Statistics Discussion
Copyright © October 2004 Ted Nissen
TABLE OF
CONTENTS
1 Introduction. 1
2 Review and More Introduction. 6
3 Central Values and the Organization of Data. 19
4 VARIABILITY. 27
5 Correlation and Regression. 29
6 SCORE TRANSFORMATIONS. 39
7 LINEAR TRANSFORMATIONS. 41
8 Theoretical Distributions
Including the Normal Distribution. 41
9 Samples and Sampling Distributions. 48
10 Differences between Means. 62
11 The t Distribution and the
t-Test 80
12 Analysis of Variance: One-Way
Classification. 116
13 Analysis of Variance: Factorial
Design. 137
14 The Chi Square Distribution. 162
15 Nonparametric Statistics. 163
16 Vista Formulas and Analysis. 186
17 Hypothesis. 197
18 Summary. 197
19 Glossary. 197
1.1
1.2 Statistics Definition
1.2.1 Algebra and Statistics
1.2.1.1
Algebra
is a generalization of arithmetic in which letters representing numbers are
combined according to the rules of arithmetic
1.2.1.2
The
product of an algebraic expression, which combines several scores, is a
statistic.[1]
1.2.2 Descriptive Statistic
1.2.2.1
1.2.3 Inferential Statistics
1.2.3.1
1.3 Purpose of Statistics
1.3.1
1.4 Terminology
1.4.1
1.4.2 Populations, Samples, and
Subsamples
1.4.2.1
A
population consists of all members of some specified group. Actually, in
statistics, a population consists of the measurements on the members and not
the members themselves. A sample is a subset of a population. A subsample is a
subset of a sample. A population is arbitrarily defined by the investigator and
includes all relevant cases.
1.4.2.2
Investigators
are always interested in some population. Populations are often so large that
not all the members can be measured. The investigator must often resort to
measuring a sample that is small enough to be manageable but still
representative of the population.
1.4.2.3
Samples
are often divided into subsamples and relationships among the subsamples
determined. The investigator would then look for similarities or differences
among the subsamples.
1.4.2.4
Resorting
to the use of samples and subsamples introduces some uncertainty into the
conclusions because different samples from the same population nearly always
differ from one another in some respects. Inferential statistics are used to
determine whether or not such differences should be attributed to chance.
1.4.3 Parameters and Statistics
1.4.3.1
A
parameter is some numerical characteristic of a population. A statistic is some
numerical characteristic of a sample or subsample. A parameter is constant; it
does not change unless the population itself changes. There is only one number
that is the mean of the population; however, it often cannot be computed,
because the population is too large to be measured. Statistics are used as
estimates of parameters, although, as we suggested above, a statistic tends to
differ from one sample to another. If you have five samples from the same
population, you will probably have five different sample means. Remember that
parameters are constant; statistics are variable.
1.4.4 Variables
1.4.4.1
A
variable is something that exists in more than one amount or in more than one
form. Memory is a variable. The Wechsler Memory Scale is used to measure
peoples memory ability, and variation is found among the memory scores of any
group of people. The essence of measurement is the assignment of numbers on the
basis of variation.
1.4.4.2
Most
variables can be classified as quantitative variables. When a quantitative
variable is measured, the scores tell you something about the amount or degree
of the variable. At the very least, a larger score indicates more of the
variable than a smaller score does.
1.4.4.3
A
score has a range consisting of an upper limit and lower limit, which defines
the range. For example, 103=102.5-103.5, the numbers 102.5 and 103.5 are called
the lower limit and the upper limit of the score. The idea is that a score can
take any fractional value between 102.5 and 103.5, but all scores in that range
are rounded off to 103.
1.4.4.4
Some
variables are qualitative variables. With such variables, the scores (number)
are simple used as names; they do not have quantitative meaning. For example,
political affiliation is a qualitative variable.
1.5 Scales of Measurement
1.5.1 Introduction
1.5.1.1
Numbers
mean different things in different situations. Numbers are assigned to objects
according to rules. You need to distinguish clearly between the thing you are
interested in and the number that symbol9izes or stands for the thing. For
example, you have had lots of experience with the numbers 2 and 4. You can
state immediately that 4 is twice as much as 2. That statement is correct if
you are dealing with numbers themselves, but it may or may not be true when
those numbers are symbols for things. The statement is true if the numbers
refer to apples; four apples are twice as many as two apples. The statement is
not true if the numbers refer to the order that runners finish in a race.
Fourth place is not twice anything in relation to second place-not twice as
slow or twice as far behind the first-place runner. The point is that the
numbers 2 and 4 are used to refer to both apples and finish places in a race,
but the numbers mean different things in those two situations.
1.5.1.2
S.
S. Stevens (1946)[2] identified
four different measurement scales that help distinguish different kings of
situation in which numbers are assigned to objects. The four scales are;
nominal, ordinal, interval, and ratio.
1.5.2 Nominal Scale
1.5.2.1
Numbers
are used simply as names and have no real quantitative value. It is the scale
used for qualitative variables. Numerals on sports uniforms are an example;
here, 45 is different from 32, but that is about all we can say. The person
represented by 45 is not more than the person represented by 32, and
certainly it would be meaningless to try to add 45 and 32. Designating
different colors, different sexes, or different political parties by numbers
will produce nominal scales. With a nominal scale, you can even reassign the
numbers and still maintain the original meaning, which as only that the
numbered things differ. All things that are alike must have the same number.
1.5.3 Ordinal Scale
1.5.3.1
An
ordinal scale, has the characteristic of the nominal scale (different numbers
mean different things) plus the characteristic of indicating greater than or
less than. In the ordinal scale, the object with the number 3 has less or
more of something than the object with the number 5. Finish places in a race
are an example of an ordinal scale. The runners finish in rank order, with 1
assigned to the winner, 2 to the runner-up, and so on. Here, 1 means less
time than 2. Other examples of ordinal scales are house number, Government
Service ranks like GS-5 and GS-7, and statements like She is a better
mathematician than he is.
1.5.4 Interval Scale
1.5.4.1
The
interval scale has properties of both the ordinal and nominal scales, plus the
additional property that intervals between the numbers are equal. Equal
interval means that the distance between the things represented by 2 and 3 is the same as the distance
between the things represented by 3 and 4. The centigrade thermometer is
based on an interval scale. The difference is temperature between 10° and 20°
is the same as the difference between 40° and 50°. The centigrade thermometer,
like all interval scales, has an arbitrary zero point. On the centigrade, this
zero point is the freezing point of water at sea level. Zero degrees on this
scale does not mean the complete absence of heat; it is simply a convenient
starting point. With interval data, we have one restriction; we may not make
simple ratio statements. We may not say that 100° is twice as hot as 50° or
that a person with an IQ of 60 is half as intelligent as a person with an IQ of
120.
1.5.5 Ratio Scale
1.5.5.1
The
fourth kind of scale, the ratio scale, has all the characteristics of the
nominal, ordinal, interval scales, plus one: it has a true zero point, which
indicates a complete absence of the thing measured. On a ratio scale, zero
means none. Height, weight, and time are measured with ratio scales. Zero
height, zero weight, and zero time mean thaqt no amount of these variables is
present. With a true zero point, you can make ratio statements like 16
kilograms is four times heavier than 4 kilograms.
1.5.6 Conclusion
1.5.6.1
Having
illustrated with examples the distinctions among these four scales-it is
sometimes difficult to classify the variables used in the social and behavioural
sciences. Very often they appear to fall between the ordinal and interval
scales. It may happen that a score provides more information than simply rank,
but equal intervals cannot be proved. Intelligence test scores are an example.
In such cases, researchers generally treat the data as if they were based on an
interval scale.
1.5.6.2
The
main reason why this section on scales of measurement is important is that the
kind of descriptive statistics you can compute on your numbers depends to some
extent upon the kind of scale of measurement the numbers represent. For
example, it is not meaningful to compute a mean on nominal data such as the
numbers on football players jerseys. If the quarterbacks number is 12 and a
running backs number is 23, the mean of the two numbers (17.5) has no meaning
at all.
1.6 Statistics and Experimental
Design
1.6.1 Introduction
1.6.1.1
Statistics
involves the manipulation of numbers and the conclusions based on those
manipulations. Experimental design deals with how to get the numbers in the first
place.
1.6.2 Independent and Dependent
Variables
1.6.2.1
In
the design of a typical simple experiment, the experimenter is interested in
the effect that one variable (called the independent variable) has on some
other variable (called the dependent variable). Much research is designed to
discover cause-and-effect relationships. In such research, differences in the
independent variable are the presumed cause for differences in the dependent
variable. The experimenter chooses values for the independent variable, administers
a different value of the independent variable to each group of subjects, and
then measures the dependent variable for each subject. If the scores on the
dependent variable differ as a result of differences in the independent
variable, the experimenter may be able to conclude that there is a
cause-and-effect relationship.
1.6.3 Extraneous (Confounding)
Variables
1.6.3.1
One
of the problems with drawing cause-and-effect conclusions is that you must be
sure that changes in the scores on the dependent variable are the result of
changes in the independent variable and not the result of changes in some other
variables. Variables other than the independent variable that can cause changes
in the dependent variable are called extraneous variables.
1.6.3.2
It
is important, then, that experimenters be aware of and control extraneous
variables that might influence their results. The simplest way to control an
extraneous variable is to be sure all subjects are equal on that variable.
1.6.3.3
Independent
variables are often referred to as treatments because the experimenter
frequently asks If I treat this group of subjects this way and treat another
group another way, will there be a difference in their behaviour? The ways
that the subjects are treated constitute the levels of the independent variable
being studied, and experiments typically have two or more levels.
1.7 Brief History of Statistics
1.7.1
2.1 Review of Fundamentals
2.1.1 This section is designed to
provide you with a quick review of the rules of arithmetic and simple algebra.
We recommend that you work the problems as you come to them, keeping the
answers covered while you work. We assume that you once knew all these rules
and procedures but that you need to refresh your memory. Thus, we do not
include much explanation. For a textbook that does include basic explanations,
see Helen M. Walker.[3]
2.1.2 Definitions
2.1.2.1
Sum
2.1.2.1.1
The
answer to an addition problem is called a sum. In Chapter
12, you will calculate a sum
of squares, a quantity that is obtained by adding together some squared
numbers.
2.1.2.2
Difference
2.1.2.2.1
The answer to a subtraction problem is called a difference.
Much of what you will learn in statistics deals with differences and the extent
to which they are significant. In Chapter 10, you will encounter a
statistic called the standard error of a
difference. Obviously, this statistic involves subtraction.
2.1.2.3
Product
2.1.2.3.1
The answer to a multiplication problem is called a product. Chapter
7 is about the product-moment
correlation coefficient, which requires multiplication. Multiplication
problems are indicated either by an x or by parentheses. Thus, 6 x 4 and (6)(4)
call for the same operation.
2.1.2.4
Quotient
2.1.2.4.1 The
answer to a division problem is called a quotient. The IQ or intelligence quotient is based on the
division of two numbers. The two ways to indicate a division problem are 
and . Thus, 9 
4 and 9/4 call for
the same operation. It is a good idea to
think of any common fraction as a division problem. The numerator is to be
divided by the denominator.
