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Standard Deviation

The standard deviation is a numerical value used to indicate how widely individuals in a group vary. If individual observations vary greatly from the group mean, the standard deviation is big; and vice versa.

It is important to distinguish between the standard deviation of a population and the standard deviation of a sample. They have different notation, and they are computed differently. The standard deviation of a population is denoted by σ and the standard deviation of a sample, by s.

The standard deviation of a population is defined by the following formula:

σ = sqrt [ Σ ( X_i - X )² / N ]

where σ is the population standard deviation, X is the population mean, X_i is the ith element from the population, and N is the number of elements in the population.

The standard deviation of a sample is defined by slightly different formula:

s = sqrt [ Σ ( x_i - x )² / ( n - 1 ) ]

where s is the sample standard deviation, x is the sample mean, x_i is the ith element from the sample, and n is the number of elements in the sample.

And finally, the standard deviation is equal to the square root of the variance.

See also:

Statistics Tutorial: Measures of Variability | AP Statistics Tutorial: Measures of Variability | Random Variable Attributes