 Translation-invariant estimator

For a location estimator, the property historically called translation invariance in this setting is more explicitly called translation equivariance:Changing the origin of the measurements changes the estimated location by exactly the same amount. Both the sample mean and sample median satisfy this identity. For example, adding $10$ to every observation adds $10$ to either estimated location.

Strict invariance would instead leave the numerical answer unchanged. The sample variance has that different property because shifting every observation also shifts their sample mean, leaving all centered differences unchanged. The distinction matters when specifying the admissible estimators in a minimax asymptotic bias problem. The older translation-invariant terminology appears in Peter J. Huber's original location analysis.^[ref]