Weighted Mean - Weighted Sample Variance

Weighted Sample Variance

Typically when a mean is calculated it is important to know the variance and standard deviation about that mean. When a weighted mean is used, the variance of the weighted sample is different from the variance of the unweighted sample. The biased weighted sample variance is defined similarly to the normal biased sample variance:

where, which is 1 for normalized weights.

For small samples, it is customary to use an unbiased estimator for the population variance. In normal unweighted samples, the N in the denominator (corresponding to the sample size) is changed to N − 1. While this is simple in unweighted samples, it is not straightforward when the sample is weighted. The unbiased estimator of a weighted population variance (assuming each is drawn from a Gaussian distribution with variance ) is given by :

where as introduced previously. The degrees of freedom of the weighted, unbiased sample variance vary accordingly from N − 1 down to 0.

The standard deviation is simply the square root of the variance above.

If all of the are drawn from the same distribution and the integer weights indicate frequency of occurrence in the sample, then the unbiased estimator of the weighted population variance is given by

If all are unique, then counts the number of unique values, and counts the number of samples.

For example, if values are drawn from the same distribution, then we can treat this set as an unweighted sample, or we can treat it as the weighted sample with corresponding weights, and we should get the same results.