2.1.3 Decimals
2.1.3.1
Addition and Subtraction of
Decimals.
2.1.3.1.1
There is only one rule about the addition and subtraction of
numbers that have decimals: keep .the
decimal points in a vertical line. The decimal point in the answer goes
directly below those in the problem. This rule is illustrated in the five
problems below.
2.1.3.1.2 Example #1
2.1.3.1.2.1 
2.1.3.2
Multiplication of Decimals
2.1.3.2.1
The basic rule for multiplying decimals is that the number of
decimal places in the answer is found by adding up the number of decimal places
in the two numbers that are being multiplied. To place the decimal point in the
product, count from the right.
2.1.3.2.2 Example #2
2.1.3.2.2.1 
2.1.3.3
Division of Decimals
2.1.3.3.1
Two methods have been used to teach division of decimals. The
older method required the student to move the decimal in the divisor (the
number you are dividing by) enough places to the right to make the divisor a
whole number. The decimal in the dividend was then moved to the right the same
number of places, and division was carried out in the usual way. The new
decimal places were identified with carets, and the decimal place in the
quotient was just above the caret in the dividend. For example,
2.1.3.3.2 Example # 3a&b
2.1.3.3.2.1 
2.1.3.3.2.2 
2.1.3.3.3 The newer method of teaching the division of
decimals is to multiply both the divisor and the dividend by the number that
will make both of them whole numbers. (Actually, this is the way the caret
method works also.) For example:
2.1.3.3.4 Example #4
2.1.3.3.4.1 
2.1.3.3.5
Both of these methods work. Use the one you are more familiar
with.
2.1.3.4
2.1.4 Fractions
2.1.4.1
In
general, there are two ways to deal with fractions
2.1.4.1.1 Convert the fraction to a decimal and
perform the operations on the decimals
2.1.4.1.2
Work directly with
the fractions, using a set of rules for each operation. The rule for addition
and subtraction is: convert the fractions to ones with common denominators, add
or subtract the numerators, and place the result over the common denominator.
The rule for multiplication is: multiply the numerators together to get the
numerator of the answer, and multiply the denominators together for the
denominator of the answer. The rule for division is: invert the divisor and multiply
the fractions.
2.1.4.1.3
For statistics problems, it is usually easier to convert the
fractions to decimals and then work with the decimals. Therefore, this is the
method that we will illustrate. However, if you are a whiz at working directly
with fractions, by all means continue with your method. To convert a fraction
to a decimal, divide the lower number into the upper one. Thus, 3/4 = .75, and
13/17 = .765
2.1.4.1.4 Examples Fractions
2.1.4.1.4.1 
2.1.5 Negative Numbers
2.1.5.1
Addition
of Negative numbers
2.1.5.1.1 Any number without a sign is understood to
be positive
2.1.5.1.2 To add a series of negative numbers, add
the numbers in the usual way, and attach a negative sign to the total
2.1.5.1.3 Example #1
2.1.5.1.3.1 
2.1.5.1.4 To add two numbers, one positive and one negative,
subtract the smaller number from the larger and attach the sign of the larger
to the result
2.1.5.1.5 Example #2
2.1.5.1.5.1 
2.1.5.1.6 To add a series of numbers, of which some
are positive and some negative, add all the positive numbers together, all the
negative numbers together (see above) and then combine the two sums (see above)
2.1.5.1.7 Example #3
2.1.5.1.7.1 
2.1.5.2
Subtraction
of Negative Numbers
2.1.5.2.1 To subtract a negative number, change it to
positive and add it. Thus
2.1.5.2.2 Example #4
2.1.5.2.2.1 
2.1.5.3
Multiplication
of Negative Numbers
2.1.5.3.1 When the two numbers to be multiplied are
both negative, the product is positive
2.1.5.3.2 (-3)(-3)=9 (-6)(-8)=48
2.1.5.3.3 When one of the number is negative and the
other is positive, the product is negative
2.1.5.3.4 (-8)(3)=-24 14 X 2= -28
2.1.5.4
Division
of Negative Numbers
2.1.5.4.1 The rule in division is the same as the
rule in multiplication. If the two numbers are both negative, the quotient is
positive
2.1.5.4.2 (-10) 
(-2)=-5 (-4) 
(-20)= .20
2.1.5.4.3 If one number is negative and the other
positive, the quotient is negative
2.1.5.4.4 (-10) 
2= -5 6 
(-18)= -.33
2.1.5.4.5 14 
(-7)= -2 (-12) 
3= -4
2.1.6 Proportions and Percents
2.1.6.1
A proportion is a part of a whole and
can be expressed as a fraction or as a decimal. Usually, proportions are
expressed as decimals. If eight students in a class of 44 received A's, we may
express 8 as a proportion of the whole (44). Thus, 8/44, or .18. The proportion
that received A's is .18.
2.1.6.2
To convert a proportion to a percent
(per one hundred), multiply by 100. Thus: .18 x 100 = 18; 18 percent of the
students received A's. As You can see proportions and percents are two ways to
express the same idea.
2.1.6.3
If
you know a proportion (or percent) and the size of the original whole, you can
find the number that the proportion represents. If .28 of the students were
absent due to illness,
and there are 50 students in all, then. 28 of the 50 were absent. (.28)(50) = 14 students who were absent. Here are some
more examples.
2.1.6.4
Example
Proportions and Percents
2.1.6.4.1 
2.1.7 Absolute Value
2.1.7.1
The absolute value of a number ignores the sign of the number. Thus, the absolute value of -6 is 6. This is expressed
with symbols as |-6| =
6. It is expressed
verbally as "the absolute value of negative six is
six. " In a similar way, the absolute value of 4 - 7 is 3. That is, |4 7| =
| - 3| = 3.
2.1.8 
Problems
2.1.8.1
A 
sign ("plus or minus"
sign) means to both add and subtract. A 
problem always has two answers.
2.1.8.2
Example
Plus-Minus Problems
2.1.8.2.1 
2.1.9 Exponents
2.1.9.1
.In the expression 52, 2 is
the exponent. The 2 means that 5 is to be multiplied by itself. Thus, 52
= 5 x 5 = 25.
2.1.9.2
In
elementary statistics, the only exponent used is 2, but it will be used
frequently. When a number has an exponent of 2, the number is said to be
squared. The expression 42 (pronounced "four squared")
means 4 x4, and the product is 16. The squares of whole numbers between 1 and
1000 can be found in Tables in the Appendix of most stats text books.
2.1.9.3
Example
Exponents
2.1.9.3.1 
2.1.10
Complex
Expressions
2.1.10.1
Two
rules will suffice for the kinds of complex expressions encountered in
statistics.
2.1.10.1.1
Perform the
operations within the parentheses first. If there are brackets in the
expression, perform the operations within the parentheses and then the
operations within the brackets.
2.1.10.1.2
Perform the
operations in the numerator and those in the denominator separately, and
finally, carry out the division.
2.1.10.1.3
Example 5
2.1.10.1.3.1
2.1.11
Simple
Algebra
2.1.11.1
To
solve a simple algebra problem, isolate the unknown (x) on one side of the
equal sign and combine the numbers on the other side. To do this, remember that
you can multiply or divide both sides of the equation by the same number without
affecting the value of the unknown. For example,
2.1.11.2
Example
6 a & b
2.1.11.2.1
2.1.11.2.2
2.1.11.3
In
a similar way, the same number can be added to or subtracted from both sides of
the equation without affecting the value of the unknown.
2.1.11.4
Example
7
2.1.11.4.1
2.1.11.5
We
will combine some of these steps in the problems we will work for you. Be sure
you see shat operation is being performed on both sides in each step
2.1.11.6
Example
8
2.1.11.6.1
2.2 Rules, Symbols, and Shortcuts
2.2.1 Rounding Numbers
2.2.1.1
There
are two parts to the rule for rounding a number. If the digit that is to be
dropped is less than 5, simply drop it. If the digit to be dropped is 5 or
greater, increase the number to the left of it by one. These are the rules
built into most electronic calculators. These two rules are illustrated below
2.2.1.2
Example
9 a & B
2.2.1.2.1 
2.2.1.2.2 
2.2.1.3
A reasonable question is "How
many decimal places should an answer in statistics have?" A good rule of
thumb in statistics is to carry all operations to three decimal places and
then, for the final answer, round back to two decimal places.
2.2.1.4
Sometimes this rule of thumb could get
you into trouble, though. For example, if half way through some work you had a
division problem of .0016 
.0074, and if you
dutifully rounded those four decimals to three (.002 
.007), you would get
an answer of .2857, which becomes .29. However, division without rounding gives
you an answer of .2162 or .22. The difference between .22 and .29 may be quite
substantial. We will often give you cues if more than two decimal places are
necessary but you will always need to be alert to the problems of rounding.
2.2.2 Square Roots
2.2.2.1
Statistics problems often require that
a square root be found. Three possible solutions to this problem are
2.2.2.1.1
A calculator with a square-root key
2.2.2.1.2
The paper-and-pencil method
2.2.2.1.3
Use a Table the back of a statistics book.
2.2.2.2
Of
the three, a calculator provides the quickest and simplest way to find a square
root. If you have a calculator, you're set. The paper -and-pencil method is
tedious and error prone, so we will not discuss it. We'll describe
the use of Tables and we recommend that you use it if you don't have access to
a calculator.
2.2.2.3
Three
Digit Numbers
2.2.2.3.1
If you need the square root of a three-digit number (000 to
999), a table will give it to you directly. Simply look in the left-hand column
for the number and ad the square root in the third column, under 
. For example;
the square root of 225 is 15.00, and 
= 8.37. Square roots
are usually carried (or rounded) to two decimal places.
2.2.2.4
Numbers
between 0 and 10
2.2.2.4.1
For numbers between 0 and 10 that have two decimal places
(.01 to 9.99), The tables will give you the square root. Find your number in the left-hand column by
thinking of its decimal point as two places to the right. Find the square root
in the 
column by moving the
decimal point one place to the left. For example, 
= 1.50. Be sure you understand how these square roots were
found: 
= 2.52, 
= .66, and 
= .28.
2.2.2.5
Numbers
between 10 and
1000 That Have Decimals
2.2.2.5.1 For
numbers between 10 and 1000 with decimals interpolation is necessary. To
interpolate a value for 
, find a value that is half way (.5 of the distance) between 
and 
. Thus, the square root of 22.5 will be
(approximately) half way between 4.69 and 4.80, which is 4.74. For a second
example, we will find 
. 
= 9.17, and 
=9.22. 
will be .35 into the interval between 
and
. That interval is .05 (9.22 - 9.17). Thus' (.35)(.05) = .02, and 
= 9.17 + .02 = 9.19. Interpolation is also necessary with numbers between 100 and 1000 that have decimals;
these can usually be estimated rather quickly because the difference between
the square roots of the whole numbers is so small. Look at the difference
between 
and 
, for example.
2.2.2.6
Numbers
Larger Than 1000
2.2.2.6.1
For numbers larger than 1000, the square root can be estimated fairly closely by using the
second column in Table A (N 2). Find the large number under N2,
and read the square root from the N column. For example, 
= 123, and 
=34. Most large
numbers you encounter will not be found in the N2 column, and you
will just have to estimate the square root as closely as possible.
