Standard Deviation

The standard deviation (abbreviated as s or SD) represents the average amount of variability in a set of scores as the average distance from the mean. The larger the standard deviation, the larger the average distance each data point is from the mean of the distribution.

The formula for computing the standard deviation is as follows:

where

s is the standard deviation,

Σ is sigma, which tells you to find the sum of what follows,

X is each individual score,

X¯ is the mean of all the scores, and

n is the sample size.

This formula finds the difference between each individual score and the mean X − X¯, squares each difference, and sums them all together. Then it divides the sum by the size of the sample (minus 1) and takes the square root of the result. As is apparent, the standard deviation is an average deviation from the mean.

Here is a sample data set for the manual computation of the standard deviation:

5, 4, 6, 7, 8, 6, 5, 7, 9, 5

Table 1 Computing the Standard Deviation
X	(X − X¯)	(X − X¯)2
5	−1.2	1.44
4	−2.2	4.84
6	−.2	.04
7	.8	.64
8	1.8	3.24
6	−.2	.04
5	−1.2	1.44
7	.8	.64
9	2.8	7.84
5	−1.2	1.44
Sum	0	21.6

To compute the standard deviation, follow these steps:

1.
List each score. It doesn't matter whether the scores are in any particular order.
2.
Compute the mean of the group.
3.
Subtract the mean from each score.
4.
Square each individual difference. The result is the column marked (X − X¯)2 in Table 1.
5.
Sum all the squared deviations about the mean. As you can see in Table 1, the total is 21.6.
6.
Divide the sum by n − 1, or 10 − 1 = 9, so then 21.6/9 = 2.40.
7.
Compute the square root of 2.4, which is 1.55. That is the standard deviation for this set of 10 scores.

What we now know from these results is that each score in this distribution differs from the mean by an average of 1.55 points.

The deviations about the mean are squared to eliminate the negative signs. The square root of the entire value is taken to return the computation to the original units.

Every statistical package available computes the standard deviation. In Figure 1, Excel's Data Analysis ToolPak was used to compute a set of descriptive statistics, including the standard deviation. [Page 941]

Summary

• The standard deviation is computed as the average distance from the mean.
• The larger the standard deviation, the more spread out the values are, and the more different they are from

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