By

Ehtesham Abbas
 The term statistics refers to a set of methods and rules
for organizing, summarizing, and interpreting
information.
 statistics is a way of gathering information and data.
 Data -> statistics ->information

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Stat is a parameter
while
Statistics is a measure that how the work have to be
done.

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 Data are everywhere
 Statistical techniques are used to make many decisions
that affect our lives
 No matter what your career, you will make
professional decisions that involve data. An
understanding of statistical methods will help you make
these decisions efectively

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 We live in the information world where most information is
determined mathematically with help of statistics.
 It means statistics keeps us informed about daily happening.
 Weather forecasting is possible due to some computer
models based on statistical concepts to predict weather.
 Researchers use statistical techniques to collect relevant
data.
 Statistics is frequently used by the researchers. They use
sstatistical techniques to collect relevant data.

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 Statistical techniques are key of how traders and
businessmen invest and make money.
 Industry tools are used in quality testing by product
managers.
 Statistics play a big role in medical field. Before any
drugs prescribed, pharmacists show statistically valid
rate of effectiveness.
 Similarly, statistics is behind all other medical studies.
Doctors predict diseases on the bases of statistical
concepts
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 Functions of Statistics

 To present facts in a definite form

 To simplify unmanageable and complex data
 To use techniques for making comparisons
 To enlarge individual experience
 To provide guidance in the formulation of policies
 To enable measurement of the magnitude of a
phenomenon.

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 Limitations of Statistics

1. The use of statistics is limited to numerical studies.

2. Statistical results might lead to fallacious conclusions.
3. Statistics relies on estimation and approximations.
4. Statistical techniques deal with population or aggregate
of individuals rather than with individuals.

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 Application of Statistics in Educational
Research

 Statistics enters in the process right from the

beginning of the research when whole plan
for the research, selection of design,
population, sample, analysis tools and
techniques etc., is prepared.

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 Descriptive and Inferential Statistics

Descriptive Statistics
Inferential Statistics

 . Inferential Statistics
 As the name implies, describes
the data. Descriptive statistics are techniques that
consist of methods for
organizing and summarizing allow a researcher to
information. study samples and then
 These are statistical procedures
that are used to organize, make generalizations
summarize, and simplify data.
about the populations
 Descriptive statistics organizes
scores in the form of a table or from which they are
a graph
selected.
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 “It is the parameter, characteristics or attribute which is
usually recorded in an experiment.”
 For example, variable is gender and attributes may be male
and female.
 The variables that yield numerical
information/measurement are called quantitative
 or
 numerical variable and the variable that yield non-
numerical information or measurement
 are called qualitative or categorical variable
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 There are two basic types of variables –
quantitative and categorical.
 Each uses different type of analysis and
measurement.
 There are four types of measurement scales:
nominal, ordinal, interval, and ratio.
 A scale of a variable gives certain structure to
the variable and also defines the meaning of
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 Types of measurement scales

Nominal Measurement
is the values “name” the attribute uniquely.

Ordinal Measurement
Ordinal level variables are nominal level variables with a
meaningful order.
True differences are not possible just order possible
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Types of measurement scales

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 Measurement Scales

 Nominal – categories
 Gender, ethnicity, etc.
 Ordinal – ordered categories
 Rank in class, order of finish, etc.
 Interval – equal intervals
 Test scores, attitude scores, etc.
 Ratio – absolute zero
 Time, height, weight, etc.
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 The Scientific Method

 Scientific method is a tool for:

 (a) forming and framing questions,
 (b) collecting information to answer those questions,
and
 (c) revising old and developing new questions.

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 The Scientific Method

 Scientific method is a process for explaining the world

to validate observations while minimizing observer
bias.
 It has many steps to describe the things we observe.
 Its goal is to conduct research in a fair, unbiased and
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 (a) forming and framing questions,

 (b) collecting information to answer those questions, and
 (c) revising old and developing new
 The Scientific Method

 i) Ask a question
 ii) Define the Problem
 iii) Forming a Hypothesis
 iv) Conducting the Experiment / Testing the Hypothesis
 v) Analyzing the Results
 vi) Drawing Conclusions
 vii) Communicate the Results
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 Basic Terms

 VARIABLES
 Variables are characteristics of an item or individual and are
 what you analyze when you use a statistical method.
 DATA
 Data are the different values associated with a variable. Data
is plural its singular is Datum.
 OPERATIONAL DEFINITIONS
 Data values are meaningless unless their variables have
 operational definitions, universally accepted meanings that
are clear to all associated with an analysis.
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 Basic Terms

