related terms here

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Internal validity

The extent to which accurate and unbiased association between the inde-
pendent variable and dependent variable was obtained in the study group.

Related Glossary Terms

Construct validity, Content validity, Hawthorne effect

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Interpretation bias

Errors in data collection that occur when knowledge of the results of one
test affect the interpretation of a second test.

Related Glossary Terms

Bias

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Interquartile range (Q)

The distance representing the range of scores between the second and third
quartile of a distribution of scores.

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Interval data

Objects classified by type or characteristic, with logical order and equal dif-
ferences between levels of data. The meaning between each level of data is
the same. Also known as scale data.

Related Glossary Terms

Descriptive statistic, Level of measurement, Nominal data, Ordinal data, Quantitative, Ratio
data

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Interval estimation

An interval of values within which we can state with a degree of confidence

that the true population parameter falls. Used in conjunction with the point
estimate. Also known as the confidence interval and is often expressed as
a margin of error.

Related Glossary Terms

Confidence interval

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Intervening variable

An independent variable that comes between another independent variable

and a dependent variable. An intervening variable suggests the original link
between the independent and dependent variables is primarily through a
third variable.

Related Glossary Terms

Antecedent variable, Dependent variable, Independent variable

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Kurtosis

The peakedness of a distribution. Leptokurtic is more peaked, Mesokurtic is

a normal distribution, and Platykurtic is a flatter distribution.

Related Glossary Terms

Frequency distribution

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Level of measurement

The mathematical characteristic of a variable and the major criterion for se-
lecting statistical techniques. Variables can be measured at the nominal, or-
dinal, interval, ratio level.

Related Glossary Terms

Interval data, Nominal data, Ordinal data, Ratio data, Scale

Index Find Term

Likert scale

A composite measure based on a collection of items that are measured on

an ordinal scale such as strongly agree to strongly disagree survey ques-
tions. Developed by Rensis Likert. Some survey questions are described as
Likert questions because they have a range of ordinal responses. A true Lik-
ert scale requires that a composite measure be created from two or more
scaled questions.

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Linear

The relationship between the independent variables and the dependent vari-
able is conducive to being fit with a straight line.

Related Glossary Terms

Positive association

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Longitudinal survey

A survey that collects data at different points in time on the same objects.

Related Glossary Terms

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Margin of error

Used to express an interval of values within which we can state with a de-
gree of confidence that the true population parameter falls. Used in conjunc-
tion with the point estimate. Also known as the confidence interval. The mar-
gin of error is commonly expressed as plus or minus a set value that repre-
sents half of the confidence interval. As an example, 50% with a margin of
error +/- 3% indicates that the true population percentage is estimated to
be between 47% and 53%. There is always a chance this interval is wrong.
If the margin of error is based on a 95% confidence level, this means there
is a 5% chance the true population value will not fall within the margin of er-
ror.

Related Glossary Terms

Confidence interval

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Marginals

The row and column subtotals (counts) reported in a bivariate contingency

table.

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Mean

The arithmetic average of the scores in a sample distribution. Mathemati-

cally represented as the sum of a set of values divided by the number of val-
ues used. The mean has an advantage over the median and mode as a
measure of central tendency. The mean includes information from every
value and uses this information to present a central balancing point or meas-
ure of central tendency.

X=
∑X i

Example:

Variable Age Also known as X

24

32 Mode

32

42 Median Xi Each score

55

60

65

n= 7 Number of scores (or cases)

310 Sum of scores

44.29 Mean

Related Glossary Terms

Central tendency, Median, Mode, Mu, Weighted mean

Index Find Term

Mean absolute deviation

This summary measure of dispersion is computed by summing the absolute

values of the deviations of each value from the mean and then dividing
these values by the number of observations in the data. Also known as
M.A.D.

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Measurement scale

A reflection of how well a variable and/or concept can be measured. Gener-

ally categorized in order of precision as nominal, ordinal, interval, and ratio
data.

Related Glossary Terms

Applied Stat: Counts-Lambda

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Median

The point on a scale of measurement below and above which fifty percent
of the observations fall. To obtain the median, the observations must be
rank ordered by their values.

