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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

What it is

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.

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.