 POPULATION
 A population consists of all individuals about which you want
to draw a conclusion. The population is the “large group.”
 SAMPLE
 A sample is the portion of a population selected for analysis.
 PARAMETER
 A parameter is a numerical measure that describes a
characteristic of a population.
 STATISTIC
 A statistic is a numerical measure that describes a sample
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 Variable

 a variable is a characteristic that varies from one person or thing to

another.
 It is a characteristic, number or quantity that increases or decreases
over time or takes different value in different situations;
 It may also be called a data item.
 . In some other words, a variable is an image, concept or a
perception that can be measured. It should be kept in mind that a
concept cannot be measured.
 Variable = A Concept that can be measured It must be converted
to some measureable form; and measureable form of a concept is
called a variable.
 Examples of variables for human are height, weight, age, number of
siblings, business income and expenses, country of birth, capital
expenditure, marital status, eye color, gender, class and grades.
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Types of Sample

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7/16/2020 1
 Educational Statistics
Lecture No. 2.
Basic Statistical Concepts
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Ehtesham Abbas 7/16/2020 2
 Learning Objectives
After completion of unit, the students will be able to:
1. Define variable and enlist commonly used variables.
2.Understand the term “data” and its types.
3. Differentiate between population and sample.
4. Know about the target population.
5. Tell the meaning & definition of assessable population
6. Write down the types of probability sampling.
7. Explain the types of non-probability sampling.
7/16/2020 3
 Variable
The word variable is something that varies or something
that is subject to variation.
It has no definite value but can assume any set of
values.
In other words we can say that a variable is a
characteristic that varies from one person to another.
It is a characteristic, number or quantity that increases
or decreases over time in different situations.
7/16/2020 4
 Variables
In some other words, a variable is an image,
concept or a perception that can be measured.
Keep in mind that a concept cannot be measured.
It must be converted to measureable form.
Examples of variables for human are height,
weight, age, number of siblings, business income
and expenses, country of birth, capital, eye color,
7/16/2020 5

expenditure, marital status, gender, class grades.

 Types of Variables
Types of Variables:
Variables can be categorized in three different
ways,
(a) The causal relationship
(b) The design of study, and
(c) The unit of measurement.
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 The Causal Relationship
In causal relationship studies four types of variables operate.
Change variables: that are responsible for bringing about
change in a phenomena;
Variables that affect link between cause and effect variables.
Outcome variables result from the effects of a change
variable.
Connecting or linking variables, which in certain situation
are important to complete relationship between cause and
7/16/2020 7

effect
Variables in Research Context
In research, change variables are referred as
independent variables while the outcome variables are
known as dependent variables.
In cause effect relationship, there are some unmeasured
variables affecting the relationship. These are called
extraneous variables.
The variables linking cause-effect relationship are called
intervening variables. 7/16/2020 8
 Design of Study
A study that investigates causation or association may be
controlled, experimental, quasi-experiment, an ex post facto or
non-experimental study.
Active Variables: these variables can be changed or
controlled; and
Attribute Variables: these variables can be controlled or
changed and refer to characteristics of study population.
Demographic features like age, gender, education,
7/16/2020 9

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 Some Common Variables
Binary Variables:
They are also called dichotomous variables. These
variables take only two values. For example, male or
female, true or false, yes or no, improved or not improved.
Polychotomous variables:
If the variables have more than two values, they are called
polychotomous variables.
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 Variables Types
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 Variable Types…..
Categorical Variables:
Usually an independent variable or predictor contains
values indicating membership in more than one possible
categories. For example, gender (male or
female), marital status (married, single, divorced, widow),
or brand of a product

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 Variable Types…..
Confounding variable: extra variables that have hidden effect
on experiment results. e. g., whether lack of exercise leads to
weight gain, then lack of exercise is your independent
variable and weight gain is dependent variable. The age factor
may change outcomes. So, age factor is confounding variable.
Continuous variable: a variable with infinite number of
values, like “time” or “weight”.
Discrete variable: a variable that has certain number of values,
as, Number of road accidents, number of children in a family
7/16/2020 13
 Variable Types…..
Control variable: a factor in an experiment held constant.
e.g., to determine whether light makes plants grow faster,
you have to control soil quality and water.
Dependent variable: called test variables, the outcome of an
experiment. When independent variable changed, what
happens to dependent variable. (achievement score)
Independent variable: variable that has power to change
dependent variable. Plotted on the x-axis. Also called
treatment variables. 7/16/2020 14
 Variable Types…..
Lurking variable: a “hidden” variable the affects the
relationship between the independent and dependent
variables. False and strong relationship in variables to
hide true result. For example, find effect of diet and
exercise on a person's blood pressure. Smoking as
lurking variables may affect blood pressure.
Measurement variable: has a number associated with
it. It’s an “amount or number” of something .
7/16/2020 15
 Variable Types…..
Ordinal variable: similar to a categorical variable, but with a
clear order. For example, income levels of low, middle, and
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Qualitative variable: a broad category for any variable that

