Bias

If a test statisticis used as an estimate for a population parameterthen in general we will not expect the value ofto equal– the estimator may be biased forThe bias of the estimator is given by

An estimatorfor a population parameteris an unbiased estimator forif

An unbiased estimator is always better than a biased estimator, and an estimator is better in general if the bias is small and/or the variance is small.

For exampleis a biased estimator forbutis an unbiased estimator forThe relationship betweenandisso that The bias ofis then

If we have two estimators for the mean of a population,andwiththenandso bothandare unbiased estimators forbut

and

The variance ofis less than the variance ofsois a better estimator for