2.2.3 Reciprocals
2.2.3.1
This section is about a professional
shortcut. This shortcut is efficient if multiplication is easier for you than
division. If you prefer to divide rather than multiply, skip this section.
2.2.3.2
A
reciprocal of a number (N) is 1/N. Multiplying a number by 1/N
is equivalent to dividing it by N. For example, 8
2 = 8 x (1/2) = 8 x .5 = 4.0; 25 -7- 5 = 25 x (1/5) = 25 x .20 = 5.0. These examples are
easy, but we can also illustrate with more difficult
problems; 541
98 =
541 x (1/98) = 541 x
.0102 = 5.52. So far, this should be clear, but there
should be one nagging question. How did we know that 1/98 = .01O2? The answer
is the versatile Table A. Table A contains a column 1/N, and, by looking
up 98, you will find that 1/98 = .0102.
2.2.3.3
If
you must do many division problems on paper, we recommend reciprocals to you.
If you have access to an electronic calculator, on the other hand, you won't
need the reciprocals in Table A.
2.2.4 Estimating Answers
2.2.4.1
Just
looking at a problem and making an estimate of the answer before you do any
calculating is a very good idea. This is referred to as eyeballing the data and
Edward Minium (1978) has captured its importance with Minium's First Law of
Statistics: "The eyeball is the statistician's most powerful
instrument."
2.2.4.2
Estimating answers should keep you
from making gross errors, such as misplacing a decimal point. For example,
31.5/5 can be estimated as a little more than 6 If
you make this estimate before you divide, you are likely to recognize that an
answer of 63.or .63 is incorrect.
2.2.4.3
The
estimated answer to the problem (21)(108) is 2000, since (20)(100) = 2000.
2.2.4.4
The
problem (.47)(.20) suggests an estimated answer of .10, since (1/2)(.20) = .10.
With .10 in mind, you are not likely to write.94 for the answer, which is .094.
Estimating answers is also important if you are finding a square root. You can
estimate that 
is about
10, since 
= 10; 
is about 1.
2.2.4.5
To
calculate a mean, eyeball the numbers and estimate the mean. If you estimate a
mean of 30 for a group of numbers that are primarily in the 20s, 30s, and 40s,
a calculated mean of 60 should arouse your suspicion that you have made an
error.
2.2.5 Statistical Symbols
2.2.5.1
Although as far as we know,
there has never been a clinical case of neoiconophobia (An extreme and unreasonable fear of
new symbols) some
students show a mild form of this behavior. Symbols like 
, 
,
and 
may cause a
grimace, a frown, or a droopy eyelid. In more severe cases, the behavior
involves avoiding a statistics course entirely. We're rather sure that you
don't have such a severe case, since you have read this far. Even so, if you
are a typical beginning student in statistics, symbols like (
, 
,
and 
are not very meaningful to you, and they may
even elicit feelings, of uneasiness. We also know from our teaching experience
that, by the end of the course, you will know what these symbols mean and be
able to approach them with an unruffled psyche-and perhaps even approach them
joyously. This section should help you over that initial, mild neoiconophobia,
if you suffer from it at all.
2.2.5.2
Below are definitions and
pronunciations of the symbols used in the next two chapters. Additional symbols will be defined
as they occur. Study this list until you know it.
2.2.5.3
Symbols
2.2.5.3.1 
2.2.5.4
Pay careful attention to symbols. They
serve as shorthand notations for the ideas and concepts you are learning. So,
each time a new symbol is introduced, concentrate on it-learn it-memorize its
definition and pronunciation. The more meaning a symbol has for you, the
better you understand the concepts it represents and, of course, the easier the
course will be.
2.2.5.5
Sometimes
we will need to distinguish between two different (
's or two X's. We will use subscripts, and the
results will look like 
1 and 
2, or X1 and X2. Later, we will use subscripts other than
numbers to identify a symbol. You will see 
x and erg. The point to learn here is
that subscripts are for identification purposes only; they never indicate
multiplication. 
does
not mean (
)(
).
2.2.5.6
Two
additional comments-to encourage and to caution you. We encourage you to do
more in this course than just read the text, work the problems, and pass the
tests, however exciting that may be. We encourage you to occasionally get
beyond this elementary text and read journal articles or short portions of
other statistics textbooks. We will indicate our recommendations with footnotes
at appropriate places. The word of caution that goes with this encouragement is
that reading statistics texts is like reading a Russian novel-the same
characters have different names in different places. For example, the mean of a
sample in some texts is symbolized M rather than 
, and, in some texts, S.D., 
and 
are used as
symbols for the standard deviation. If you expect such differences, it will be
less difficult for you to make the necessary translations.
3.1 Summary
3.1.1 A typical or representative
score from the sample population is a measure of central tendency.
3.1.2 Mode (Mo)
3.1.2.1
The
most frequently occurring score in the distribution.
3.1.2.2
Extreme
scores in the distribution do not affect the mode.
3.1.3 Median (Md)
3.1.3.1
This
score cuts the distribution of scores in half. That is half the scores in the
distribution fall above the middle score and half fall below the middle
score. The steps involved in computing
the median are
3.1.3.1.1 Rank the scores from lowest to highest
3.1.3.1.2 In the case of an odd number of scores pick
the middle score that divides the scores so that an equal number of scores are
above that score and an equal number are below that score. Example
3.1.3.1.2.1 2 5 7 8 12 14 18= 8 would be the median in
the aforementioned distribution of scores.
3.1.3.1.3 In the case of an even number of scores
pick the two middle scores which divide the scores so that an equal number of
scores are above those scores and an equal number of scores are below those
scores. Then add the two middle scores and divide the product by two. Example
3.1.3.1.3.1 2 5 7 8 12 14 18 20=8+12=20/2=10 would be
the median in the aforementioned distribution.
3.1.3.2
Extreme
scores in the distribution do not affect the median.
3.1.4 Mean 
(Average)
3.1.4.1
The
mean is the sum of scores divided by the number of scores.
3.1.4.1.1 Formula
3.1.4.1.1.1 
3.1.4.1.1.2 In the above formula X= the sum of the
scores and N= the number of scores.
3.1.4.2
Extreme
scores in the distribution will affect the mean.
3.1.4.3
The
term average is often used to describe the mean and is usually accurate.
Sometimes however the word average is used to describe other measures of
central tendency such as mode and median.
3.2 Introduction
3.2.1
Now that the preliminaries are out of
the way, you are ready to start on the basics of descriptive statistics. The
starting point is an unorganized group of scores or measures, all obtained from
the same test or procedure. In an experiment, the scores are measurements on
the dependent variable. Measures of central value (often called measures of
central tendency) give you one score or measure that represents or is typical
of, the entire group You will recall that in Chapter 1 we discussed the mean
(arithmetic average). This is one of the three central value statistics.
Recall from Chapter 1 that for every statistic there is also a parameter.
Statistics are characteristics of samples and parameters are characteristics of
population. Fortunately, in the case of the mean, the calculation of the parameter is identical to
the calculation of the statistic. This is not true for the standard deviation. (Chapter 4) Throughout this
book, we will refer to the sample mean (a statistic) with the symbol 
pronounced
"ex-bar"-and to the population mean a parameter with the symbol
pronounced
"mew."
3.2.2
However,
a mean based on a population is interpreted differently from a mean based on a sample. For a
population, there is only
one mean, 
. Any sample, however, is only
one of many possible samples, and 
will vary from sample to
sample. A population mean is obviously better than a sample mean, but often it
is impossible to measure the entire population. Most of the time, then, we must resort to a sample and use 
as an estimate of 
.
3.2.3
In this chapter you will learn to
3.2.3.1
Organize data gathered on a dependent
measure,
3.2.3.2
Calculate central values from the
organized data and determine whether they are statistics or parameters, and
3.2.3.3
Present the data graphically.
3.3 Finding the mean of Unorganized Data
3.3.1
Table 3.1 presents the scores of 100
fourth-grade students on an arithmetic achievement test. These scores were
taken from an alphabetical list of the students' names; therefore, the scores
themselves are in no meaningful order. You probably already know how to compute
the mean of this set of scores. To find the mean, add the scores and divide
that sum by the number of scores.
3.3.2 Formula Mean
3.3.2.1
3.3.3 Table 3.1
3.3.3.1
3.3.4
If these 100 scores are a population,
then 39.43 would be a 
, but if the 100 scores are a sample from some larger
population, 39.43 would be the sample mean, 
.
3.3.5
This mean provides a valuable bit of
information. Since a score of 40 on this test is considered average (according
to the test manual that accompanies it), this group of youngsters, whose mean
score is 39.43, is about average in arithmetic achievement.
3.4 Arranging Scores in Descending
Order and Finding the Median
3.4.1
Look again at Table 3.1. If you knew
that a score of 40 were considered average, could you tell just by looking that
this group is about average? Probably not. Often, in research, so many
measurements are made on so many subjects that just looking at all those
numbers is a mind-boggling experience. Although you can do many computations
using unorganized data, it is often very helpful to organize the numbers in
some way. Meaningful organization will permit you to get some general
impressions about characteristics of the scores by simply' 'eyeballing"
the data (looking at it carefully). In addition, organization is almost a
necessity for finding a second central value-the median.
3.4.2
One way of making some order out of
the chaos in Table 3.1 is to rearrange the numbers into a list, from
highest to lowest. Table 3.2 presents this rearrangement of the arithmetic
achievement scores. (It is usual in statistical tables to put the high numbers
at the top and the low numbers at the bottom.) Compare the unorganized data of
Table 3.1 with the rearranged data of Table 3.2. The ordering from high to low
permits you to quickly gain some insights that would have been very difficult
to glean from the unorganized data. For example, by simply looking at
the center of the table, you get an idea of what the central value is. The
highest and lowest scores are readily apparent and you get the
impression that there are large differences in the achievement levels of these
children. You can see that some scores (such as 44) were achieved by several
people and that some (such as 33) were not achieved by anyone. All this
information is gleaned simply by quickly eyeballing the rearranged data.
3.4.3 Table 3.2
3.4.3.1
3.4.4 Error Detection
3.4.4.1
Eyeballing data is a valuable means of
avoiding large errors. If the answers you calculate differ from what you expect
on the basis of eyeballing, wisdom dictates that you try to reconcile the
difference. You have either overlooked something when eyeballing or made a
mistake in your computations.
3.4.5
This simple rearrangement of data also
permits you to find easily another central value statistic, which can be found
only with extreme difficulty from Table 3.1. This statistic is called the
median. The median is defined as the point (Note that the median, like the mean, is a
point and not necessarily an actual score.) on the scale
of scores above, which half the scores fall and below which half the scores
fall. That is, half of the scores are larger than the median, and half
are smaller. Like the mean, the sample median is calculated exactly the same as
the population median. Only the interpretations differ.