can’t be counted (has no numerical value). Nominal and
ordinal variables fall under this umbrella term.
Quantitative variable: The variable that can be counted, or
has a numerical value value. Examples of variables that fall
into this category include discrete variables and ratio variables.
7/16/2020 16
 Variable Types…..
Random variables are associated with random processes and
give numbers to outcomes of random events.
A ranked variable is an ordinal variable; a variable where
every data point can be put in order (1st, 2nd, 3rd, etc.).
Ratio variables: similar to interval variables, but has a
meaningful zero.

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 Variable Types…..
Active Variable: a variable that is manipulated by the
researcher.
Antecedent Variable: a variable that comes before the
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Attribute variable: another name for a categorical

variable (in statistical software) or a variable that isn’t
manipulated (in design of experiments).
7/16/2020 18
 Variable Types…..
Covariate variable: similar to an independent variable, it has
an effect on the dependent variable but not variable of interest.
Criterion variable: another name for a dependent variable,
when the variable is used in non-experimental situations.
Dummy Variables: used in regression analysis when you want
to assign relationships to unconnected categorical variables.
For example, if you had the categories “has dogs” and “owns a
car” you might assign a 1 to mean “has dogs” and 0 to mean
“owns a car.” 7/16/2020 19
 Variable Types…..
Endogenous variable: similar to dependent variables, they are
affected by other variables in the system.
Exogenous variable: variables that affect others in the system.
Explanatory Variable: a type of independent variable. When a
variable is independent, it is not affected at all by any other
variables. When a variable isn’t independent for certain, it’s an
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Extraneous variables are any variables that you are not

intentionally studying in your experiment or test.
 Variable Types…..
A grouping variable (called coding variable) sorts data within
data files into categories or groups.
Identifier Variables: variables used to uniquely identify
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Indicator variable: another name for a dummy variable.
Interval variable: a meaningful measurement between two
variables. Same as continuous variable.
Intervening variable: a variable that is used to explain the
7/16/2020 21

relationship between variables.

 Variable Types…..
Latent Variable: a hidden variable that can’t be measured or
observed directly.
Manifest variable: a variable that can be directly observed or
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Manipulated variable: another name for independent variable.

Mediating variable or intervening variable: variables that
explain how the relationship between variables happens. For
example, it could explain the difference between the predictor
7/16/2020 22

and criterion.
 Variable Types…..
Moderating variable: changes the strength of an effect
between independent and dependent variables. For example,
psychotherapy may reduce stress levels for women more than
men, so sex moderates the effect between psychotherapy and
stress levels.
Nuisance Variable: an extraneous variable that increases
variability overall.
Outcome variable: similar in meaning to a dependent variable,
but used in a non-experimental study.7/16/2020 23
 Variable Types…..
Predictor variable: similar as independent variable, but used
in regression and in non-experimental studies.
Responding variable: an informal term for dependent
variable, usually used in science fairs.
Scale Variable: same as measurement variable.
Study Variable (Research Variable): can mean any variable
used in a study, but does have a more formal definition when
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 Data
Data is any information collected by the researcher.
Primary Data is originated by the researcher for the first time
for addressing his research problem.
The data can be collected using various methods like survey,
observations, physical testing, mailed questionnaire,
questionnaire filled and sent by enumerators, personal
interviews, telephonic interviews, focus groups discussion,
case studies.
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 Secondary Data
The information already collected and recorded by any other
person with a purpose not relating to current research problem.
It is readily available form of data and saves time and cast of
the researcher. This data may be limited in a number of ways
like relevance and accuracy.
Examples of secondary data are censuses data, publications,
internal records of the organizations, reports, books, websites,
journal articles.
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Diff. bet. Primary & Secondary Data
The Primary data refers to the data originated by the researcher
for the first time.
Secondary data is already existing data, collected by other
researchers, agencies, and organizations.
Primary data is collected to address the problem in hand.
Collection of primary data is a laborious process.
On the other hand, collection of secondary data is easy and
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Diff. bet. Primary & Secondary Data
Sources of primary data are survey, observations, physical
testing, mailed questionnaire, personal interviews, telephonic
interviews.
On the other hand, sources of secondary are censuses data,
publications, internal records, reports, journal articles, websites
Collection of primary data requires large amount of resources.
On the other hand collection of secondary data is expensive
and easily available.
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Diff. bet. Primary & Secondary Data
Primary data is specific to the researcher’s needs. He can
control the quality of research.
On the other hand, secondary data is neither specific to
researcher needs nor has he control over the quality of data.
Primary data is available in the raw form while secondary data
has undergone statistical procedures and is in refined from.
Data collected from primary sources are more reliable and
accurate than the secondary sources.
7/16/2020 29
 Population and Sample
A research Population is a large collection of individuals or
objects to which the researcher wants to apply the study results.
Population is the main focus of a research question. Aresearch
population is also known as a well-defined collection of
individuals or objects known to have similar characteristics.
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A small and representative part of individuals having same