Example:

Variable Age

24

32

32

42 Median

55

60

65

Related Glossary Terms

Central tendency, Mean, Mode

Index Find Term

Missing data

Observations that don't report data for a variable. As an example, a survey

respondent may answer several survey questions but refuse to indicate their
age. For the age variable, the data are "missing" for this particular case.

Related Glossary Terms

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Mode

The most frequently occurring score in a distribution. Also referred to as the

modal value. It is possible to have bimodal distributions (two values with
equal counts) or multimodal distributions (three or more values with equal
counts).

Example:

Variable Age

24

32
Mode
32

42

55

60

65

Related Glossary Terms

Central tendency, Mean, Median

Index Find Term

Mu

The arithmetic average of the scores in a population.

Related Glossary Terms

Mean

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Multicolinearity

Occurs when one of the independent variables has a substantial linear rela-
tionship with another independent variable in a multiple regression model. It
occurs in any model and is more a matter of degree of colinearity rather
than whether it exists or not.

Related Glossary Terms

Applied Stat: Regression-OLS Multiple

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Multiple correlation coefficient

A statistic that indicates strength of association between a dependent vari-

able and two or more independent variables included in a regression model.
Computed by taking the square root of the coefficient of multiple determina-
tion (R squared).

Related Glossary Terms

Applied Stat: Regression-OLS Multiple

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Multiplication rule of probability

The probability of the joint occurrence of two or more outcomes across two
or more events. The probability of the joint occurrence of outcome A and
outcome B will be equal to the product of the probabilities of A and B.

Example:

Out of a deck of 52 playing cards, what is the probability of picking an Ace

and a Jack from two draws (events)? Assume a well shuffled deck and mu-
tually exclusive event possibilities (i.e., you cannot pull a Jack and an Ace at
the same time).

Multiplication Rule: When two events, A and B, are mutually exclusive, the
probability that A and B will occur is the product of the probability of each
event. (The example assumes a card is not returned to the deck after selec-
tion.)

P(A and B) = P(A) * P(B)

or

P(Ace and Jack) = 4/52 * 4/51 = 16/2652

There is about a 0.6% chance of pulling an Ace and a Jack with two draws
from a deck of playing cards.

Related Glossary Terms

Addition rule of probability

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Mutually exclusive

No two outcomes can both occur in any given trial. As an example, males
and females are mutually exclusive groups in the characteristic sex. The ran-
dom selection of one individual from a population cannot (normally) produce
a person who is both male and female.

Related Glossary Terms

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N

Represents the number of cases (observations) in a population. Lower case

"n" represents the number of cases (observations) in a sample.

Related Glossary Terms

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Naturalistic study

A non-experimental study where relationships are simply observed as they

occur or are exhibited in a natural environment (i.e., without investigator con-
trol). Common examples are social and attitudinal surveys.

Related Glossary Terms

Drag related terms here

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Negative association

A relationship where two variables vary in opposite directions. As values in

one variable increase, values in the other variable decrease.

Related Glossary Terms

Association

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Nominal data

Objects classified by type or characteristic. There is no specific order. Exam-

ples of nominal variables include religion, sex, political affiliation, medical
treatment. Nominal data are also known as nominal scale data.

Related Glossary Terms

Interval data, Level of measurement, Ordinal data, Qualitative, Ratio data

Index Find Term

Non-orthogonal

Changes in one variable are accompanied by changes in another variable.

This is one of the major objectives of statistical analysis. The question an-
swered by association is whether knowledge of one set of data allows us to
infer or predict the characteristics of another set of data.

Related Glossary Terms

Association

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Nonparametric

A distribution free test where the assumption of a normal sampling distribu-

tion is not needed. Chi-square is considered to be a nonparametric test as
are tests of medians.

Related Glossary Terms

Applied Stat: Counts-Chi-square

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Normal curve

A theoretical distribution of scores that is symmetrical and unimodal (com-

monly called bell shaped). A standard normal curve has a mean of 0 and a
standard deviation of 1.