3.4.6
In Table 3.2 there are 100 scores;
therefore, the median will be a point above which there are 50 scores and below
which there are 50 scores. This point is somewhere among the scores of 39. Remember
from Chapter 1, that any number actually stands for a range of numbers that has
a lower and upper limit. This number, 39, has a lower limit of 38.5 and an
upper limit of 39.5. To find the exact median somewhere within the range of
38.5-39.5 you use a procedure called interpolation. We will give you the procedure
and the reasoning that goes with it at the same time. Study it until you
understand it. It will come up again.
3.4.7
There are 42 scores below 39. You will
need eight more (50 - 42 = 8) scores to reach the median. Since there are ten
scores of 39, you need 8/10 of them to reach the median. Assume that those ten
scores of 39 are distributed evenly throughout the interval of 38.5 and 39.5 and that,
therefore, the median is 8/10 of the way through the
interval. Adding. 8 to the lower limit of the interval, 38.5, gives you 39,3,
which is the median for these scores.
3.4.8
There are occasions when you will need
the median of a small number of scores. In such cases, the method we have just
given you will work, but it usually is not necessary to go through that whole
procedure. For example, if N is an odd number and the middle score has a
frequency of 1, then it is the median. In the five scores 2, 3, 4, 12, 15, the
median is 4. If there had been more than one 4, interpolation would have to be
used.
3.4.9
When N is an even number, as in
the six scores 2, 3, 4, 5, 12, 15, the point dividing the scores into two equal
halves will lie halfway between 4 and 5. The median, then, is 4.5. If there had
been more than one 4 or 5, interpolation would have to be used. Sometimes the
distance between the two middle numbers will be larger, as in the scores 2, 3,
7, 11. The same principle holds: the median is halfway between 3 and 7. One-way
of finding that point is to take the mean of the two numbers: (3 + 7) / 2 = 5, which
is the median.
3.4.10
There is no accepted symbol to
differentiate the median of a population from the median of a sample. When we
need to make this distinction, we do it with words.
3.5 The Simple Frequency
Distribution
3.5.1
A more common (and often more useful)
method of organizing data is to construct a simple frequency distribution. Table
3.3 is a simple frequency distribution for the arithmetic achievement data in
Table 3.1.
3.5.2
The most efficient way to reduce
unorganized data like Table 3.1 into a simple frequency distribution like Table
3.3 is to follow these steps:
3.5.2.1
Find the highest and lowest scores. In
Table 3.1, the highest score is 65 and the lowest score is 23.
3.5.2.2
In column form, write down in
descending order all possible scores between the highest score (65) and the
lowest score (23). Head this column with the letter X.
3.5.2.3
Start
with the number in the upper-left-hand comer of the unorganized scores (a score
of 40 in Table 3.1),
draw a line through it, and place a tally
mark beside 40 in your frequency distribution.
3.5.2.4
Continue
this process through all the scores.
3.5.2.5
Count
the number of tallies by each score and place that number beside the tallies in
the column headed . Add up the
numbers in the column to be sure they equal N You
have now constructed a simple frequency distribution.
3.5.2.6
0ften, when simple frequency
distributions are presented formally, the tally marks and all scores with a
frequency of zero are deleted.
3.5.3
Don't
worry about the X column in Table 3.3 yet. It is not part of a simple
frequency distribution, and we will discuss it in the next section.
3.5.4 Table 3.3
3.5.4.1
3.6 Finding Central Values of a
Simple Frequency Distribution
3.6.1 Mean
3.6.1.1
Computation of the mean from a simple
frequency distribution is illustrated in table 3.3. Remember that the numbers
in the column represent the number of people
making each of the scores. To get N, you must add the numbers in the f
column because that's where the people are represented. If you are a
devotee of shortcut arithmetic, you may already have discovered or may already
know the basic idea behind the procedure: multiplication is shortcut addition.
In Table 3.3, the column headed X means what it says algebraically: multiply f (the number of
people making a score) times X, (the score they made) for each of the scores.
The reason this is done. is that everyone who made a particular score must be
taken into account in the computation of the mean. Since only one person made a
score of 65, multiply 1 x 65, and put a 65 in the
X column. No one made a score of 64 and 0 x 64 = 0;
put a zero in the X column. Since
four people had scores of 55, multiply 4 x 55 to get 220. After X is computed for all scores, obtain 
X by adding up the X
column. Notice that 
X in the simple frequency distribution is
exactly the same as 
:X in Table 3.1. To compute the mean from a simple frequency
distribution, use the formula
3.6.1.2
Mean
Frequency distribution
3.6.1.2.1 
3.6.2 Median
3.6.2.1
The procedure for finding the median
of scores arranged in a simple frequency distribution is the same as that for
scores arranged in descending order, except that you must now use the frequency
column to find the number of people making each score
3.6.2.2
The median is still the point with
half the scores above and half below it, and is the same point whether you
start from the bottom of the distribution or from the top. If you start from
the top of Table 3.3, you find that 48 people have scores of 40 or above. Two
more are needed to get to 50, the halfway point in the distribution. There are
ten scores of 39, and you need two of them. Thus 2/10 should be subtracted from
39.5 (the lower limit of the score of 40); 39.5 - .2 = 39.3.
3.6.2.3
Error
Detection
3.6.2.3.1 Calculating the median by starting from the
top of the distribution will produce the same answer as calculating it by
starting from the bottom.
3.6.3 Mode
3.6.3.1
You may also find the third
central-value statistic from the simple frequency distribution. This statistic is
called the mode.
The mode is the score made by the greatest number
of people-the score with the greatest frequency.
3.6.3.2
Distribution may have more than one
mode. A bimodal distribution is one with two high frequency scores separated by
one or more low frequency scores. However, although a distribution may have
more than one mode, it can have only one mean and one median.
3.6.3.3
A sample mode and a population mode
are determined in the same way.
3.6.3.4
In Table 3.3, more people had a score
of 39 than any other score, so 39 is the mode. You will note, however, that it
was close. Ten people scored 39, but nine scored 34 and eight scored 41. A few
lucky guesses by children taking the achievement test could have caused
significant changes in the mode. This instability of the mode limits its
usefulness. .
3.7 The Grouped Frequency
Distribution
3.7.1
There is a way of condensing the data
of Table 3.1 even further. The result of such a condensation is called a grouped frequency
distribution and Table 3.4 is an example of such a
distribution, again using the arithmetic achievement-test scores.5
3.7.2 A formal grouped frequency
distribution does not include the tally marks or the X and X columns.
3.7.3
The grouping of data began as a-way of
simplifying computations in the days before the invention of all these
marvellous computational aids such as computers and calculators. Today, most
researchers group their data only when they want to construct a graph or when N's
are very large. These two occasions happen often enough to make it
important for you to learn about it.
3.7.4 In the grouped frequency distribution, X values are grouped into
ranges called class
intervals. In Table 3.4, the entire range of
scores, from 65 to 23 has been reduced to 15 class intervals, each interval
covers three scores and, the size of the interval (the number of scores
covered) is indicated by i. For Table 3.4, i = 3. The midpoint of each interval
represents all scores in that interval for example, there were nine children who
had scores of 33, 34 or 35. The midpoint of the class interval 33-35 is 34. All
nine children are represented by 34. Obviously, this procedure may introduce
some inaccuracy into computations; however, the amount of error introduced is
usually very slight. For example, the mean computed from Table 3.4 is 39.40. The
mean computed from
ungrouped data is 39.43.
3.7.5
Class intervals have upper and lower
limits, much like simple scores obtained by measuring a quantitative
variable. A class interval of 33-35 has a lower limit of 32.5 and an upper
limit of 35.5. Similarly, a class interval of 40-49 has a lower limit of 39.5
and an upper limit of 49.5.
3.7.6 Table 3.4
3.7.6.1
3.7.7 Establishing Class Intervals
3.7.7.1
There are three conventions that are
usually followed in establishing class intervals. We call them
conventions because they are customs rather than hard-and-fast rules. There are
two justifications for these conventions. First, they allow you to get
maximum information from your data with minimum effort. Second, they
provide some standardization of procedures, which aids in communication among
scientists. These conventions are
3.7.7.2
Data
should be grouped into not fewer than 10 and not more than 20 class
intervals.
3.7.7.2.1
The primary purpose of grouping data is to provide a clearer
picture of trends in the data and to make computations easier. (For
example, Table 3.4 shows that there are normally frequencies near the center of
the distribution with fewer and fewer as the upper and lower ends of the
distribution are approached. If the data are grouped into fewer than 10
intervals, such trends are not as apparent. In Table 3.5, the same scores are
grouped into only five class intervals. The concentration of frequencies in the
center of the distribution is not nearly so apparent.
3.7.7.2.2
Another reason for using at least 10 class intervals is that,
as you reduce the number of class intervals, the errors caused by grouping
increase. With fewer than 10 class intervals, the errors may no longer be
minor. For example, the mean computed from Table 3.4 was 39.40-only .03 points
away from the exact mean of 39.43 computed from ungrouped data. The mean
computed from Table 3.5, however, is 39.00-an error of .43 points.
3.7.7.2.3
On the other hand, the use of
more than 20 class intervals may tend to exaggerate fluctuations in the data that are really due. to chance occurrences. . You
also sacrifice much of the ease of computation, with little gain in
control over errors. So, the convention
is: use 10 to 20 class intervals.
3.7.7.3
The size of the class intervals (i)
should be an odd number or 10 or a multiple of 10. (Some writers include i= 2 as acceptable. Some
also object to the use of i= 7 or 9. In actual practice, the most frequently
seen i's are 3, 5, 10, and multiples of 10.)
3.7.7.3.1
The reason for this is simply computational ease. The
midpoint of the interval is used as representative of all scores in the
interval; and if i is an odd number,
the midpoint will be a whole number. If hs an even number, the midpoint will
be a decimal number. In the interval 12-14 (i = 3), the midpoint is the whole
number 13. In an interval 12 to 15 (i = 4), the midpoint is the decimal number
13.5. However, if the range of scores is so great that you cannot include all
of them in 20 groups with i = 9 or less, it is conventional to
place 10 scores or a multiple of 10 in each class interval.
3.7.7.4
Begin each class interval with
a multiple of .i.
3.7.7.4.1
For example, if the lowest score is 44 and i = 5, the first
class interval should be 40-44 because 40 is a multiple of 5. This convention
is violated fairly often. However, the practice is followed more often than not.
A violation that seems to be justified occurs when i = 5. When the interval
size is 5, it may be more convenient to begin the interval such that multiples
of 5 will fall at the midpoint, since multiples of 5 are easier to manipulate.
For example, an interval 23-27 has 25 as its midpoint, while an interval 25-29
has 27 as its midpoint. Multiplying by 25 is easier than multiplying by 27.
3.7.7.4.2
In addition to these three conventions, remember that the
highest scores go at the top and the lowest scores at the bottom.
3.7.8 Converting Unorganized Data into
a Grouped Frequency Distribution
3.7.8.1
Now that you know the conventions for
establishing class intervals, we will go through the steps for converting
a mass of data like that in Table 3.1 into a grouped frequency distribution
like Table 3.4:
3.7.8.2
Find the highest and lowest scores. In
Table 3.1, the highest score is 65, and the lowest score is 23.
