026. Variance and the Standard Deviation for a Sample

The variance and standard deviation for a sample still represent measures of dispersion around the mean. They are calculated quite similarly to those for a population. The Sample variance is

, (3.14)

Or , (3.14a)

Where N – 1 are degrees of freedom.

The Sample standard deviation is

. (3.15)

The Number of degrees of freedom in Formula (3.14) in any statistical operation is equal to the number of observations minus any constraints placed on those observations. A Constraint is any value that must be computed from the observations.

Another reason for deviation by “N – 1” in (3.14) is that a sample is generally a little less dispersed that the population from which it was taken. There is therefore a tendency for the sample standard deviation S to be a little less that the population standard deviation σ.

Formulas (3.12a) and (3.14a) reduces the required arithmetic, but provide no insight into the nature of a variance.

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