Descriptive statistics generally characterizes or describes a set of data

elements by graphically displaying the information or describing its central

tendancies and how it is distributed.

The last half of the course will cover inferential statistics.

Inferential statistics tries to infer information about a population

by using information gathered by sampling.
Statistics: The collection of methods used in planning an experiment
and analyzing data in order to draw accurate conclusions.

General Terms Used Throughout Statistics

Population: The complete set of data elements is termed the population.

The term population will vary widely with its application. Examples could be
any of the following proper subsets: animals; primates; human beings; homo
sapiens; U.S. citizens; who are attending Andrews University, as graduate
students, in the School of Education, as Masters students, female, last name
starting with S, who web registered.

Sample: A sample is a portion of a population selected for further analysis.

How samples are obtained or types of sampling will be studied in lesson 7.

Most any of the examples above for population could serve as a sample for
the next higher level data set.

Parameter: A parameter is a characteristic of the whole population.

Statistic: A statistic is a characteristic of a sample, presumably measurable.

The plural of statistic just above is another basic meaning of statistics.

Assume there are 8 students in a particular statistics class, with 1 student

being male. Since 1 is 12.5% of 8, we can say 13% are male. The 13%
represents a parameter (not a statistic) of the class because it is based on
the entire population. If we assume this class is representative of all classes,
and we treat this 1 student as a sample drawn from a larger population, then
the 13% becomes a statistic.

Remember: Parameter is to Population as Statistic is
to Sample.

Inferential statistics is used to draw conclusions about a population by

studying a sample. It is not guesswork! We test hypotheses about a
parameter's value with a certain risk of being wrong. That risk is carefully
specified. Also, descriptive and inferential statistics are not mutually
exclusive. The inferences made about a population from a sample help
describe that population. We also tend to use Roman letters for statistics and
Greek letters for parameters.

Basic Mathematics for Statistics

This course will avoid complex models utilizing complicated mathematics.

You will need to be familiar with, however, the fundamental arithmetic
operations, elementary algebra, and some basic symbolism.

An interesting subset of the natural numbers generated by addition are

called Triangular Numbers. These are so called because these are the total
number of dots, if we arrange the dots in a triangle with one additional dot in
each layer.

•
• •
• • •
• • • •
The triangular numbers thus are: 0, 1, 3, 6, 10, 15, 21, ....

Suppose we wish to add together the first 100 natural numbers, which is
equivalent to finding the 100th triangular number. One way to do this is by
grouping them as follows:

(1 +100) + (2+99) + (3 + 98) + ... +

T100 =
(50 + 51)
=101• 50
=101• 100/2

In general we write:   where

mathematicians use the capital Greek letter   (sigma) to
represent summation. Your teacher has a particular fondness for this
symbol since the first computer he had much access to had that nickname.

There are three important rules for using the summation operator:
 1. Since multiplication distributes over addition, the sum of a constant
times a set of numbers is the same as the constant times the sum of
the set of numbers.

Example: Cx1 + Cx2 + ... + Cxn = C(x1 + x2 + ...+ xn)

2. The sum of a series of constants is the same as N times the constant,

where N represents how many constants there are.

Example: 4 + 4 + 4 + 4 + 4 = 5 × 4 = 20.

3. Since addition is commutative, the total sum of two or more scores for
several individuals can be achieved either by summing the scores
separately and then combining them or by summing an individual's
scores and then combining them.

Example: Joe got scores of 500 and 550 for his verbal and quantitative
SAT scores whereas Jim got scores of 520 and 510, respectively. 500 +
550 + 520 + 510 = 1050 + 1030 = 500 + 520 + 550 + 510 = 1020 +
1060 = 2090.

In addition to the operations of addition, subtraction, multiplication, and

division, several other arithmetic operators often
appear. Exponentiation and absolute value are two such. Also, various
symbols of inclusion (parentheses, brackets, braces, vincula) are used.

Exponentiation is a general term which includes squaring (122=144), cubing

(63=216), and square roots (16½= (16)=4. When the square root symbol
(surd and symbol of inclusion, in recent history a vinculum, but historically
parentheses) is used, we general (although not quite always) mean only the
positive square root.

The absolute value operator indicates the distance (always non-negative) a

number is from the origin (zero). The symbol used is a vertical line on either
side of the operand. Thus, if x>0, then |x|=x, if x<0, then |x|=-x, and if x=0,
|x|=0.  (x2)=|x|.

There is a proscribed order for arithmetic operations to be performed.

Example: If we write 4 × 5 + 3 it is conventional to multiply the 4 and 5

together before adding the 3 and thus obtain 23. Some calculators are
algebraic and handle this appropriately, others do not.
Parentheses and other symbols of inclusion are used to modify the normal
order of operations. We say these symbols of inclusion have the highest
priority or precidence.

Exponentiation is done next. There is confusion when exponents are stacked

which we will not deal with here except to say computer scientists tend to do
it from left to right while mathematicians know that is wrong.

Multiplication and Division are done next, in order, from left to right.

Addition and Subtraction are done next, in order, from left to right.

A mnemonic such as Please Eat Miss Daisy's Apple Sauce can be useful for

remembering the proper order of operation.

Accuracy vs. Precision

The distinction between accuracy and precision, reviewed in Numbers lesson

9, is very important.

This ties in with significant figures, and proper rounding of results. I have
several major concerns regarding significant digits.