3.7.8.3
Find the range of scores by
subtracting the lowest score from the highest and adding 1: 65 - 23 + I =
43. The 1 is added so that the upper limit of the highest score and the lower
limit of the lowest score will be included.
3.7.8.4
Determine i by a trial-and-error
procedure. Remember that there are to be 10 to 20 class intervals and that the
interval size should be odd, 10, or a multiple of 10. Dividing the range by a
potential i value tells the number of class intervals that will result. For
example, dividing the range of 43 by 5 provides a quotient of 8.60. Thus, i =
5 produces 8.6 or 9 class intervals. That does not satisfy the rule calling for
at least 10 intervals, but it is close and might be acceptable. In most such
cases, however it is better to use a smaller I and get a larger number of
intervals. Dividing the range by 3 (43/3) gives you 14.33 or 15 class
intervals. It sometimes happens that this process results in an extra class
interval. This occurs when the lowest score is such that extra scores must be
added to the bottom of the distribution to start the interval with a multiple
of i. For the data in Table 3.1, the most appropriate interval size is 3,
resulting in 15 class intervals.
3.7.8.5
Begin the bottom interval with the
lowest score. if it is a multiple of i. If the lowest score is not a multiple
of i, begin the interval with the next lower number that is a multiple of i. In
the data of Table 3.1, the lowest score, 23, is not a multiple of i. Begin the
interval with 21. since it is a multiple of 3. The lowest class interval, then,
is 21-23. From there on, it's easy. Simply' begin the next interval with the
next number and end it such that it includes three numbers (24-26). Look at the
class intervals in Table 3.4. Notice that each interval begins with a number
evenly divisible by 3.
3.7.8.6
3.7.8.7
Table
3-5
3.7.8.7.1 
3.7.8.8
The
rest of the process is the same as for a simple frequency distribution. For
each score in the unorganized data, put a tally mark beside its class interval
and cross out the score. Count the tally marks and put the number into the
frequency column. Add the frequency column to be sure that: 
= N.
3.7.8.9
Clue
to the Future
3.7.8.9.1 The distributions that you have been
constructing are empirical distributions based on scores actually gathered in
experiments. This chapter and the next two are about these empirical frequency
distributions. Starting with Chapter 8, and throughout the rest of the book,
you will also make use of theoretical distributions-distributions based on
mathematical formulas and logic rather than on actual observations.
3.8 Finding Central Values of a
Grouped Frequency Distribution
3.8.1 Mean
3.8.1.1
The procedure for finding the mean of
a grouped frequency distribution is similar to that for the simple frequency
distribution. In the grouped distribution, however, the midpoint of each
interval represents all the scores in the interval. Look again at Table 3.4.
Notice the column headed with the letter X. The numbers in that column are the
midpoints of the intervals. Assume that the scores in the interval are evenly
distributed throughout the interval. Thus, X is the mean for all scores within
the interval. After the X column is filled, multiply each X by its value in order to include all frequencies in that interval. Place
the product in the X column.
Summing the X ..column
provides 
X, which, when
divided-by N, yields the mean. In terms of a formula,
3.8.1.2
Formula
3.8.1.2.1 
3.8.2 Median
3.8.2.1
Finding the median of a grouped
distribution requires interpolation within the interval containing the median.
We will use the data in Table 3.4 to illustrate the procedure. Remember that
the median is the point in the distribution that has half the frequencies
above it and half the frequencies below it. Since N= 100, the median
will have 50 frequencies above it and 50 below it. Adding frequencies from the
bottom of the distribution, you find that there are 42 who scored below the
interval 39-41. You need
8 more frequencies (50 - 42 = 8) to find the median. Since 23 people scored
in the interval 39-41, you need 8 of these 23
frequencies or 8/23. Again, you assume that the 23 people in the
interval are evenly distributed through the interval. Thus, you need the
same proportion of score points in the interval as you have frequencies-that
is, 8/23 or, 35 of the 3 score points in the interval. Since .35 x 3 = 1.05,
you must go 1.05 score points into the interval to reach the median. Since the
lower limit of the interval is 38.5, add 1.05 to find the median, which is
39.55. Figure 3.1 illustrates this procedure.
3.8.2.2
In summary, the steps for finding the
median in a grouped frequency distribution are as follows.
3.8.2.3
Divide N by 2
3.8.2.4
Starting at the bottom of the
distribution, add the frequencies until you find the interval containing the
median
3.8.2.5
Subtract
from N/2 the total frequencies of all intervals below the interval
containing the median.
3.8.2.6
Divide
the difference found in step 3 by the number of frequencies in the interval
containing the median.
3.8.2.7
Multiply
the proportion found in step 4 by i
3.8.2.8
Add
the product found in step 5 to the lower limit of the interval containing the
median. That sum is the median.
3.8.2.9
Figure
3.1
3.8.2.9.1 
3.8.3 Mode
3.8.3.1
The third central value, the mode, is
the midpoint of the interval having the greatest number of frequencies. In Table 3.4, the interval 39-41 has the greatest number of frequencies-23. The midpoint of
that interval, 40, is the mode.
3.9 Graphic Presentation of Data
3.9.1
In order to better communicate your
findings to colleagues (and to understand them better yourself), you will often
find it useful to present the results in the form of a graph. It has been said,
with considerable truth, that one picture is worth a thousand words; and a
graph is a type of picture. Almost any data can be presented graphically. The
major purpose of a graph is to get a clear, overall picture of the data.
3.9.2
Graphs are composed of a
horizontal axis (variously called the baseline, X axis or abscissa) and
a vertical axis called the Y-axis or ordinate. We will take
what seems to be the simplest course and use the terms X and Y.
3.9.3 We will describe two kinds of
graphs. The first kind is used to present frequency distributions like those
you have been constructing. Frequency polygons, Histograms, and bar graphs are
examples of this first kind of graph. The second kind we will describe is the
line graph, which is used to present the relationship
between two different variables.
3.9.4 Illustration XY Axis
3.9.4.1
3.9.5 Presenting Frequency
Distributions
3.9.5.1
Whether you use a frequency polygon, a
histogram, or a bar graph to present a frequency distribution depends on the
kind of variable you have measured. A frequency polygon or histogram is used
for quantitative data, and the bar graph is used for qualitative data. It is
not wrong to use a bar graph for quantitative data; but most researchers follow
the rule given above. Qualitative data, however, should not be presented
with a frequency polygon or a histogram. The arithmetic achievement scores
(Table 3.1) are an example of quantitative data.
3.9.5.2
Frequency
Polygon
3.9.5.2.1 Figure
3.2 shows a-frequency polygon based on the frequency distribution in Table 3.4.
We will use it to demonstrate the characteristics of all frequency polygons. On
the X-axis we placed the midpoints of the class intervals. Notice that the
midpoints are spaced at equal intervals, with the smallest midpoint at the left
and the largest midpoint at the right. The Y-axis
is labeled "Frequencies and is also marked off into equal intervals.
3.9.5.2.2
Graphs are designed to "look right." They look right
if the height of the figure is 60 percent
to 75 percent of its length. Since
the midpoints must be plotted along the X axis, you must divide the Y axis into units that will satisfy
this rule. Usually, this requires a little juggling on your part. Darrell Huff
(1954) offers an excellent demonstration of the misleading effects that occur
when this convention is violated.
3.9.5.2.3 The
intersection of the X and Y axes is
considered the zero point for both variables. For the Y-axis in Figure 3.2, this is indeed the case. The distance on the Y axis is the same from zero to two as
from two to four, and so on. On the X axis, however, that is not the case.
Here, the scale jumps from zero to 19 and then is divided into equal units of
three. It is conventional to indicate a break in the measuring scale by
breaking the axis with slash marks between zero and the lowest score used, as
we did in Figure 3.2. It is also conventional to close a polygon at both ends by
connecting the curve to the X-axis.
3.9.5.2.4
Each
point of the frequency polygon represents two numbers; the class midpoint
directly below it on the X-axis and
the frequency of that class directly across from it on the Y-axis. By looking at the points in Figure 3.2, you can readily see
that three people are represented by the midpoint 22, nine people by each of
the midpoints 31, 34, and 37, 23 people by the midpoint 40, and so on.
3.9.5.2.5 The
major purpose of the frequency polygon is to gain an overall view of the
distribution of scores. Figure 3.2 makes it clear, for example, that the
frequencies are greater for the lower scores than for the higher ones. It also
illustrates rather dramatically that the greatest number of children scored in
the center of the distribution.
3.9.5.3
Figure
3-2
3.9.5.3.1 
3.9.5.4
Histogram
3.9.5.4.1
Figure 3.3 is a histogram constructed from the same data that
were used for the frequency polygon of Figure 3.2. Researchers may choose
either of these methods for a given distribution of quantitative data, but the
frequency polygon is usually preferred for several reasons: it is easier to
construct, gives a generally clearer picture of trends in the data, and can be
used to compare different distribution, on the same graph. However frequencies
are easier to read from a histogram.
3.9.5.4.2
Figure 3-3
3.9.5.4.2.1
3.9.5.4.3 Actually, the two figures are very
similar. They differ only in that the histogram is made by raising bars from
the X axis to the appropriate frequencies instead of plotting points above the
midpoints. The width of a bar is from the lower to the upper limit of its class
interval. Notice that there is no space between the bars.
3.9.5.5
Bar
Graph
3.9.5.5.1
The third type of graph that presents frequency distributions
is the bar graph. A bar graph presents frequencies of the categories of a
qualitative variable. An example of a qualitative variable is laundry
detergent; the there are many different brands (types of the variable), but the
brands don't tell you the order they go in, for example
3.9.5.5.2 With quantitative variables, the measurements
of the variable impose an order on themselves. Arithmetic achievement scores
of 43 and 51 tell you the order they belong in. "Tide" and
"Lux" do not signify any order.
3.9.5.5.3
Figure 3.4 is an example of a bar graph. Notice that each bar
is separated by a small space. This bar graph was constructed by a grocery store manager who had a
practical problem to solve. One side of an aisle in his store was stocked with
laundry detergent, and he had no more space for this kind of product. How much
of the available space should he allot for each brand? For one week, he kept a
record of the number of boxes of each brand sold. From this frequency
distribution of scores on a qualitative variable, he constructed the bar graph
in Figure 3.4. (He, of course, used the names of the brands. We wouldn't dare!)
3.9.5.5.4 Brands
E, H, and K are obviously the big sellers and should get the greatest amount of
space. Brands A and D need very little space. The other brands fall between
these. The grocer, of course, would probably consider the relative profits from
the sale of the different brands in order to determine just how much space to
allot to each. Our purpose here is only to illustrate the use of the bar graph
to present qualitative data.
3.9.6 The Line Graph
3.9.6.1
Perhaps the most frequently used graph
in scientific books and journal articles is the line graph. A line graph is
used to present the relationship between two variables.
3.9.6.2
. A point on a line -graph represents
the two scores made by' one person on each of the two variables. Often, the
mean of a group is used rather than one person, but the idea is the same: a
group with a mean score of X on one variable had a mean score of Y on
the other variable. The point on the graph represents the means of that group
on both variables.
