Another Measure of Variability: Standard Deviation
The standard deviation, like the range, is a measure of the variability of a set of data. While the range is based only on the maximum and minimum of the data, the standard deviation is calculated using all the data. The mean and standard deviation are often used together.

Consider these two sets of data, A and B, represented both in table form and dot plots.

Data
range, r
stdev, s
set A
1
1
5
5
4
2.3
set B
1
3
3
5
4
1.6

Both data sets have the same range (4), but set A has all values at the high and low ends. Set B has two values that are in the middle; this set is less dispersed. The greater dispersion of A compared B is reflected in the higher standard deviation 2.3 for A, against 1.6 for B. In this case, the standard deviation distinguishes a difference that the range did not.

The standard deviation of identical values, for example 2.3, 2.3, 2.3, 2.3 is zero.

Standard deviation does not have a simple formula. It is the square root of the average squared distance of the data from the mean. Click on formula for more information.

The standard deviation formula is difficult to evaluate by hand, but a calculator or computer program can do it easily. Both a lower-case s and the greek letter sigma () denote standard deviation. Use s for the standard deviation of a sample and for the standard deviation of a population. Use the s key if you are working with a sample. Use the key if you have every member of the population in your data. If in doubt use s.

In the Excel spreadsheet program, the standard deviation of a sample is given by =stdev( ); for a population use stdevp( ).

#17 Calculate the standard deviations of the Line 1 or Line 2 using a calculator or computer. Note that both groups have the same mean.

Descriptive statistics for both production lines are:

Line 1: = 14.5 oz. s = 1.27 Line 2: = 14.5 oz. s = 1.84

#18 In what way does the standard deviation for Line 1 and Line 2 reflect the histograms?

#19 As both the manufacturer and the customer which pattern of variability would you rather have?