The Sample Standard Deviation: A Measure of Precision

Equation 1.2 must be modified when it is applied to a small sample of data. Thus, the sample standard deviation s is given by the equation

                       (1.1)

where the quantity (xi - x̅ ) represents the deviation di of value xi from the mean x.

Note that Equation 1.1 differs from Equation 1.2 in two ways.

                                        (1.2)

 First, the sample mean, x, appears in the numerator in place of the population mean, m. Second, N in Equation 1.2 is replaced by the number of degrees of freedom (N - 1). When N - 1 is used instead of N, s is said to be an unbiased estimator of the population standard deviation s. If this substitution is not used, the calculated s will be less on average than the true standard deviation s, that is, s will have a negative bias. The sample variance s² is also of importance in statistical calculations.