Properties of a normal distribution. This is a theoretical construct; no popula-

tion or sample will be perfectly normal.

1. Forms a symmetric bell-shaped curve.

2. 50% of the scores lie above and 50% below the midpoint of the dis-
tribution.

3. The curve is asymptotic to the x axis (i.e., never touches the x axis).

4. The mean, median, and mode are located at the midpoint of the x
axis.

Related Glossary Terms

Normal distribution

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Normal distribution

A frequency distribution of scores that is symmetric about the mean, me-

dian, and mode. Distributions that are not symmetric are called skewed. A
normal distribution is also sometimes referred to as a bell-shaped distribu-
tion.

Related Glossary Terms

Frequency distribution, Frequency polygon, Histogram, Normal curve, Skewness

Index Find Term

Null hypothesis

This is a statement that asserts there is no difference between two popula-

tion parameters. A statistically significant finding rejects the null hypothesis
and supports the alternate hypothesis.

Example:

Null Hypothesis: There is no statistically significant difference between the

historical proportion of clients reporting poor service and the current propor-
tion of clients reporting poor service.

Alternate Hypothesis: There is a statistically significant difference between

the historical proportion of clients reporting poor service and the current pro-
portion of clients reporting poor service.

Related Glossary Terms

Alternative hypothesis

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Odds ratio

The odds ratio is used to determine if an event is the same for two groups.
The odds of an outcome represents the probability that the outcome does
occur divided by the probability that the outcome does not occur. An odds
ratio of 1 means the event is equally likely in both groups. An odds ratio
greater than one suggests the event is more likely in one group. A ratio less
than one suggests the event is less likely in one group as compared to an-
other.

Related Glossary Terms

Relative risk

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One-tailed test

A hypothesis test used when the direction of the difference can be predicted
with reasonable confidence or the focus of the test is only one tail of the
sampling distribution.

Related Glossary Terms

Two-tailed test

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Operationalize

Involves determining how an abstract concept will be measured. As an ex-

ample, temperature is an abstract concept that could be measured by de-
gree Celsius or simple statements such as cold, warm, hot.

Related Glossary Terms

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Ordinal data

Objects classified by type or characteristic with some logical order. Also re-
ferred to as ordinal scale data. Examples of ordinal variables include military
rank, letter grade, class standing.

Related Glossary Terms

Interval data, Level of measurement, Nominal data, Qualitative, Ratio data

Index Find Term

P-value

The probability of a Type I error. Type I error is rejecting a true null hypothe-
sis. Commonly interpreted as the probability of being wrong when conclud-
ing there is statistical significance.

Related Glossary Terms

Alpha, Paradigm, Statistical inference, Statistical significance, Type I error

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Paradigm

The larger frame of understanding shared by the profession and/or scientific

community; part of the core set of assumptions from which we may base
our inquiry.

Related Glossary Terms

P-value, Theory

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Parameter

The measure of a population characteristic. This requires that observations

must be collected for every member of a population.

Related Glossary Terms

Population, Sample, Statistic

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Partial correlation

A multivariate technique that produces a correlation coefficient that repre-

sents a bivariate relationship when controlling for other variables.

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Partial slope

Reported in multiple regression, a partial slope represents the relationship

(slope) between one independent variable and the dependent variable while
controlling for all other independent variables included in the regression
model.

Related Glossary Terms

Applied Stat: Regression-OLS Multiple

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Partial tables

Bivariate tables resulting from controlling by a third variable. There will be

one bivariate table for each value (category) in the control variable.

Related Glossary Terms

Contingency table, Control variable

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Pearson's C

Pearson's Contingency Coefficient (C) is interpreted as a measure of the rela-

tive (strength) of an association between two variables. The coefficient will
always be less than 1 and varies according to the number of rows and col-
umns.

________________

Software Output Example: Data from survey of U.S. Adults.