attributes and characteristics is called sample. A sample is
7/16/2020 30

simply a subset or subgroup of population

Population and Sample

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7/16/2020 31
 Types of Population
The Target Population is referred to the entire group of
individuals or objects to which a researcher is interested
to generalize the conclusions. This type of population
usually has varying degree of characteristics.
The Accessible Population is also known as the study
population. It is the population to which a researcher can
apply the conclusions of the study. This population is a
subset of the target population. 7/16/2020 32
 Sampling Methods
Sampling
Methods

Non-
Probability
Probability
Sampling
Sampling
7/16/2020 33
 Probability Sampling
In probability sampling, each individual in chosen with a
known probability.
This type of sampling is also known as random sampling
or representative sampling; and depends on objective
judgment.
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7/16/2020 34
 Sampling Methods
Probability Sampling
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Multistage Simple Random Stratified Systematic

Random Sampling Sampling Sampling Sampling

Cluster Sampling

7/16/2020 35
 Simple Random Sampling
In random sampling each member of the population has an
equal chance of being selected as subject.
Each member is selected independently of the other member
of population.
In a commonly used method each member of the population is
assigned a unique number. All assigned numbers are placed in
bowl and mixed thoroughly.
The researcher, then blind-folds and picks numbered tags from
7/16/2020 36

the bowl.
Pros & Cons of Random Sampling
Advantages of Simple Random Sampling
It is an easy way of selecting a sample from a given population. This
method is free from personal bias. As each member of the population
is given equal opportunities of being selected so it a fair way and one
can get representative sample.
Disadvantages of Simple Random Sampling
One of the most obvious limitations of random sampling method is its
need of complete list of all members of population. For larger
population, usually this list is not available. In such case, it is better to
use other sampling techniques. 7/16/2020 37
 Systematic Sampling
In systematic random sampling, the researcher first randomly
picks the first item or the subject from the population. Then he
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The procedure involved in this sampling is easy and can be

done manually. The sample drawn using this procedure is
representative unless certain characteristics of the population
are repeated for every nth member, which is highly risky.
7/16/2020 38
Advantages of Systematic Sampling
The main advantage of using this technique is its simplicity.
It allows researcher to add a degree of system or process into
the random selection of subjects.
Another advantage is its assurance that the population will be
evenly sampled.
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Disadvantages of Systematic Sampling
Systematic sampling assumes that the size of the population is
available or can be approximated.
Suppose a researcher wants to study the behavior of monkeys
of a particular area.
If he does not have any idea of how many monkeys there are,
he cannot systematically select a starting point or interval size.
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 Stratified Sampling
In this type of sampling, the whole population is divided into
disjoint subgroups.
These subgroups are called stratum. From each stratum a
sample of pre-specified size is drawn independently in
different strata, using simple random sampling.
The collection of these samples constitutes a stratified sample.
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 Pros & Cons of Stratified Sampling
Advantages
This type of sampling is appropriate when the
population has diversified social or ethnic subgroups.
Disadvantages
While using this type of sampling, there is greater
chance of overrepresentation of subgroups in the
sample.
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 Cluster Sampling
In this type each sampling unit is a collection or cluster, or
groups. For example, a researcher who wants to study students
may first sample groups or cluster of students such as classes,
and then, select the sample of students from among clusters.
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Appropriate for larger population. It saves time and resources.

Disadvantages
7/16/2020 43

A greater chance of selecting a non-representative sample.

