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

SECTION 4.2 | Defining Standard Deviation and Variance 105
(see page 72). Thus, the total for the positive deviations is exactly equal to the total for the
negative deviations, and the complete set of deviations always adds up to zero.
Because the sum of the deviations is always zero, the mean of the deviations is also zero
and is of no value as a measure of variability. Specifically, the mean of the deviations is
zero if the scores are closely clustered and it is zero if the scores are widely scattered. (You
should note, however, that the constant value of zero is useful in other ways. Whenever you
are working with deviation scores, you can check your calculations by making sure that the
deviation scores add up to zero.)
STEP 3
The average of the deviation scores will not work as a measure of variability because it is
always zero. Clearly, this problem results from the positive and negative values canceling
each other out. The solution is to get rid of the signs (1 and 2). The standard procedure
for accomplishing this is to square each deviation score. Using the squared values, you then
compute the mean squared deviation, which is called variance.
DEFINITION
Variance equals the mean of the squared deviations. Variance is the average
squared distance from the mean.
Note that the process of squaring deviation scores does more than simply get rid of
plus and minus signs. It results in a measure of variability based on squared distances.
Although variance is valuable for some of the inferential statistical methods covered
later, the concept of squared distance is not an intuitive or easy to understand descriptive
measure. For example, it is not particularly useful to know that the squared distance
from New York City to Boston is 26,244 miles squared. The squared value becomes
meaningful, however, if you take the square root. Therefore, we continue the process
one more step.
STEP 4
Remember that our goal is to compute a measure of the standard distance from the mean.
Variance, which measures the average squared distance from the mean, is not exactly what
we want. The final step simply takes the square root of the variance to obtain the standard
deviation, which measures the standard distance from the mean.
DEFINITION
Standard deviation is the square root of the variance and provides a measure of
the standard, or average distance from the mean.
Standard deviation 5 Ïvariance
Figure 4.2 shows the overall process of computing variance and standard deviation.
Remember that our goal is to measure variability by finding the standard distance from
the mean. However, we cannot simply calculate the average of the distances because this
value will always be zero. Therefore, we begin by squaring each distance, then we find the
average of the squared distances, and finally we take the square root to obtain a measure of
the standard distance. Technically, the standard deviation is the square root of the average
squared deviation. Conceptually, however, the standard deviation provides a measure of the
average distance from the mean.
Although we still have not presented any formulas for variance or standard deviation,
you should be able to compute these two statistical values from their definitions. The following
example demonstrates this process.