3.9.6.3
Figure
3-4 & 3-5
3.9.6.3.1 
3.9.6.4
Figure
3-6
3.9.6.4.1 
3.9.6.5
Figure
3-7
3.9.6.5.1 
3.9.6.6
Figure
3-8
3.9.6.6.1 
3.9.6.7
Figure 3.5 is an example of a line
graph of the relationship between subjects scores on an anxiety test and their
scores on a difficult problem-solving task. Many studies have discovered this
general relationship. Notice that performance on the task is better and better
for subjects with higher and higher anxiety scores up to the middle range of
anxiety. But as anxiety scores continue to increase, performance scores decrease.
Chapter 5, "Correlation and Regression," will make extensive use of a
version of this type of line graph.
3.9.6.8
A variation of the line graph places
performance scores on the Y-axis and some condition of training on the
X-axis. Examples of such training conditions are: number of trials, hours of food
deprivation, year in school, and: amount of reinforcement. The "score on
the training condition is assigned by the experimenter.
3.9.6.9
Figure 3.6 is a generalized learning
curve with a performance measure (scores) on Y axis and number of
reinforced trials on the .X axis. Early in training (after only one or two
trials), performance is poor. As trials continue, performance improves rapidly
at first and then more and more slowly. Finally, at the extreme right-hand
portion of the graph, performance has levelled off; continued trials do not
produce further changes in the scores.
3.9.6.10
A line graph, then, presents a picture
of the relationship between two variables. By looking at the line, you can tell
what changes take place in the Y variable as the value of the X variable
changes.
3.10 Skewed Distributions
3.10.1
Look
back at Table 3.4 graphed as Figure 3.2. Notice that the largest frequencies
are found in the middle of the distribution. The same thing is true in Problem
3 of this chapter. These distributions are not badly skewed; they are
reasonably symmetrical ln some data, however, the largest frequencies are found at one end of
the distribution rather than in the middle. Such distributions are said to he
skewed.
3.10.2
The word skew is similar to the
word skewer, the name of the cooking implement used !n making shish
kebab. A skewer is long and pointed and is thicker at one end than the other
(not symmetrical). Although skewed distributions do not function like skewers
(you would have a terrible time poking one through a chunk of lamb), the, name
does help you remember that a skewed distribution has a thin point on one side.
3.10.3
Figures 3.7 and 3.8 are illustrations
of skewed distributions. Figure 3.7 is positive skewed; the thin point is toward the high scores, and
the most frequent scores are low ones. Figure 3.8 is negatively skewed; the
thin point or skinny end is toward .the low scores and most frequent scores are
high ones.
3.10.4
There
is a mathematical of measuring the degree of skewness that is more precise
than eyeballing, but it is beyond the scope of this book. However, figuring the
relationship of the mean to the median is an objective way to determine the direction
of the skew. When the mean is numerically smaller than the median, there is some amount of negative
skew.
3.10.5
Figure
3-9
3.10.5.1
3.10.6
When the mean is larger than the
median there is positive skew. The reason for this is that the mean is affected by the size of the numbers
and is pulled in the direction
of the extreme scores. The median is not influenced by the size of the
scores. The relationship between the mean and the
median is illustrated by Figure 3.9. The size of the difference between the
mean and the median gives you an indication of how much the distribution is
skewed.
3.11 The Mean, Median, and Mode Compared
3.11.1
A common question is [Which measure of
central value should I use?" The general answer is "Given a choice,
use the mean. " Sometimes, however, the data give you no choice. For example if the frequency
distribution is for a nominal variable the mode is the
only appropriate measure of central value.
3.11.2
Figure 3-10
3.11.2.1
3.11.3
It is meaningless to find a median or
to add up the scores and divide to find a mean for data based on a nominal
scale. For the data from the voting-behavior experiment the mode is the only
measure of central value that is meaningful. For a frequency distribution of an
ordinal variable, the median or the mode is appropriate. For data based on
interval or ratio data, the mean, median or mode may be used-you have a choice.
3.11.4
Even if you have interval or ratio
data, there are two situations in which the mean is inappropriate because it
gives an erroneous impression of the distribution. The first situation is the
case of a severely skewed distribution. The following story demonstrates why
the mean is inappropriate for severely skewed distributions.
3.11.5
The developer of Swampy Acres
Retirement Home sites is attempting, with a computer-selected mailing list, to
sell the lots in his southern paradise to northern buyers. The marks express
concern that flooding might occur. The developer reassures them by explaining
that the average elevation of his lots is 78.5 feet and that the water has
never exceeded 25 feet in that area. On the average, he has told the truth; but
this average truth is misleading. Look at the actual lay of the land in Figure
3.10 and examine the frequency distribution in Table 3.6, which summarizes the
picture.
3.11.6
The mean elevation as the developer
said is 78.5 feet; however, only 20 lots, all on a cliff, are out of the flood
zone. The other 80 lots are, on the average, under water. The mean, in this
case, is misleading. In this instance, the central value that describes the
typical case is the median because it is unaffected by the size of the few
extreme lots on the cliff. The median elevation is 12.5 feet, well below the
high-water mark.
3.11.7
Darrell Huff's delightful and
informative book, How to Lie with Statistics (1954) gives a
number of such examples. We heartily recommend this book to you. It provides many
cautions concerning misinformation conveyed through the use of the inappropriate
statistic. A more recent and equally delightful book is Flaws and Fallacies
in Statistical Thinking by Stephen Campbell (1974).
3.11.8
There is another instance that
requires a median, even though you have a symmetrical distribution. This is
when the class interval with the largest (or smallest) scores is not limited.
In such a case, you do not nave a midpoint and, therefore, cannot compute
a mean. For example, age data are sometimes reported with the highest category
as "75 and over. " The mean cannot be computed. Thus, when one or
both of the extreme, class intervals is not limited, the median is the
appropriate measure of central value. To reiterate: given a choice, use the mean.
3.11.9
Table 3-6
3.11.9.1
3.12 The Mean of a Set of Means
3.12.1
Occasions arise in which means are
available from several samples taken from the same population. If these means
are combined, the mean of the set of means will give you the best estimate of
the population parameter, 
. If every sample has the same N. you can
compute the average mean simply by adding the means and dividing; by the number
of means. If, however, the means to be averaged have varying N 's, it is
essential that you take into account the various sample sizes by multiplying
each mean by its own N
before summing. Table 3.7 illustrates this procedure. Four means. are
presented, along with two hypothetical sample sizes for each mean. In the
left-hand table, the four sample sizes are equal. In the right-hand table, the
four sample sizes are not equal. Notice that 18.50 is the mean of the means
when the separate means are simply added and the sum divided by the number of
means. This gives the correct answer when the sample sizes are equal. However,
when sample sizes differ, 18.50 is wrong. Each mean must be multiplied by its
respective N, and the mean of the means is 17.60. When N 's are
unequal, averaging the means without accounting for sample frequencies always
causes an error.
3.12.2
Table
3-7
3.12.2.1
3.12.3
Clue
to The Future
3.12.3.1
In Chapter 9 you will learn
a most important concept-a concept called a sampling distribution of the mean.
The mean of a set of means is an inherent part of that concept.
3.13 Skewed Distributions and
Measures of Central Tendency
3.13.1
Introduction
3.13.1.1
Distributions
when the mean median and mode are represented graphically may demonstrate
varying degrees of Skewness, which refers to the degree of asymmetry of the
graphical curve.
3.13.2
Symmetrical
Distribution
3.13.2.1
In
a symmetrical distribution the mean, median and mode all fall in the same
point.
3.13.2.2
3.13.3
Bimodal
Symmetrical Distribution
3.13.3.1
If
there are two modes (bi-modal) even though the mean, median fall in the same
point the two modes will represent the highest points of the distribution. This
is considered a bimodal symmetrical distribution
3.13.3.2
3.13.4
Skewed
Distributions
3.13.4.1
Introduction
3.13.4.1.1
In a
symmetrical distribution the largest frequencies are found in the middle
whereas in a skewed distribution the largest frequencies are found at one end
of the distribution rather than in the middle.
3.13.4.1.2
The word skew
is similar to the word skewer which is long and pointed and is thicker at one
end than the other (not symmetrical). A skewed distribution has a thin point on
one side.
3.13.4.1.3
In a
positively skewed; the thin point is toward the high scores, and the most
frequent scores are low ones. In the negatively skewed, the thin point or
skinny end is toward the low scores, and the most frequent scores are high
ones. There are mathematical ways of measuring the degree of skewness that are
more precise than eyeballing, but you can figure the relationship of the mean
to the median and this provides an objective way to determine the direction of
the skew. When the mean is numerically smaller than the median, there is some
amount of negative skew. When the mean is larger than the median there is
positive skew. The reason for this is that the mean is affected by the size of
the numbers and is pulled in the direction of extreme scores. The median is not
influenced by the size of the scores. The relationship between the mean and the
median is illustrated in the picture below. The size of the difference between
the mean and the median gives you an indication of how much the distribution is
skewed.
3.13.4.2
Illustration
3.13.4.2.1
3.13.5
Positively
Skewed Distributions
3.13.5.1
The
positively skewed distribution below demonstrates an asymmetrical pattern. In
this case the mode is smaller than the median, which is smaller than the mean.
3.13.5.2
3.13.5.3
This
relationship exists between the mode, median and mean because each statistic describes
the distribution differently.
3.13.5.4
The
mode represents the most frequently occurring score and thus is the highest
point on the X axis in a frequency distribution. The median cuts the
distribution in half so that 50% of the scores are on either side.
3.13.5.5
3.13.5.6
The
mean unlike the median and mode is affected by larger scores since it is the
product of the additive score values divided by their number. The mean
represents the balance point in the distribution. Because of this it is drawn
towards the skewness and in positively skewed towards the larger values.
3.13.5.7
3.13.6
Negatively
Skewed Distribution
3.13.6.1
This
distribution is also asymmetrical but with the opposite order of the mean,
median, and mode. The mean is smaller than the median, which is smaller than
the mode.
3.13.6.2
3.13.6.3
The
mode which has the highest value in a frequency distribution points the
skewness in a negative direction.
3.13.6.4
3.14 The Mean of a Set of Means (
)
3.14.1
5 VARIABILITY
5.1 The spread or dispersion of
scores is known as variability. If the distribution of scores fall within a
narrow range there is little variability. Conversely scores that vary widely
connote a distribution that is highly variable.
5.2 Range
5.2.1 The range is the difference
between the largest score and the smallest score.
5.3 Standard Deviation
5.4
5.5 Standard Deviation (s) as an
Estimate of Population Variability
5.5.1
5.5.2 Deviation Scores
5.5.3
5.5.4 Deviation-Score Method of
Computing s from Ungrouped Data
5.5.5
5.5.6 Deviation-Score Method of
Computing s from Grouped Data
5.5.7
5.5.8 The Raw-Score Method of
Computing s from Ungrouped Data
5.5.9
5.5.10
Raw-Score
Method of Computing s from Grouped Data
5.5.11
5.6 The Other Two Standard
Deviations, 
and S
5.7
5.8 Variance
5.9
5.10 z Scores
5.10.1
Introduction
5.10.1.1
You
have used measures of central value and measures of variability to describe a
distribution of scores. The next statistic, z, is used to describe a single
score.