Crosstabulation: GUNLAW (Rows) by SEX (Columns)

Column Variable Label: Respondent's Sex

Row Variable Label: Gun permits

Count |
Row % |
Col % |
Total % | Male | Female | Total
--------------------------------------------
Favor | 314| 497| 811
| 38.72| 61.28|
| 73.88| 88.91| 82.42
| 31.91| 50.51|
--------------------------------------------
Oppose | 111| 62| 173
| 64.16| 35.84|
| 26.12| 11.09| 17.58
| 11.28| 6.30|
--------------------------------------------
| 425| 559| 984
Total | 43.19| 56.81| 100.00

Chi-square Value DF p <

-----------------------------------------------------------
Pearson 37.622 1 0.0000 [a]
Likelihood Ratio 37.417 1 0.0000
Yate's Correction 36.592 1 0.0000 [b]

Measures of Association
-------------------------------------
Cramer's V .196 [c]
Pearson C .192
Lambda Symmetric .082 [d]
Lambda Dependent=Column .000
Lambda Dependent=Row .115 [e]

Note: 00.00% of the cells have an expected frequency <5

Interpretation

[a] Statistically significant (most common measure used for significance).

[b] When sample sizes are small, the continuous chi-square value tends to be too
large. The Yates continuity correction adjusts for this bias in 2x2 contingency tables.
Regardless of sample size, it is a preferred measure for chi-square tests on 2x2 ta-
bles.

[c] Weak association (both Cramer’s V and Pearson C).

[d] A symmetric lambda is used when identification of independent and dependent

variables is not useful.

[e] Knowing a person’s sex can reduce prediction error by 11.5%.

Related Glossary Terms

Applied Stat: Counts-Chi-square, Association
Percentile

The point below which a specific percentage of cases fall.

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Phi

A chi-square based measure of association for nominal level variables dis-

played in a 2 by 2 contingency table (2 rows and 2 columns). Phi can range
from 0 to 1.

Related Glossary Terms

Applied Stat: Counts-Chi-square, Association

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Pie chart

Commonly used for nominal data. It is a circular figure with slices that repre-
sent the relative frequency (proportion) of a value of a variable.

Example:

Related Glossary Terms

Bar chart, Frequency polygon

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Point estimation

A statistic based on a sample of observations that represents our best possi-

ble estimate of the true population parameter. A point estimate does not
provide us with any information on how close our estimate is to the parame-
ter.

Related Glossary Terms

Confidence interval

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Pooled estimate

An estimate of the standard deviation of the sampling distribution of the dif-

ference in sample means based on the standard deviations from both sam-
ples. The pooled estimate is a standard error.

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Population

Contains all members of a group. Also referred to as a universe. As an ex-

ample, everyone living in the state of Indiana would be the population for
data used to represent all persons living in Indiana. In comparison, a sample
represents a subset of a population.

Related Glossary Terms

Parameter, Universe

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Positive association

Two variables vary in the same direction. As the values of one variable in-
crease the values of a second variable also increase. High values in one
variable are associated with high values in the second variable.

Related Glossary Terms

Association, Linear

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Post hoc test

A statistical technique used to determine which pairs of means from several

possible pairings are significantly different. Associated with analysis of vari-
ance (ANOVA).

Related Glossary Terms

Applied Stat: Mean-Bonferroni post hoc

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Power

The probability of correctly finding statistical significance. A common value

for power is .80. To estimate power we need to know four factors: 1) size of
the difference between parameters (effect size), 2) significance level and
whether 1 or 2-tailed test, 3) standard deviation of each sampled popula-
tion, and 4) sample size of each population.

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Predictive validity

The ability of an indicator to correctly predict (or correlate with) an outcome

(GRE and performance in a graduate program).

Related Glossary Terms

Construct validity, Content validity, Criterion-related validity, Validity

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Principle of least squares

Used in simple and multiple regression techniques. Assumes a linear rela-

tionship between variables where a line can be fit to the data to represent
the point at which the deviations of all observations from the regression line
are at a minimum.

Related Glossary Terms

Applied Stat: Regression-OLS Multiple, Applied Stat: Regression-OLS Simple, Regression
model

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Probability distribution

The simple outcomes of a sample space and their associated probabilities

comprise a probability distribution. Sample space is the set of probabilities
for all possible mutually exclusive and exhaustive outcomes. Probability dis-
tributions are used in inferential statistics by providing critical values and
their associated probabilities.