 Non-Probability Sampling
This technique depends on subjective judgment.
It is a process where probabilities cannot be assigned to the
individuals objectively.
It means that samples are gathered in a way does not give all
individuals in the population equal chances of being
selected.
Choose these methods could result in biased data or a limited
ability to make general inferences based on the findings.
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 Sampling Methods
Non-Probability Sampling

Judgmental Quota Extensive Snowball

Sampling Sampling Sampling Sampling
Convenient
Sampling
7/16/2020 45
 Convenient Sampling
In this technique a researcher relies on available subjects, such as
stopping peoples in the markets or on street corners as they pass by.
This method is extremely risky and does not allow the researcher to
have any control over the representativeness of the sample.
It is useful when the researcher wants to know the opinion of the
masses on a current issue; or the characteristics of people passing by
on streets at a certain point of time; or
if time and resources are limited in such a way that the research
would not be possible otherwise.
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Judgmental or Purposive Sampling
In this technique a sample is selected on the bases of the
knowledge of population and the purpose of the study.
For example, when an educational psychologist wants
to study the emotional and psychological effects of corporal
punishment, he will create a sample that will include only
those students who ever had received corporal punishment.

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 Snowball Sampling
This type of sampling is appropriate when the members of the populationare
difficult to locate, such as homeless industry workers, undocumented
immigrants etc.
In snowball sample, the researcher collects data on a few members of the
target population he or she can locate, then asks to locate those individuals to
provide information needed to locate other members of that population whom
they know.
For example, if a researcher wants to interview undocumented immigrants
from Afghanistan, he might interview a few undocumented individuals he
knows or can locate and from them take the address or location of those
7/16/2020 48
individuals.
 Quota Sampling
A quota sample is one in which units are selected into a
sample on the basis of pre-specified characteristics so that the
total sample has the same distribution of characteristics
assumed to exist in the population.
For example, if a researcher wants a national quota sample, he
might need to know what proportion of the population is male
and what proportion is the female, as well as what proportion
of each gender fall into different age category and educational
category. 7/16/2020 49
 References
www.statisticshowto.datasciencecentral.com
Educational Statistics Unit 2. Online Soft Books - Allama Iqbal Open University. www.aiou.edu.pk › Books
Soft

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7/16/2020 50
Self Assessment Activity
Q. 1. Define variable. Write commonly used types of variable?
Q. 2. What do you understand by the term “data”?
Q. 3. Write down the types of data?
Q. 4. What is population?
Q. 5. What do you understand by the target population?
Q. 6. What do you mean by the assessable population?
Q. 7. What do you mean by the term “sample”?
Q. 8. Write down the types of probability sampling.
Q. 9. Write down the types of non-probability sampling.
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7/16/2020 52
 BY:
Ehtesham Abbas
 Graphical representation of data is for the
purpose of easier interpretation.
 Facts and figures as such do not catch our
attention unless they are presented in an
interesting way.
 Graphical representation of data is the most
commonly used interesting modes of
presentation.
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 Bar charts are commonly used graphical
representations of data used to visually display to
compare values.
 They are easy to create and interpret.
 Data for a bar chart are entered in columns.
 Each numeric data value becomes a bar.
 X-axis represents the different categories, the y-axis
does have a scale and indicates the units of
measurement.

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 A pictogram is a kind of graph that uses pictures
instead of bars to represent data under analysis.
 A pictogram may include a symbol plus graphic
elements such as border, back pattern, or color that is
intended to covey specific information.
 Pictograms form a part of our daily lives. They are
used in transport, medication, education, computers.

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 A histogram is a type of graph that provides a visual
interpretation of numerical data by indicating the
number of data points that lie within the range value.
These range values are called classes.
 A histogram looks similar to bar charts. The height of
the bar corresponds to the relative frequency of the
amount of data in the class. The higher the bar is, the
greater the frequency of the data will bean vice versa.
The main difference between these graphs is the level
of measurement of the data. Bar graphs are used for
data at nominal level of measurement.
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 A bimodal shape has two peaks. This shape
may show that the data has come from two
different systems. Often in a single system,
there may be two modes in the data set
 Some histograms will show a skewed
distribution to the right, as shown below. A
distribution skewed to the right is said to be
positively skewed.

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 A distribution skewed to the left is said to be
negatively skewed. This kind of distribution
has a large number of occurrences in the
upper value cells (right side) and few in the
lower value cells (left side).
 A uniform distribution provides little information about
the system. It may describe a distribution which has
several modes (peaks).
 A uniform distribution often means that the number of
classes is too small.
 The frequency polygon is as graph that displays data by
using lines that connect points plotted for the
frequencies.
 This graph is useful for understanding the shape of
distribution.
 A frequency polygon is similar to histogram.
 The difference is that histogram tends to be rectangles
while a frequency polygon resembles a line graph.

