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# Standard Deviation Measures Variability

The standard deviation is a measure of variability of a set of numbers. For example, consider the numbers: 6, 10, 12, 8, 14, 16, 13 and 9. The sum of the eight numbers 88 and the average is 11.

The standard deviation is based on the deviation of each of the eight numbers from the average of the numbers. For example, the number 6 is 5 less than the average of 11 so its deviation is -5 (6-11). The deviation of 10 from the average is -1 (10-11). The deviations of the remaining numbers are 1, -3, 3, 5, 2, and -2 respectively. The sum of the eight deviations is zero.

To compute the standard deviation do the following:

1. Multiply each deviation by itself (square the deviation). For the example the squared deviations are: 25, 1, 1, 9, 9, 25, 4 and 4.
2. Compute the sum of the squared deviations. For the example the sum of the deviations squared is 78.
3. Divide the sum of the deviations squared by the number of observations minus 1. This quotient is called the variance. For the example the variance is 78/7 = 11.14.
4. Take the square root of the variance to obtain the standard deviation. For the example the standard deviation is the square root of 11.14 = 3.34.

The Excel spreadsheet STDEV function easily computes the standard deviation of a set of numbers.

Compare the Variability of Sets of Numbers

You can use the standard deviation to compare the variability between sets of numbers. For example, consider two sets:

Set 1: 6, 10, 12, 8, 14, 16, 13 and 9

Set 2: 8, -4, 5, 3, 35, 10, 25 and 6

The average of both sets is 11. But the standard deviations of the two sets are very different. Because the variability of the numbers in the second set is much more than the numbers in the first set, the second set's standard deviation is 12.72 compared with 3.34 for the first set.

See Coefficient of Variation Measures Variability.