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Proportional reduction in error (PRE)

An approach to measuring association that compares the number of errors

made in predicting values in the dependent variable without considering the
independent variable to the reduction in errors made in predicting values
when the independent variable is considered. Values can range from 0 to 1
and in some uses can range from -1 to 1. Lambda and Gamma are two PRE
measures of association.

Related Glossary Terms

Applied Stat: Counts-Chi-square, Association

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Qualitative

Variables measured on a non-metric nominal or ordinal scale are commonly

referred to as qualitative. Examples can include blood type, religion, social
class.

Related Glossary Terms

Nominal data, Ordinal data

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Quantitative

Variables measured on a metric interval or ratio scale are commonly referred

to as quantitative. Examples include GPD, income, age, IQ.

Related Glossary Terms

Interval data, Ratio data

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Quartiles

The points that represent a division of distribution into quarters.

Related Glossary Terms

Drag related terms here

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Quasi-experimental

This study design does not have the ability to employ the controls employed
in an experimental design. Although internal validity is less than the experi-
mental design, external validity is generally better and statistical controls are
used to compensate for extraneous variables.

Related Glossary Terms

Experimental design, External validity

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Random error

Error that can randomly occur in the measurement and data collection proc-
ess.

Related Glossary Terms

Random sample, Random sampling, Random variable

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Random sample

A subgroup of a population or universe that is based on the principle that

every member of the population has an equal chance of being included in
the sample.

Related Glossary Terms

Random error, Random sampling, Random variable

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Random sampling

Each and every element in a population has an equal opportunity of being

selected.

Related Glossary Terms

Random error, Random sample, Random variable

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Random variable

A variable whose values are not pre-determined. A measure where any par-
ticular value is based on chance through random sampling. A random vari-
able requires that a researcher have no influence on a particular observa-
tion's value.

Related Glossary Terms

Random error, Random sample, Random sampling

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Range

The simplest measure of variation. Range represents the difference between

the lowest and highest observed value in a collection of data.

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Ratio data

Objects classified by type or characteristic, with logical order and equal dif-
ferences between levels, and having a true zero starting point. A ratio vari-
able includes more information than nominal, ordinal, or interval scale data.
Measures using this scale are considered to be quantitative variables. Exam-
ples include time, distance, income, age. Other examples include frequency
counts or percentage scales.

Related Glossary Terms

Interval data, Level of measurement, Nominal data, Ordinal data, Quantitative

Index Find Term

Raw score

The actual value of an observation in a data set.

Related Glossary Terms

Data

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Regression line

A straight line that best displays the relationship between two variables.
The equation for a straight line is used to fit the line to the data points.
"Best" is defined as the regression line that minimizes the error exhibited be-
tween the observed data points and those predicted with the regression
line.

Related Glossary Terms

Applied Stat: Regression-OLS Simple, Regression model

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Regression model

A regression model represents the dependent and independent variables in-

cluded in the analysis.

Related Glossary Terms

Autocorrelation, Beta-weight, Bias, Biased estimate, Heteroskedasticity, Homoscedasticity,
Instrumental variable, Intercept (a), Principle of least squares, Regression line, Slope (b),
Specification error, Standardized partial slopes

Index Find Term

Relative frequency

Observed frequencies that have been converted into percentages. Percent-

ages are based on the total number of observations. As an example, if 50
out of 100 subjects are female, the relative frequency would be 50% (50/
100). Most statistical software provides this by default when you run a fre-
quency procedure.

Related Glossary Terms

Applied Stat: Frequencies, Applied Stat: Percentage, Applied Stat: Proportion, Frequency
distribution

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Relative risk

The ratio of two conditional probabilities. Relative risk is defined as the ratio
of risk in the exposed group divided by the risk in the unexposed group.
Relative risk equals 1 when an event is equally probable in both groups. A
relative risk greater than 1 suggests the event is more likely in the one group
as compared to another. A relative risk less than 1 suggest the event is less
likely.

