Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

126 CHAPTER 4 | Variability
Except for minor changes in notation, the calculation
of SS is identical for samples and populations. There
are two methods for calculating SS:
I. By definition, you can find SS using the following
steps:
a. Find the deviation (X 2 ) for each score.
b. Square each deviation.
c. Add the squared deviations.
This process can be summarized in a formula as
follows:
Definitional formula: SS 5 S(X 2 ) 2
II. The sum of the squared deviations can also be found
using a computational formula, which is especially
useful when the mean is not a whole number:
Computational formula: SS 5SX 2 2 sSXd2
N
3. Variance is the mean squared deviation and is
obtained by finding the sum of the squared deviations
and then dividing by the number of scores. For a
population, variance is
s 2 5 SS
N
For a sample, only n 2 1 of the scores are free to
vary (degrees of freedom or df 5 n 2 1), so sample
variance is
s 2 5
SS
n 2 1 5 SS
df
Using n 2 1 in the sample formula makes the sample
variance an accurate and unbiased estimate of the
population variance.
4. Standard deviation is the square root of the variance.
For a population, this is
s5Î SS
N
Sample standard deviation is
s 5Î SS
n 2 1
Î 5 SS
df
5. Adding a constant value to every score in a distribution
does not change the standard deviation. Multiplying
every score by a constant, however, causes
the standard deviation to be multiplied by the same
constant.
6. Because the mean identifies the center of a distribution
and the standard deviation describes the average
distance from the mean, these two values should
allow you to create a reasonably accurate image of
the entire distribution. Knowing the mean and standard
deviation should also allow you to describe the
relative location of any individual score within the
distribution.
7. Large variance can obscure patterns in the data
and, therefore, can create a problem for inferential
statistics.
KEY TERMS
variability (101)
range (102)
deviation score (104)
population variance (s 2 ) (110)
population standard deviation (s) (110)
sum of squares (SS) (108)
sample variance (s 2 ) (113)
sample standard deviation (s) (113)
degrees of freedom (df) (115)
biased statistic (117)
unbiased statistic (117)
SPSS ®
General instructions for using SPSS are presented in Appendix D. Following are detailed
instructions for using SPSS to compute the Range, Standard Deviation, and Variance for a
sample of scores.
Data Entry
1. Enter all of the scores in one column of the data editor, probably VAR00001.
Data Analysis
1. Click Analyze on the tool bar, select Descriptive Statistics, and click on Descriptives.