Standard, Deviation, Variance and Bias

Given a flock of sheep, we can measure the masses of a sample of sheep and calculate the mean mass and the standard deviation of the masses of the sheep in the sample.

Suppose the masses of the sheep are

50, 55, 52, 60, 54, 49, 59, 58

The sample mean is

The sum of the squares of the masses is

The variance of the sample, called the sample variance is thenand the sample standard deviation

This is not that same as the variance of the masses of the whole population of sheep in general, and is not even a very good estimate.

A better estimate is given by the formula

This is said to be an unbiased estimator for the population standard deviation, and is generally a better estimate.

For the sample aboveand an unbiased estimator for the standard deviation is