Related Glossary Terms

Odds ratio

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Reliability

The extent to which a measure obtains similar results over repeat trials.

Related Glossary Terms

Validity

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Replication

Duplicating a study or experiment to confirm results or expose error.

Related Glossary Terms

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Research question

Defines the purpose of the study by clearly identifying the relationship(s) the
researcher intends to investigate.

Related Glossary Terms

Abstract concept

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Response bias

Errors in data collection caused by differing patterns and completeness of

data collection that are dominated by a specific subgroup within the sam-
ple.

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Response variable

The measure not controlled in an experiment. Commonly known as the de-

pendent variable.

Related Glossary Terms

Dependent variable

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Sample

A subset of a population. In inferential statistics, sample data are used to de-

scribe (estimate) the population value.

Related Glossary Terms

Parameter, Sample distribution, Sampling distribution, Sampling frame

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Sample distribution

A frequency distribution of data from one sample.

Related Glossary Terms

Sample, Sampling distribution, Sampling frame, Skewness

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Sampling distribution

A probability distribution representing an infinite number of sample distribu-

tions for a given sample size drawn from the population.

Related Glossary Terms

Sample, Sample distribution, Sampling frame, Skewness

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Sampling frame

A list of all possible members of a population under study. For a sample to

be representative of a population, the sampling frame must include as best
possible all members of the population.

Related Glossary Terms

Sample, Sample distribution, Sampling distribution

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Scale

A scale is a numerical representation of the values of a variable. In statis-

tics, we find four general types of scales (nominal, ordinal, interval, and ra-
tio). Scale is also sometimes used to refer only to continuous variables (not
ordinal or nominal).

Related Glossary Terms

Level of measurement

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Secondary analysis

Research conducted on data collected by another researcher who probably

collected the data for a purpose other than your primary interest.

Related Glossary Terms

Data

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Skew

The extent to which a distribution is affected by a few extremely large or

small scores. A few extremely large scores will result in a positively skewed
distribution. A few extremely small scores will result in a negatively skewed
distribution.

Related Glossary Terms

Skewness

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Skewness

Skewness provides an indication of the how asymmetric the distribution is

for a given sample. Knowing the mean and median will provide enough in-
formation to determine if there is a positive or negative skew. If the mean is
greater than the median, there is a positive skew. If the mean is less than
the median, there is a negative skew.

Related Glossary Terms

Normal distribution, Sample distribution, Sampling distribution, Skew

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Slope (b)

Represents the average change in Y associated with a one unit change in X.

Related Glossary Terms

Applied Stat: Regression-OLS Multiple, Applied Stat: Regression-OLS Simple, Regression
model

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Spearman's rho

A measure of association for two ordinal level variables that have numerous
rank ordered categories that are similar in form to a continuous variable.
Spearman's rho can range from -1 (strong negative association) to 1 (strong
positive association). Zero represents no association.

Related Glossary Terms

Association

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Specification error

Specification error occurs when the functional form of a relationship does

not match the statistical technique. Using simple regression as an example,
if the relationship between X and Y cannot be modeled by a straight line, we
may fail to find significance because our technique was not appropriate for
the data and/or relationship we are attempting to model.

Related Glossary Terms

Applied Stat: Regression-OLS Simple, Regression model

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Spurious association

Association between two variables that can be better explained by or de-

pends greatly upon a third variable. Suggests there is not a relationship be-
tween the two variables. Instead, the two variables are caused or strongly
related to a third (control) variable.

Related Glossary Terms

Association

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Standard deviation

A measure of dispersion that is represented in units of measurement consis-

tent with the raw data. It is the square root of the variance. Standard devia-
tion is also known as the root mean square (RMS), since it is the square root
of the mean squared deviations.

