Skip to main content
Edlintics

Mean squared error

Mean squared error (MSE) measures estimator accuracy by summing two components: the estimator's variance and its squared bias. For an unbiased estimator, MSE equals variance alone.

Also known asMSE

ByHoang TruongUpdated

See it move

Loading infographic...

The infographic is a tree diagram showing that MSE(θ̂) decomposes additively into two branches: Variance, which captures the scatter of repeated estimates around their own mean, and Bias², which measures the squared systematic offset of the estimator from the true parameter value. The equality MSE = Variance + Bias² means a single number summarises both dimensions of estimation quality. An estimator can be low-variance but highly biased, or unbiased but imprecise, and the MSE reveals the combined cost of both failings.

Where it fits
SubjectData Analysis & StatisticsAdvancedTopicEstimation & Sampling DistributionsAdvanced

The formula

LaTeX
MSE(θ^)=E[(θ^−θ)2]\text{MSE}(\hat{\theta}) = E[(\hat{\theta} - \theta)^2]

Variables

Estimator of the true parameter θ
True (unknown) population parameter

Defining form: the expected squared distance between the estimator and the true value.

LaTeX
MSE(θ^)=Var(θ^)+[Bias(θ^)]2\text{MSE}(\hat{\theta}) = \text{Var}(\hat{\theta}) + [\text{Bias}(\hat{\theta})]^2

Variables

Variance of the estimator across repeated samples
Systematic deviation of the estimator from the true parameter: E[θ̂] − θ

Bias-variance decomposition: for an unbiased estimator, Bias = 0 and MSE reduces to variance alone.