---
tag: 01BD2TFFP4
kind: definition
name: Bias of an estimator
msc: [62F10]
defines: 0BWGTN92FG
uses: [0TTNRP5PFQ, 0DCT0QZVQ1]
sources:
  - url: https://en.wikipedia.org/wiki/Bias_of_an_estimator
    locator: Definition
---

## Definition

Let $\hat{\theta}$ be an [[estimator|0TTNRP5PFQ]] of a real parameter $\theta$, with $E_\theta[|\hat{\theta}|] < \infty$ for each $\theta$. Here $E_\theta$ is the expectation when $\theta$ is the true parameter.
The bias of $\hat{\theta}$ at $\theta$ is
$$\operatorname{Bias}_\theta(\hat{\theta}) = E_\theta[\hat{\theta}] - \theta.$$
The estimator is unbiased if $\operatorname{Bias}_\theta(\hat{\theta}) = 0$ for each $\theta$.

## Remarks

For an i.i.d. sample with mean $\mu$, the sample mean is an unbiased estimator of $\mu$.
The sample variance with divisor $n$ has bias $-\sigma^2/n$. With divisor $n - 1$, it is unbiased.
An unbiased estimator need not be good. The [[mean squared error|0V4VSCBXP3]] is variance plus squared bias, and a biased estimator can have a smaller mean squared error.
Unbiasedness is not preserved by nonlinear maps. For example, the sample standard deviation $S_n$ is a biased estimator of $\sigma$, even though $S_n^2$ is unbiased for $\sigma^2$.

## Explanations

### L1 human

An estimator is unbiased if it is right on average, over many repetitions of the experiment.
Bias is a systematic error: a scale that always shows $1$ kg too much has a bias of $1$ kg.