2
Σ( X i − µ )
σ=
Population Standard Deviation: N

S=
(
Σ Xi − X )
 Sample Standard Deviation: n −1

Example:

Variable (Xi) Age (Xi-mean) (Xi-mean)* (Xi-mean)

24 -20.29 411.68
32 -12.29 151.04
32 -12.29 151.04 squared deviations
42 -2.29 5.24
55 10.71 114.70
60 15.71 246.80
65 20.71 428.90

n= 7 1509.43 sum of squares

mean= 44.29 251.57 sample variance
15.86 sample standard deviation

Related Glossary Terms

Standard error of the mean, Variance

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Standard error of the mean

A measure reflecting the dispersion in a sampling distribution (not a sample

distribution). Also known as the standard deviation of the sampling distribu-
tion.

s
sx =
n

Note: assumes sample standard deviation was calculated using:

Related Glossary Terms

Standard deviation

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Standard score

Represents standard deviations from the mean. It is very useful in describ-

ing an object's relative position in a data distribution. A positive standard
score indicates an individual object's score is above the mean of the group.
Similarly, a negative standard score indicates the object's value is less than
the mean. The standard score value represents how many standard devia-
tions a particular raw score is above or below the mean. Also called stan-
dardized values or z-scores.

Example: (where mean=44.29 and standard deviation=15.86)

Variable (Xi) Age (Xi-mean) (Xi-mean)/s Z

Doug 24 -20.29 -20.29/15.86 -1.28

Jenny 32 -12.29 -12.29/15.86 -0.77

John 55 10.71 10.71/15.86 0.68

Beth 60 15.71 15.71/15.86 0.99

Ed 65 20.71 20.71/15.86 1.31 [a]

Interpretation

[a] As an example of how to interpret z-scores, Ed is 1.31 standard deviations

above the mean age for those represented in the sample. Another simple example is
exam scores from two history classes with the same content but difference instruc-
tors and different test formats. To adequately compare student A's score from class A
with Student B's score from class B you need to adjust the scores by the variation
(standard deviation) of scores in each class and the distance of each student's score
from the average (mean) for the class.

Related Glossary Terms

Test statistic, Z score

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Standardized partial slopes

Also known as beta-weights. Standardized regression partial slope coeffi-

cients or beta-weights indicate the amount of change in the standardized
scores of Y (dependent variable) for a one unit change in the standardized
scores of each independent variable, controlling for the effects of all inde-
pendent variables included in the regression model. Beta-weights in the
same regression can be compared to one another to evaluate relative effect
on the dependent variable. Beta-weights will also indicate the direction of
the relationship (positive or negative). Regardless of whether negative or
positive, larger beta-weight absolute values indicate a stronger relationship.

Related Glossary Terms

Applied Stat: Regression-OLS Multiple, Regression model

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Statistic

A measure of a sample characteristic.

Related Glossary Terms

Parameter

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Statistical analysis

A field of inquiry that is concerned with the collection, organization, and in-
terpretation of data according to well-defined procedures. It is the key to
every scientific discipline that studies the behavioral and biological charac-
teristics of human beings or our physical environment.

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Statistical control

Holding the value of one variable constant in order to clarify associations

among other variables.

Related Glossary Terms

Control variable

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Statistical inference

Reaching a conclusion about a population on the basis of information ob-

tained from a random sample drawn from that population. There are two
such methods, statistical estimation and hypothesis testing.

Related Glossary Terms

P-value

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Statistical significance

Interpreted as the probability of a Type I error. Test statistics that meet or ex-
ceed a critical value are interpreted as evidence that the differences exhib-
ited in the sample statistics are not due to random sampling error and there-
fore are evidence supporting the conclusion there is a real difference in the
populations from which the sample data were obtained.

Related Glossary Terms

P-value

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Statistics

Mathematical techniques used for organizing and analyzing data.

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Stochastic element

Unobserved random variable representing the error of a model.

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Stratification

The grouping of units representing similar (homogeneous) segments of a

population. Each group is referred to as a stratum.

Related Glossary Terms

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Substantive Importance

The real world implications of statistical significance. A statistically signifi-

cant finding may not be substantively important in practical application (i.e.,
does a statistically significant difference of one or two percent matter? -- it
may but the context of the research is very important).

The substantive importance of a statistical result is a common problem

when dealing with very large sample sizes where small differences in groups
can be statistically significant because of the mathematics behind statistical
testing (probability distributions and the affect of sample size on measures
of variation).