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 A scatter plot is used to plot data in XY- plane to show how
much one variable or data set is affected by another.
 It has points that show the relationship between two
variables or two sets of data.
 These points are sometimes called markers and position of
these points depends on the values in the columns sets on
the XY axis.
 Scatter plot gives good visual picture of the relationship or
association between two variables or data sets, and aids to
interpretation of the correlation coefficient or regression
model
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 i) When there is paired numerical data.
 ii) When the dependent variable have multiple values
for each value of independent variable.
 iii) When the researcher tries to determine whether the
two variables are related

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 A pie chart displays data in an easy pie-slice format
with varying sizes.
 The size of a slice tells how much data exists in one
element.
 The bigger the slice, the more of that particular data
was gathered and vice versa.
 Pie charts are mainly used to show comparison
among various segments of data

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 These charts are used for displaying data that are
classified into nominal or ordinal categories
Educational Statistics
Descriptive Data Analysis

By:
Ehtesham Abbas
 An average is a single value, which represents
the set of data as whole. Since the average
tends to lie in the center of distribution they
are also called measure of central tendency.
There are three methods of measuring the
center of any data.
 Arithmetic mean
 The Median
 The Mode
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 It is defined as the sum of all the observations
divided by the number of observations. It is
denoted by X.
When to use Arithmetic Mean:
 We use arithmetic mean, when we are required
to study social, economic and commercial
problems like production, price, export and
import. It helps in getting average income,
average price, average production etc.
 5,
10, 12, 16, 8, 42, 25, 15, 10, 7
Solution:
5+10+12+16+8+42+25+15+10+7=150/10
Mean = 15

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 Median is the middle most value of a set of data
when the data is arranged in order of magnitude. If
the number of observations is in odd form, then
median is the mid value and if the number of
observations is even form, then median is the average
of two middle values.
When we Apply Median:
 We apply median to the situations, when the direct
measurements of variables are not possible like
poverty, beauty and intelligence etc.
 Example: 12,15, 10, 20, 18, 25, 45, 30,
26
 We need to make order of the data
 10, 12, 15, 18, 20, 25, 26, 30, 45
 So Mean = 20

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 The most frequent value that occurs in the set
of data is called mode. A set of data may have
more than one mode or no mode. When it has
one mode it is called uni-modal. When it has
two or three modes it is called bi-modal or tri-
modal respectively.
 Example:
 12, 24, 15, 18, 30, 48, 20, 24
 So Mode = 24
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 Measures of central tendency estimate normal
or central value of a data set, while measures
of dispersion are important for describing the
spread of the data, or its variation around a
central value.

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 A measure of dispersion indicates the scattering of
data. In other words, dispersion is the extent to
which values in a distribution differ from the average
of the distribution. It gives us an idea about the extent
to which individual items vary from one another, and
from the central value.

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 Measure # 1. Range:
 Measure # 2. Quartile Deviation:
 Measure # 3. Average Deviation (A.D.) or
Mean Deviation (M.D.):
 Measure # 4. Standard Deviation or S.D. and
Variance:

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 The range is the simplest measure of spread
and is the difference between the highest and
lowest scores in a data set. In other words we
can say that range is the distance between
largest score and the smallest score in the
distribution.

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The values that divide the given set of data into four
equal parts is called quartiles, and is denoted by Q1,
Q2, and Q3.

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MAD= mean average deviation
 Standard deviation is the most commonly used and
the most important measure of variation. It
determines whether the scores are generally near or
far from the mean.

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 It cannot be negative.
 It is only used to measure spread or
dispersion around the mean of a data set.
 For data with almost the same mean, the
greater the spread, the greater the
standard deviation.

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 Variancedescribes how much a
random variable differs from its
expected value.

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Mp = middle point

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 One way of presenting out how data
are distributed is to plot them in a
graph.
 If the data is evenly distributed, our
graph will come across a curve.
 In statistics this curve is called a
normal curve.
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 Skewness tells us about the amount and
direction of the variation of the data set.
 It is a measure of symmetry (evenness).
 A distribution or data set is symmetric if
it looks the same to the left and right of
the central point.

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 Kurtosis is a parameter that describes the shape of
variation.
 It is a measurement that tells us how the graph of the
set of data is peaked and how high the graph is around
the mean. In other words we can say that kurtosis
measures the shape of the distribution.
 The concept of kurtosis is very useful in decision-
making.

