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

110 CHAPTER 4 | Variability
X X 2
1 1
0 0
6 36
1 1
SX 5 8 SX 2 5 38
SS 5SX 2 2 sSXd2
N
5 38 2 s8d2
4
5 38 2 64 4
5 38 2 16
5 22
■
Note that the two formulas produce exactly the same value for SS. Although the formulas
look different, they are in fact equivalent. The definitional formula provides the most
direct representation of the concept of SS; however, this formula can be awkward to use,
especially if the mean includes a fraction or decimal value. If you have a small group of
scores and the mean is a whole number, then the definitional formula is fine; otherwise, the
computational formula is usually easier to use.
In the same way that
sum of squares, or SS, is
used to refer to the sum
of squared deviations,
the term mean square,
or MS, is often used to
refer to variance, which
is the mean squared
deviation.
■ Final Formulas and Notation
With the definition and calculation of SS behind you, the equations for variance and standard
deviation become relatively simple. Remember that variance is defined as the mean
squared deviation. The mean is the sum of the squared deviations divided by N, so the
equation for the population variance is
variance 5 SS
N
Standard deviation is the square root of variance, so the equation for the population
standard deviation is
standard deviation 5Î SS
N
There is one final bit of notation before we work completely through an example computing
SS, variance, and standard deviation. Like the mean (m), variance and standard
deviation are parameters of a population and are identified by Greek letters. To identify the
standard deviation, we use the Greek letter sigma (the Greek letter s, standing for standard
deviation). The capital letter sigma (S) has been used already, so we now use the lowercase
sigma, s, as the symbol for the population standard deviation. To emphasize the relationship
between standard deviation and variance, we use s 2 as the symbol for population variance
(standard deviation is the square root of the variance). Thus,
population standard deviation 5s5Ïs 2 5Î
population variance 5s 2 5 SS
N
SS
N
(4.3)
(4.4)
DEFINITIONS
Population variance is represented by the symbol s 2 and equals the mean squared
distance from the mean. Population variance is obtained by dividing the sum of
squares by N.
Population standard deviation is represented by the symbol s and equals the
square root of the population variance.