# Bias of an estimator

## Statistical property / From Wikipedia, the free encyclopedia

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In statistics, the **bias of an estimator** (or **bias function**) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called * unbiased*. In statistics, "bias" is an

*objective*property of an estimator. Bias is a distinct concept from consistency: consistent estimators converge in probability to the true value of the parameter, but may be biased or unbiased; see bias versus consistency for more.

All else being equal, an unbiased estimator is preferable to a biased estimator, although in practice, biased estimators (with generally small bias) are frequently used. When a biased estimator is used, bounds of the bias are calculated. A biased estimator may be used for various reasons: because an unbiased estimator does not exist without further assumptions about a population; because an estimator is difficult to compute (as in unbiased estimation of standard deviation); because a biased estimator may be unbiased with respect to different measures of central tendency; because a biased estimator gives a lower value of some loss function (particularly mean squared error) compared with unbiased estimators (notably in shrinkage estimators); or because in some cases being unbiased is too strong a condition, and the only unbiased estimators are not useful.

Bias can also be measured with respect to the median, rather than the mean (expected value), in which case one distinguishes *median*-unbiased from the usual *mean*-unbiasedness property.
Mean-unbiasedness is not preserved under non-linear transformations, though median-unbiasedness is (see § Effect of transformations); for example, the sample variance is a biased estimator for the population variance. These are all illustrated below.

An unbiased estimator for a parameter need not always exist. For example, there is no unbiased estimator for the reciprocal of the parameter of a binomial random variable.^{[1]}