Standard Deviation

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A useful and commonly used measure of precision is the experimental standard deviation defined by the International Vocabulary of Basic and General Terms in Metrology (VIM) as... "for a series of n measurements of the same measurand, the quantity s characterizing the dispersion of the results and given by the formula: $$s = [ \sum {(x_i- \overline X)^2 \over (n-1)} ]^{1/2} \tag{1}$$ $x_i$ being the result of the i-th measurement and being the arithmetic mean of the n results considered." The above definition is for estimating the standard deviation for n values of a sample of a population and is always calculated using $n-1$. The standard deviation of a population is symbolized as s and is calculated using n. Unless the entire population is examined, s cannot be known and is estimated from samples randomly selected from it.
For example, an analyst may make four measurements upon a given production lot of material (population). The standard deviation of the set ($n=4$) of measurements would be estimated using ($n-1$). If this analysis was repeated several times to produce several sample sets (four each) of data, it would be expected that each set of measurements would have a different mean and a different estimate of the standard deviation. The experimental standard deviations of the mean for each set is calculated using the following expression: $$ {s \over (n)^{1/2}} \tag{2}$$ Using the example 1, where values of 1004, 1005, and 1001 were considered acceptable for the calculation of the mean and the experimental standard deviation the mean would be 1003, the experimental standard deviation would be 2 and the standard deviation of the mean would be 1.

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