1. There needs to be sufficient (not to few). Slide rule accuracy or three

significant digits has a long-standing precident in science. We are not
doing science here so two may suffice, but rarely one.
2. There should not be too many significant digits. Generally, more than 5
is probably a joke, especially in the "softer" sciences. Thus
representing 1/3 or 1/7 with infinite precision (by indicating the
repeated unit) should not occur.
3. Care must be taken so that a primary statistics (such as variance) is
not incorrectly derived from a secondary statistic (such as standard
deviation) in such a way that accuracy is lost. We will discuss this more
in textbook Chapter 3.
4. A mean and standard deviation or mean and margin of error should be
given to compatible precision.
5. There are proper rules, but they are difficult to explain to the general
public. Thus every statistics book gives its own heuristic.

Uses and Abuses of Statistics

Most of the time, samples are used to infer something (draw conclusions)
about the population. If an experiment or study was done cautiously and
results were interpreted without bias, then the conclusions would be
accurate. However, occasionally the conclusions are inaccurate or
inaccurately portrayed for the following reasons:

 Sample is too small.

 Even a large sample may not represent the population.
 Unauthorized personnel are giving wrong information that the public
will take as truth. A possibility is a company sponsoring a statistics
research to prove that their company is better.
 Visual aids may be correct, but emphasize different aspects. Specific
examples include graphs which don't start at zero thus exaggerating
small differences and charts which misuse area to represent
proportions. Often a chart will use a symbol which is both twice as long
and twice as high to represent something twice as much. The area, in
this case however, is four times as much!
 Precise statisitics or parameters may incorrectly convey a sense of
high accuracy.
 Misleading or unclear percentages are often used.

Statistics are often abused. Many examples could be added, (even books
have been written) but it will be more instructive and fun to find them on
your own.

Types of Data

A dictionary defines data as facts or figures from which conclusions may be

drawn. Thus, technically, it is a collective, or plural noun. Some recent
dictionaries acknowledge popular usage of the word data with a singular
verb. However we intend to adhere to the traditional "English" teacher
mentality in our grammar usage—sorry if "data are" just doesn't sound quite
right! (My mother and step-mother were both English teachers, so clearly no
offense is intended above.) Datum is the singular form of the noun data.
Data can be classified as either numeric or nonnumeric. Specific terms are
used as follows:

1.

Qualitative data are nonnumeric.

2. {Poor, Fair, Good, Better, Best}, colors (ignoring any physical causes),
and types of material {straw, sticks, bricks} are examples of
qualitative data.
3. Qualitative data are often termed catagorical data. Some books use
the terms individual and variable to reference the objects and
characteristics described by a set of data. They also stress the
importance of exact definitions of these variables, including what units
 they are recorded in. The reason the data were collected is also
important.
4.

Quantitative data are numeric.

5. Quantitative data are further classified as either discrete or

continuous.
o

Discrete data are numeric data that have a finite number of

possible values.

o A classic example of discrete data is a finite subset of the

counting numbers, {1,2,3,4,5} perhaps corresponding to
{Strongly Disagree... Strongly Agree}.
o Another classic is the spin or electric charge of a single
electron. Quantum Mechanics, the field of physics which deals
with the very small, is much concerned with discrete values.
o When data represent counts, they are discrete. An example
might be how many students were absent on a given day.
Counts are usually considered exact and integer. Consider,
however, if three tradies make an absence, then aren't two
tardies equal to 0.67 absences?
o

Continuous data have infinite possibilities: 1.4, 1.41, 1.414,

1.4142, 1.141421...
The real numbers are continuous with no gaps or interruptions. Physically
measureable quantities of length, volume, time, mass, etc. are generally
considered continuous. At the physical level (microscopically), especially for
mass, this may not be true, but for normal life situations is a valid
assumption.

The structure and nature of data will greatly affect our choice of analysis
method. By structure we are referring to the fact that, for example, the data
might be pairs of measurements. Consider the legend of Galileo dropping
weights from the leaning tower of Pisa. The times for each item would be
paired with the mass (and surface area) of the item. Something which Galileo
clearly did was measure the time it took a pendulum to swing with various
amplitudes. (Galileo Galilei is considered a founder of the experimental
method.)

Levels of Measurement
The experimental (scientific) method depends on physically measuring
things. The concept of measurement has been developed in conjunction with
the concepts of numbers and units of measurement. Statisticians categorize
measurements according to levels. Each level corresponds to how this
measurement can be treated mathematically.

1.

Nominal: Nominal data have no order and thus only gives names or

labels to various categories.
3.

Ordinal: Ordinal data have order, but the interval between

measurements is not meaningful.
5.

Interval: Interval data have meaningful intervals between

measurements, but there is no true starting point (zero).
7.

Ratio: Ratio data have the highest level of measurement. Ratios

between measurements as well as intervals are meaningful because
there is a starting point (zero).

Nominal comes from the Latin root nomen meaning name. Nomenclature,

nominative, and nominee are related words. Gender is nominal. (Gender is
something you are born with, whereas sexis something you should get a
license for.)

Example 1: Colors
To most people, the colors: black, brown, red, orange, yellow, green, blue,
violet, gray, and white are just names of colors.

To an electronics student familiar with color-coded resistors, this data is in

ascending order and thus represents at least ordinal data.

To a physicist, the colors: red, orange, yellow, green, blue, and violet
correspond to specific wavelengths of light and would be an example of ratio
data.