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Sum of squares

The sum of the squared deviations between each observed value and the
group mean. Used to calculate variance and standard deviation.

Related Glossary Terms

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Summative evaluation

A method of judging the value of a program at the end of the program activi-
ties. The focus is on the outcome.

Related Glossary Terms

Formative evaluation

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Test statistic

The value computed from sample statistics that is converted to a standard-

ized score for comparison to the critical value of a sampling distribution. If
the test statistic equals or exceeds the critical value, statistical significance
is assumed and the null hypothesis is rejected.

Related Glossary Terms

Standard score

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Theoretical frequency distribution

A theoretical frequency distribution is identical in design to a frequency distri-

bution but is based on the idea of an infinite number of observations. As the
number of observations increases, a histogram of a theoretical frequency
distribution begins to take on a smooth curve as compared to the normal
bar representation. This smooth curve is considered a representation of a
continuous distribution. In addition, the y axis represents the height of the
curve instead of the count.

Related Glossary Terms

Central limit theorem

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Theory

One or more propositions that suggest why an event occurs that provides a
framework for further analysis.

Related Glossary Terms

Conceptual model, Deduction, Paradigm

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Two-tailed test

A type of hypothesis test used when the direction of the difference cannot
be reasonably predicted a priori (before collecting or analyzing the data) or
the focus is on both tails of the sampling distribution. Most statistical test-
ing employs a two-tailed test. Statistical software typically reports p-values
based on a two-tailed test. Divide the p-value by 2 to get the one-tailed p-
value.

Related Glossary Terms

One-tailed test

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Type I error

Rejecting a true null hypothesis. Commonly interpreted as the probability of

being wrong when concluding there is statistical significance. Also referred
to as p-value or significance.

Related Glossary Terms

Alpha, P-value, Type II error

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Type II error

Retaining a false null hypothesis. Also referred to as Beta.

Related Glossary Terms

Alpha, Type I error

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Unit of analysis

The object under study. This could be people, schools, cities, etc.

Related Glossary Terms

Drag related terms here

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Univariate analysis

The study of one variable characteristic. Examples include test scores, vot-
ing record, salaries, education obtained, and medical test results.

Related Glossary Terms

Applied Stat: Bivariate Analysis-Counts

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Universe

Contains all members of a group. Also referred to as a population. In com-

parison, a sample represents a subset of a population.

Related Glossary Terms

Population

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Validity

The extent to which a measure accurately represents an abstract concept.

There are various forms of validity (see links below).

Related Glossary Terms

Construct validity, Content validity, Criterion-related validity, External validity, Face validity,
Predictive validity, Reliability

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Variable

A characteristic that can form different values from one observation to an-
other. If a characteristic is not a variable, it is a constant.

Related Glossary Terms

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Variance

A measure of dispersion that involves summing the squared deviations of

each value from the mean and dividing by the number of observations (n).

2
Σ( X i − µ )
σ2 =
Population Variance: N

S2 =
(
Σ Xi − X )
Sample Variance: n −1

Related Glossary Terms

Dispersion, Standard deviation

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Weighted mean

A mean based on weighted observations. This weight can be some meas-

ure of importance, or in the case of obtaining a grand mean from several
means with different sample sizes, the weight is the count for the subgroup
producing the mean.

Example:

Wrong Method:

Correct Method:

Related Glossary Terms

Mean

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Weighting

Used to adjust a sample that was created from an unequal representation of

a population. Once weighted the sample is expected to be representative of
the population. Weighting is used when over sampling is employed to col-
lect enough subjects from small groups and is also used to compensate for
disproportionate response rates.

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Z score

A score that standardizes individual observation values in one distribution of

scores that fit a normal distribution. A z score represents the distance of an
individual score from the mean, the direction of the distance, and the varia-
tion of scores in the distribution.

Related Glossary Terms

Standard score

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Zero-order correlation

A correlation coefficient for bivariate comparisons when no other variable is

used as a control.

Related Glossary Terms

Applied Stat: Correlation

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