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Kurtosis has three types,
 mesokurtic, platykurtic, and leptokurtic.
 If the distribution has kurtosis of zero, then the graph
is nearly normal. This nearly normal distribution is
called mesokurtic.
 If the distribution has negative kurtosis, it is called
platykurtic.
 If the distribution has positive kurtosis, it is called
leptokurtic.
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 The coefficient of variation is another useful
statistics for measuring dispersion of a data
set.

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BY: Ehtesham Abbas
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 Inferential statistics is of vital importance in educational
research.
 It is used to make inferences about the population on
the bases of data obtained from the sample.
 inferential statistics helps researchers to make
generalization about a population based on the data
obtained from the sample.
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 Descriptive statistics only gives us the central values,
dispersion or the variability of the data
 But
 inferential statistics leads us to take a decision about the whole
population and in the end to any conclusion
 Inferential statistics enables us to infer from the data obtained
the sample what the population might think

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 Inferential statistics has two broad areas
 i) Estimating Parameter
 This means taking a statistics from the sample data (e.g. the sample mean)
and
 saying something about population parameter (e.g. the population mean).
 ii) Hypothesis testing
 This is where a researcher can use sample data to answer research
questions.
Inferential statistics deals with two or more than two variables. If in an
analysis there are two variables it is called bivariate analysis and if the
variables are more than two it is called multivariate analysis
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 The logic of inferential statistics starts with a
hypothesis about a relationship of two or more attributes
or concepts observed in a population

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 The purpose of inferential statistics is to
determine whether the findings from the sample
can generalize - or be applied - to the entire
population. There will always be differences in
scores between groups in a research study
 Inferential statistics allows us to
use what we've learned from descriptive statistics.
 Using data from the past and the resulting descriptive stats I.e.,
mean, mode, median, standard deviation, etc. infential statistics
extend beyond the immediate data.
 Inferential stats are used to infer from the sample data what the
population might think.
 Hypothesis testing is a statistical method that uses
sample data to evaluate a hypothesis aboutt a population
parameter (Gravetter & Wallnau, 2002)
 A hypothesis test is usually used in context of a research
study. Depending on the type of research and the type of
data, the details of the hypothesis test will change from
one situation to another.
 Hypothesis testing is a formalized procedure that
follows a standard series of operations
Inferential
statistics
 The process of hypothesis testing goes through
following four steps.
 i) Stating the Hypothesis
 The process of hypothesis testing begins by stating a
hypothesis about the unknown population.
 Usually, a researcher states two opposing hypotheses.
And both
hypotheses are stated in terms of population parameters.
 The first and most important of two hypotheses is called null hypothesis.
A null hypothesis states that the treatment has no effect.
 The null hypothesis is denoted by the symbol Ho (H stands for
hypothesis and 0 denotes that this is zero effect).
 The null hypothesis (Ho) states that in the general population there is no
change, no difference, or no relationship. In an experimental study, null
hypothesis (Ho) predicts that the independent variable (treatment) will
have no effect on the dependent variable for the population.
 The alternative hypothesis (H1) states that
there is a change, a difference, or a
relationship for the general population. In an
experiment, H1 predicts that the independent
variable (treatment) will have an effect on
the dependent variable.
In a common practice, a researcher uses the data from the sample to evaluate
the authority of null hypothesis.
The data will either support or deny the null hypothesis.
To formalize the decision process, a researcher will use null hypothesis
 to predict exactly what kind of sample should be obtained if the treatment
has no effect.
 In particular, a researcher will examine all the possible sample means that
could be obtained if the null hypothesis is true.
 The next step in hypothesis testing is to obtain the sample data.
 The raw data are summarized with appropriate statistics such as
mean, standard deviation etc.
 Then it is possible for the researcher to compare the sample
mean with the null hypothesis.
 In the final step the researcher decides, in the light of analysis
of data, whether to accept or reject the null hypothesis. If
analysis of data supports the null hypothesis, he accepts it and
vice versa.
 As hypothesis test uses information obtained from the
Sample in this situation, there is always the probability
of reaching incorrect conclusion.
 Generally two kinds of errors can be made
 i) Type I Errors
 A type I error occurs when a researcher rejects a null hypothesis that is actually
true. It means that the researcher concludes that the treatment does have an effect
when in fact the treatment has no effect.
 Type I error is not a stupid mistake in the sense that the researcher is overlooking
 something that should be perfectly obvious.
 ii) Type II Errors
 A type II error occurs when a researcher fails to reject the null hypothesis that is
really false. It means that a treatment effect really exists, but the hypothesis test
has failed to detect it. This type of error occurs when the effect of the treatment is
relatively small.
 A t-test is a useful statistical technique used for comparing
mean values of two data sets obtained from two groups.
 The comparison tells us whether these data sets are different
from each other.
 It further tells us how significant the differences are and if
these differences could have happened by chance.
 There are a number of t-test available but two main types
independent sample t-test and
paired sample t-test are most commonly used.
 i) Independent sample t-test
 Independent sample t-test is used when there are two different
independent groups
of people and the researcher is interested to compare their scores.
In this case the
 researcher collects information from two different groups of
people on only one occasion.
 ii)Paired sample t-test
 Paired sample t-test is also called repeated
measures.
 It is used when the researcher is interested in
comparing changes in the scores of the same group
tested at two different occasions
 A correlation is a relationship between two variables. The
purpose of using correlation in research is to determine the
degree to which a relationship exists between two or more
variables.
 observed as they naturally exist in the environment.
 Correlation requires two separate scores for each individual
(one score from each of two variables). These scores are
normally identified as X and Y and can be presented in a table
or in a graph.
 A correlation measures three characteristics of the relationship between X
and Y.
 i) The Direction of the Relationship
 The direction of the relationship can be classified into two basic categories:
positive and negative
 Inn a positive correlation both variables tend to change into same direction.
When variable X increases, the variable Y also increases. And if the
variable X decreases, the variable Y also decreases. In other words we can
say that both variables are directly proportional to each other
 In a negative correlation both variables do not tend to change into same
direction.
 They go in opposite direction of each other. When the variable X
increases, the variable Y decreases. And if the variable X decreases, the
variable Y increases. In other words we can say that both variables are
indirectly proportional to each other.
 The direction of the relationship is identified by the sign of the
correlation. A positive sign (+) indicates positive relationship. A negative
sign (−) indicates negative relationship.
 ii) The form of the Relationship
 The form of correlation measures how well the data fit the specific form being
considered.
 For example, a linear correlation measures how well the data points fit
 on a straight line
 iii) The Degree of the Relationship
 The degree of relationship is measured by the numerical value of the correlation.
 This value varies from 1.00 to – 1.00.
 A perfect correlation is always identified by a correlation of 1.00 and indicates a
perfect fit. + 1.00 will indicate perfect positive
 correlation and –1.00 will indicate perfect negative correlation.
 A correlation of 0 indicates no correlation or no fit at all.
 The most commonly used correlation is the Pearson Correlation. It is also known
as Pearson product-moment Correlation. It measures the degree and the
direction of linear relationship of between two variables. It is denoted by r, and r
= degree to which X and Y vary together / degree to which X and Y vary
separately = co-variability of X and Y / variability of X and Y vary separately
 To calculate the Pearson correlation we use the formula