5.10.1.2
A
z score is a mathematical way to change a raw score so that it reflects its
relationship to the mean and standard deviation of its fellow scores.
5.10.1.3
Any
distribution of raw scores can be converted to a distribution of z scores; for
each raw score, there is a z score. Raw scores above the mean will have
positive z scores; those below the mean will have negative z scores.
5.10.1.4
A
z score is also called a standard score because it is a deviation score
expressed in standard deviation units. It is the number of standard deviations
a score is above or below the mean. A z score tells you the relative position
of a raw score in the distribution. (z) scores are also used for inferential
purposes. Much larger z scores may occur then.
5.10.2
Formula
and Procedure
5.10.2.1
Formula
5.10.2.1.1
5.10.2.2
Variables
Defined
5.10.2.2.1.1
(z)=z score
5.10.2.2.1.2
S=standard
deviation of a sample of scores
5.10.2.2.1.3
(x)=individual
raw score
5.10.2.2.1.4
=mean
of a sample
5.10.2.2.2
5.10.2.3
Procedure
5.10.2.3.1
Find the
difference between the raw score and the mean
5.10.2.3.2
Divide that
difference by the standard deviation of the sample
5.10.3
Use
of z Scores
5.10.3.1
(z)
scores are used to compare two scores in the same distribution. They are also
used to compare two scores from different distributions, even when the
distributions are measuring different things.
5.11 Variance and Standard Deviation
5.11.1
Variance
5.11.1.1
S2
is the symbol for variance and is a measure of variability from the mean
of the distribution of scores.
5.11.1.2
5.11.1.3
Find
the mean of the scores.
5.11.1.4
Subtract
the mean from every score.
5.11.1.5
Square
the results of step two.
5.11.1.6
Sum
the results of step three.
5.11.1.7
Divide
the results of step four by N (The number of scores)-1.
5.11.1.8
Example
5.11.1.8.1
5.11.1.8.2
Find the mean
of the scores. 
= 50 / 5
= 10
5.11.1.8.3
Subtract the
mean from every score. The second column above
5.11.1.8.4
Square the
results of step two. The third column above
5.11.1.8.5
Sum the results
of step three. 22
5.11.1.8.6
Divide the
results of step four by N (# of scores)-1. s2 = 22 / (5-1) =
22/4=5.5
5.11.1.8.7
Note that the
sum of column *2* is zero. This must be the case if the calculations are
performed correctly up to that point.
5.11.2
Standard
Deviation
5.11.2.1
S
is the symbol for standard deviation and it is the square root of the variance.
5.11.2.2
The
standard deviation is the preferred measure of variability.
5.11.2.3
Formula
5.11.2.3.1
5.11.2.4
Example
5.11.2.4.1
Take the
square of the variance above. Square root of 5.5=2.35
6
Correlation and Regression
6.1 Introduction
6.1.1 Sir Francis Galton (1822-1911)
in England conducted some of the earliest investigations making use of
statistical analysis. Galton was concerned with the general question of whether
people of the same family were more alike than people of different families.
Galton needed a method that would describe the degree to which, for example,
heights of fathers and their sons were alike. The method he invented for this
purpose is called correlation (co-relation). With it, Galton could also measure
the degree to which the heights of unrelated men were alike. He could then
compare these two results and thus answer his question.
6.1.2 Galtons student Karl Pearson
(1857-1936), with Galtons aid, later developed a formula that yielded a
statistic known as a correlation coefficient. Pearsons product-moment
coefficient, and other correlation coefficients based on Pearsons work, have
been widely used in statistical studies in psychology, education, sociology,
medicine, and many other areas.
6.2 Concept of Correlation
6.2.1 In order to compute a
correlation, you must have two variables, with values of one variable (X)
paired in some logical way with values of the second variable (Y). Such an
organization of data is referred to as a bivariate (two-variable) distribution.
6.2.2 Examples
6.2.2.1
Same
group of people may take two tests and the score results of both tests can be
compared.
6.2.2.2
Family
relationships may be organized as bivariate distribution such as height of
fathers is one variable, X and height of sons is another variable Y.
6.3 Positive Correlation
6.3.1 In the case of a positive
correlation between two variables, high measurements on one variable tend to be
associated with high measurements on the other and low measurements on one with
low measurements on the other. In other words, the two variables vary together
in the same direction. A perfect positive correlation is 1.00. A scatterplot is
used to visualize this relationship with each point in the scatterplot
representing a pair of scores represented on the X and Y axis of the chart. The
line that runs through the points is called a regression line or line of best
fit. When there is perfect correlation (+ -1.00), all points fall exactly on
the line. When the points are scattered away from the line, correlation is less
than perfect and the correlation coefficient falls between .00 (No correlation)
and 1.00 (Perfect correlation). It was when Galton cast his data in the form of
a scatterplot that he conceived the idea of a correlationship between the
variables. It is from the term regression that we get the symbol r for
correlation. Galton chose the term regression because it was descriptive of a
phenomenon that he discovered in his data on inheritance. He found, for
example, that tall fathers had sons somewhat shorter than themselves and that
short fathers had sons somewhat taller than themselves. From such data, he
conceived his law of universal regression, which states that there exists a
tendency for each generation to regress, or move toward, the mean of the general
population.
6.3.2 Today, the term regression also
has a second meaning. It refers to a statistical method that is used to fit a
straight line to bivariate data and to predict scores on one variable from
scores on a second variable.
6.3.3 It is not necessary that the numbers
on the two variables be exactly the same in order to have perfect correlation.
The only requirement is that the differences between pairs of scores be all the
same. The relationship must be such that all points in a scatterplot will lie
on the regression line. If this requirement is met, correlation will be
perfect, and an exact prediction can be made.
6.3.4 Nature, of course, is not so
accommodating as to permit such perfect prediction, at least at sciences
present state of knowledge. People cannot peredict their son heights
precisely. The points do not all fall on the regression line; some miss it
badly. However, as Galton found , there is some positive relationship; the
correlation coefficient between father and son height is r=.50. The correlation
between math and reading skills is r=.54. Predictions made from these
correlations although far from perfect would be far better than a random guess.
6.4 Negative Correlation
6.4.1 Negative correlation occurs
where high scores of one variable are associated with low scores of the other.
The two variables thus tend to vary together but in opposite directions. The regression line runs from the upper
left of the graph to the lower right. Negative correlation could be changed to
positive by changing the type of score plotted on one of the variables.
6.4.2 Perfect negative correlation
exists, as does perfect positive correlation, when all points are on the
regression line. The correlation coefficient in such a case is 1.00. For
example, there is a perfect negative relationship between the amount of money
in your checking account and the amount of money you have written check for (if
you ignore service charge and deposits). As the amount of money you write
checks for increases, your balance decreases by exactly the same amount.
6.4.3 Other examples of negative
correlation (less than perfect) are;
6.4.3.1
Temperature
and inches of snow at the top of a mountain, measured at noon each day in May
6.4.3.2
Hours
of sunshine and inches of rainfall per day at Miami, Florida
6.4.3.3
Number
of pounds lost and number of calories consumed per day by a person on a strict
diet
6.4.4 Negative correlation permits
prediction in the same way that positive correlation does. With correlation,
positive is not better than negative. In both cases, the size of the
correlation coefficient indicates the strength of the relation ship-the larger
the absolute value of the number, the stronger the relationship. The algebraic
sign (+ or -) indicates the direction of the relationship.
6.5 Zero Correlation
6.5.1 A zero correlation means that
there is no relationship between the two variables. High and low scores on the
two variables are not associated in any predictable manner. In the case of zero
correlation, the best prediction from any X score is the mean of the Y scores.
The regression line, then, runs parallel to the X axis at the height of Y on
the Y axis.
6.6 Computation of the correlation
Coefficient
6.6.1
6.7 Computational Formulas
6.7.1
6.7.2 Blanched Formula
6.7.2.1
This
procedure requires you to find the means and standard deviations of both X and
Y before computing r.
6.7.2.2
Formula
6.7.2.2.1 r=(j(X(each value)*Y(each
value))/N)-((X(Mean))*((Y)(Mean))/(Sx)* (Sy)
6.7.2.3
Variables
Defined
6.7.2.3.1 j=Sum
6.7.2.3.2 XY=Product of each X value multiplied by
its paired Y value
6.7.2.3.3 X(mean)=Mean of variable X
6.7.2.3.4 Y(mean)=Mean of variable Y
6.7.2.3.5 Sx=Standard deviation of
variable X
6.7.2.3.6 Sy= Standard deviation of
variable Y
6.7.2.3.7 N=Number of pairs of observations
6.7.2.4
Procedure
6.7.2.4.1 Multiply each paired X and Y score
6.7.2.4.2 Sum the products of X*Y
6.7.2.4.3 Divide the summed products of X*Y by the
number of paired scores (N)
6.7.2.4.4 Multiply the mean of the X scores X(mean)
by the mean of the Y scores Y(mean)
6.7.2.4.5 Minus the product of X(mean)*Y(mean) from
the product of the division in step 3
6.7.2.4.6 Multiply the standard deviation of X
scores Sx by the standard
deviation of Y scores Sy.
6.7.2.4.7 Divide the product of step 5 by the product
of step 6
6.7.3 Raw Score Formula
6.7.3.1
With
this formula, you start with the raw scores and obtain r without having to
compute means and standard deviations
6.7.3.2
Formula
6.7.3.2.1 r=(N*(j(X(each value)*Y(each value))))-(( jX)(*(
jY))/Square
Root [(N*(jX2 )-( jX)2]*[
(N*(jY2 )-( jY)
2]
6.7.3.3
Variables
Defined
6.7.3.3.1 j=Sum
6.7.3.3.2 XY=Product of each X value multiplied by
its paired Y value
6.7.3.3.3 X(mean)=Mean of variable X
6.7.3.3.4 Y(mean)=Mean of variable Y
6.7.3.3.5 N=Number of pairs of observations
6.7.3.4
Procedure
6.7.3.4.1 Multiply each paired X and Y score
6.7.3.4.2 Sum the products of X*Y
6.7.3.4.3 Multiply the summed products by the number
of paired observations.
6.7.3.4.4 Sum the X scores
6.7.3.4.5 Sum the Y scores
6.7.3.4.6 Multiply the summed X scores by the Summed
Y scores
6.7.3.4.7 Minus the product of step 6 (Summed X
scores*Summed Y scores) from the product of step 4 (summed products*N)
6.7.3.4.8 Square each X score (X2) and sum
the products
6.7.3.4.9 Multiply the product of step 8 (Summed
products of X*X (X2)) by the number of paired scores.
6.7.3.4.10
Sum the X
scores and square the product (jX*jX) or (jX) 2.
6.7.3.4.11
Minus the
product of step 10 ((jX) 2) from the product of step 9
(N*(jX2)
6.7.3.4.12
Square each Y
score (Y 2) and sum the products
6.7.3.4.13
Multiply the
product of step 12 (Summed products of Y*Y (Y 2)) by the number of
paired scores.
6.7.3.4.14
Sum the Y
scores and square the product (jY *j Y) or (jY) 2.
