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A standard deviation is obtained by taking the positive square root (with positive sign) of the variance. The population standard deviation and the sample standard deviation are defined as follows:

(2.5.10)

(2.5.11)

Example 2.5.10 (Lengths of certain chips) The following data give the length (in millimeters) of material chips removed during a machining operation:

4, 2, 5, 1, 3, 6, 2, 4, 3, 5

Determine the variance and the standard deviation for these data.

Solution: There are three simple steps to calculate the variance of any data set.

1 Step 1. Calculate , the sum of all the data values, that is,

2 Step 2. Calculate , the sum of squares of all the observations, that is,

3 Step 3. Since the sample size is , by inserting the values and , calculated in Step 1 and Step 2 in formula (2.5.9), the sample variance is given by

The standard deviation is obtained by taking the square root of the variance, that is

Note: It is important to remember the value of , and therefore of , is always greater than zero, except when all the data values are equal, in which case .