 where SP is the sum of the product of deviation

 SS is sum of squares, SSx is the sum of squares of the variable X and SSy is the
sum of squares of variable Y
 i) Prediction
 If two variables are known to be related in some systematic way, it is
possible to use one variable to make prediction about the other.
 For example, when a student seeks admission in a college, he is required
to submit a great deal of personal information, including his scores in
SSC annual/supplementary examination. The college officials want this
information so that they can predict that student’s chance of success in
college.
 ii) Validity
One common technique for demonstrating validity is to use correlation.
iii) Reliability
A measurement procedure is reliable if it produces stable and consistent
measurement. It means a reliable measurement procedure will produce the
same (or
nearly same) scores when the same individuals are measured under the same
Conditions.
 iv) Theory Verification
 Many psychological theories make specific predictions about the
relationship between two variables.
 For example, a theory may predict a relationship between
brain size and learning ability; between the parent IQ and the child IQ etc. In
each
case, the prediction of the theory could be tested by determining the
correlation
between two variables
 iv) Theory Verification
 Many psychological theories make specific predictions about the
relationship between two variables.
 For example, a theory may predict a relationship between
brain size and learning ability; between the parent IQ and the child IQ etc. In
each
case, the prediction of the theory could be tested by determining the
correlation
between two variables
 The most commonly used measure of relationship is the Pearson
correlation. It measures the degree of linear relationship between two
variables and is used with interval or ratio data.
 i) The Spearman correlation is designed to measure the relationship
between variables measured on an ordinal scale of measurement.
 ii) The Spearman correlation is used when the researcher wants to
measure the consistency of a relationship between the variables X and Y
 .