6.7.3.4.15
Minus the
product of step 14 ((jY) 2) from the product of step
13 (N*(j Y 2)
6.7.3.4.16
Multiply the
product of step 15 [N*(j Y 2)- ((jY)
2)] by the product of step 11 [N*(j X 2)- ((jX)
2)]
6.7.3.4.17
Obtain the
square root of step 16 [N*(j X 2)- ((jX)
2)]* [N*(j Y 2)- ((jY)
2)]
6.7.3.4.18
Divide the
product of step 7 [(N*jXY)-(( jX)*( jY))] by the product of step 17 [SQUARE
ROOT[N*(j X 2)- ((jX)
2)]* [N*(j Y 2)- ((jY)
2)]]
6.8 The Meaning Of r
6.8.1 (r)=is a descriptive statistic
or summary index number, like the mean and standard deviation and is used to
describe a set of data.
6.8.2 A correlation coefficient is a
measure of the relationship between two variables. It describes a the tendency
of two variables to vary together (covary); that is, it describes the tendency
of high or low values of one variable to be regularly associated with either
high or low values of the other variable. The absolute size of the coefficient
(from 0 to 1.00) indicates the strength of that tendency to covary.
6.8.3 Illustration
6.8.3.1
6.8.4 The above scatterplot shows the
correlational relationships of r=.20, .40, .60, and .80. Notice that as the
size of the correlation coefficient gets larger, the points cluster more and
more closely to the regression line; that is, the envelope containing the
points becomes thinner and thinner. This means that a stronger and stronger tendency
to covary exists as r becomes larger and larger. It also means that predictions
made about values of the Y variable from values of the X variable will be more
accurate when r is larger.
6.8.5 The algebraic sign tells the
direction of the covariation. When the sign is positive, high values of X are
associated with high values of Y, and low values of X are associated with low
values of Y. When the sign is negative, high values of X are associated with
low values of Y, and low values of X are associated with high values of Y.
Knowledge of the size and direction of r, then, permits some prediction of the
value of one variable if the value of the other variable is known.
6.8.6 Correlation vs. Causation
6.8.6.1
A
correlation coefficient does not tell you whether or not one of the variables
is causing the variation in the other. Quite possibly some third variable is
responsible for the variation in both.
6.8.6.2
A
correlation coefficient alone cannot establish a causal relationship.
6.8.7 Coefficient of Determination
6.8.7.1
This
is an overall index that specifies the proportion of variance that two
variables have in common.
6.8.7.2
Formula
6.8.7.2.1 COD=r2
6.8.7.3
Variables
Defined
6.8.7.3.1 COD=Coefficient of Determination
6.8.7.3.2 ( r )=Pearson product-moment correlation
coefficient
6.8.7.4
Procedure
6.8.7.4.1 Multiply r * r (r2)
6.8.7.5
It
could be argued that the proportion of variance the two variables have in
common can be attributed to the same cause. Or that this is the percentage of
variance which adheres most closely to the regression line.
6.8.7.6
Note
what happens to a fairly strong correlation of .70 when it is interpreted in
terms of variance. Only 49 % of the variance is held in common.
6.8.7.7
The
coefficient is useful in comparing correlation coefficients. When one compares
an r of .80 with an r of .40, the tendency is to think of the .80 as being
twice as high as .40, but that is not the case. Correlation coefficients are
compared in terms of the amount of common variance. .802=.64, .402=.16,
.64/.16=4 Thus, two variables that are correlated with r=.80 have four times as
much variance as two variables correlated with r=.40
6.8.8 Practical Significance of r
6.8.8.1
How
high must a correlation coefficient be before it is of use? How low must it be
before we conclude it is useless? Correlation is useful if it improves
prediction over guessing. In this sense, any reliable correlation other than
zero, whether positive or negative, is of some value because it will reduce to
some extent the incorrect predictions that might other wise be made. Very low
correlations allow little improvement over guessing in prediction. Such poor
prediction usually is not worth the costs involved in practical situations.
Generally, researchers are satisfied with lower correlations in theoretical
work but require higher ones in practical situations.
6.9 Correlation and Linearity
6.9.1 For r to be a meaningful
statistic, the best fitting line through the scatterplot of points must be a
straight line. If a curved regression line fits the data better than a straight
lie, r will be low, not reflecting the true relationship between the two
variables. The product-moment correlation coefficient is not appropriate as a
measure of curved relationships. Special non-linear correlation techniques for
such relationships do exist and are described elsewhere.[4]
[5]
6.10 Other Kinds of Correlation Coefficients
6.10.1
Dichotomous
Variables
6.10.1.1
Correlations
may be computed on data for which one or both of the variables are dichotomous
(having only two possible values). An example is the correlation of the
dichotomous variable sex and the quantitative variable grade-point average.
6.10.2
Multiple
correlation
6.10.2.1
Several
variables can be combined, and the resulting combination can be correlated with
one variable. With this technique, called multiple correlation a more precise
prediction can be made. Performance in school or on the job can usually be
predicted better by using several measures of a person rather than just one.
6.10.3
Partial
correlation
6.10.3.1
A
technique called partial correlation allows you to separate or partial out the
effects of one variable from the correlation of two other variables. For example,
if we want to know the true correlation between achievement-test scores in two
school subjects it will probably be necessary to partial out the effects of
intelligence since IQ and achievement are correlated.
6.10.4
Rho
for Ranked data
6.10.4.1
Rho
is used when the data are ranks rather than raw scores.
6.10.5
Non-linear
correlation
6.10.5.1
If
the relationship between two variables is curved rather than linear, the
correlation ratio, eta gives the degree of association.
6.10.6
Intermediate-level
statistic text books
6.10.6.1
The
above correlation techniques are covered in intermediate level text books. [6]
[7]
6.11 Correlation and Regression
6.11.1
6.12 Regression Equation 109
6.12.1
6.12.2
Formula
6.12.2.1
Y
=a+bX
6.12.3
Variables
Defined
6.12.3.1
Y
=the Y value predicted from a particular X value (Y is pronounced
y prime).
6.12.3.2
a=the
point at which the regression line intersects the Y axis
6.12.3.3
b=the
slope of the regression line--that is, the amount Y is increasing for each
increase of one unit in X
6.12.3.4
X=the
X value used to predict Y .
6.12.3.5
Regression
Coefficients
6.12.3.5.1
The symbols X
and Y can be assigned arbitrarily in correlation, but, in a regression
equation, Y is assigned to the variable you wish to predict. To make
predictions of Y using the regression equation, you need to calculate the
values of the constants a and b, which are called regression coefficients.
6.12.3.5.2
Formula
6.12.3.5.2.1
b=r*(Sy/Sx)
6.12.3.5.3
Variables
Defined
6.12.3.5.3.1
r=correlation
coefficient for X and Y
6.12.3.5.3.2
Sy
=the standard deviation of the Y variable
6.12.3.5.3.3
Sx
=the standard deviation of the X variable
6.12.3.5.3.4
Notice that
for positive correlation b will be a positive number. For negative correlation
b will be negative
6.12.3.5.4
Formula
6.12.3.5.4.1
a=Y(mean)-b*X(mean)
6.12.3.5.5
Variables
Defined
6.12.3.5.5.1
Y(mean)=Mean
of the Y scores
6.12.3.5.5.2
b=regression
coefficient computed above
6.12.3.5.5.3
X(mean)=mean
of the x scores
6.12.4
Procedure
6.12.4.1
Calculate
b
6.12.4.1.1
Divide Sy
(standard deviation of y) by Sx (standard deviation of x)
6.12.4.1.2
Multiply the
product of step 1 (Sy/Sx) by r (correlation coefficient
for X and Y)
6.12.4.2
Calculate
a
6.12.4.2.1
Multiply
X(mean) (mean of the x scores) by b (regression coefficient computed in step 1
above)
6.12.4.2.2
Minus the product
of the previous step above from Y(mean)
(Mean of the Y scores)
6.12.4.3
Calculate
the predicted Y score
6.12.4.3.1
Multiply X
(value used to predict Y) by b (calculated in step 1 above)
6.12.4.3.2
Add the
product of the previous step above to a (product of step 2 above)
6.12.5
Drawing
a Regression Line
6.12.5.1
6.12.6
Predicting
a Y Score
6.12.6.1
6.13 Rank Order Correlation
6.13.1
6.13.2
Web
6.13.3
6.14 r Distribution Tables
6.14.1
Web
6.14.1.2
7.1 A raw score does not reveal its
relationship to other scores and must be transformed into a score that reveals
these relationships. There are two types of score transformations; percentile
ranks and linear transformations.
7.2 Purpose
7.2.1 A relationship between scores is
revealed increasing the amount of information for analytical interpretation.
7.2.2 Allows two scores to be
compared.
7.3 Percentile Ranks Based On The
Sample
7.3.1 The percentile rank is the
percentage of scores that fall below a given score.
7.3.2 Procedure
7.3.2.1
Rate
the scores from lowest to highest and determine total number of scores.
7.3.2.1.1 Example
7.3.2.1.1.1 33 28 29 37 31 33 25 33 29 32 35
7.3.2.1.1.2 25 28 29 29 31 32 33 33 33 35 37=Total
number of scores=11
7.3.2.2
Determine
the number of scores falling below the selected score
7.3.2.2.1 Example Number=31
7.3.2.2.1.1 Number of scores below=4
7.3.2.3
Determine
the percentage of scores which fall below the selected score by dividing the
number of scores below by the total number of scores and multiplying by 100.
7.3.2.3.1 Example=4/11=.364*100=36.4%
7.3.2.4
Determine
the percentage of scores which fall at the selected score by dividing that
number by the total number of scores and multiplying by 100.
7.3.2.4.1 Example=1/11=.09009*100=9.09
7.3.2.5
Divide
the percentage of scores at the selected score by 2 and add the product to the
percentage of scores below the selected score.
7.3.2.5.1 Example=9.09/2=4.55+36.4=40.95%
7.3.2.5.2 This would mean that the percentage of
scores falling below the score of 31would be 40.95% and that would be the
scores percentile rank.
7.3.2.6
Brief
Summary of Process
7.3.2.6.1 Rank the scores from lowest to highest
7.3.2.6.2 Add the percentage of scores that fall
below the score to one-half the percentage of scores that fall at the score.
7.3.2.6.3 The result is the percentile rank of that
score which is the percentage of scores which fall below the selected score.
7.3.2.7
Another
example=Selected Score=33
7.3.2.7.1 ((6/11)+((3/11)/2))*100=68.18%
7.3.2.7.2 This would mean that the percentage of
scores falling below the score of 33 would be 68.18% and that would be the
scores percentile rank.
7.3.2.8
Example
formula
7.3.2.8.1 
7.3.2.9
Example
of the algebraic procedure applied to the selected numbers of 31 and 33.
7.3.2.9.1 31
7.3.2.9.1.1 
7.3.2.9.2 33
7.3.2.9.2.1 
7.4 Percentile Ranks Based On The
Normal Curve
7.4.1
8.1
8.2 ADDITIVE COMPONENT
8.2.1
8.3 MULTIPLICATIVE COMPONENT
8.3.1
8.4
