Understanding Psychology (10th Ed)

standard deviation
An index of the 
average deviation of a set of scores 
from the center of the distribution.
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A-12 Appendix 
Going by the Numbers: Statistics in Psychology
FIGURE 2 
How to calculate a standard deviation.
Calculating a Standard Deviation
1. The calculation of a standard deviation begins with the calculation of the mean of 
distribution. In the following distribution of scores on a psychology student’s weekly 
quizzes, the mean is 84.5: 82, 88, 71, 86, 96, 84. (As you recall, the mean is the sum 
of the scores divided by the number of scores in the distribution, or 507
4
6
5
84.5.)
2. The next step is to produce a deviation score for each score in the distribution. A 
deviation score is simply an original score minus the mean of all the scores in a 
distribution. This has been done in the second column below:

 
Deviation 
Score
Original Score 
Deviation Score 
Squared
82 
2
2.5 6.25
88 
3.5 
12.25
71 
2
13.5 182.25
86 
1.5 
2.25
96 
11.5 
132.25
84 
2
.5 .25
3. In the third step, the deviation scores are squared (multiplied by themselves) to 
eliminate negative numbers. This has been carried out in the third column above.
4. The squared deviation scores are then added together; and this sum is divided by 
the number of scores. In the example above, the sum of the squared deviation 
scores is 6.25
1
12.25
1
182.25
1
2.25
1
132.25
1
.25
5
335.50, and 335.50
4
6
5
55.916.
5.
The fi nal step is to take the square root of the resulting number. The square root of 
55.916 is 7.4777—which is the standard deviation of the distribution of scores.
6. To summarize, the standard deviation is calculated using the formula 
o
(score 
2
mean)
2
N
√
Original score minus the mean of 84.5
(Note: Because this formula provides the standard deviation for a sample, the sum of 
the deviation scores is divided by the number of scores, 
N
. However, in some cases, in 
which we might wish to generalize beyond the specifi c sample to a larger population, 
the standard deviation is calculated by using the number of scores minus 1, or 
N
2
1.)
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