Editing Mean squared error (section)

{{Short description|Measure of the error of an estimator}}
{{distinguish-redirect|Mean squared deviation|Mean squared displacement}}
In [[statistics]], the '''mean squared error''' ('''MSE''')<ref name=":1">{{Cite web|title=Mean Squared Error (MSE)|url=https://www.probabilitycourse.com/chapter9/9_1_5_mean_squared_error_MSE.php|access-date=2020-09-12|website=www.probabilitycourse.com}}</ref> or '''mean squared deviation''' ('''MSD''') of an [[estimator]] (of a procedure for estimating an unobserved quantity) measures the [[expected value|average]] of the squares of the [[Error (statistics)|errors]]—that is, the average squared difference between the estimated values and the [[true value]]. MSE is a [[risk function]], corresponding to the [[expected value]] of the [[squared error loss]].<ref>{{cite book |title=Mathematical Statistics: Basic Ideas and Selected Topics |volume=I |edition=Second |last1=Bickel |first1=Peter J. |authorlink1=Peter J. Bickel |last2=Doksum |first2=Kjell A. |year=2015 |page=20 |quotation="If we use quadratic loss, our risk function is called the ''mean squared error'' (MSE) ..."}}</ref> The fact that MSE is almost always strictly positive (and not zero) is because of [[randomness]] or because the estimator [[Omitted-variable bias|does not account for information]] that could produce a more accurate estimate.<ref name="pointEstimation">{{cite book |first1=E. L. |last1=Lehmann |first2=George |last2=Casella  |title=Theory of Point Estimation |publisher=Springer |location=New York |year=1998 |edition=2nd |isbn=978-0-387-98502-2 |mr=1639875}}</ref> In [[machine learning]], specifically [[empirical risk minimization]], MSE may refer to the ''empirical'' risk (the average loss on an observed data set), as an estimate of the true MSE (the true risk: the average loss on the actual population distribution).

The MSE is a measure of the quality of an estimator.  As it is derived from the square of [[Euclidean distance]], it is always a positive value that decreases as the error approaches zero.

The MSE is the second [[moment (mathematics)|moment]] (about the origin) of the error, and thus incorporates both the [[variance]] of the estimator (how widely spread the estimates are from one [[data sample]] to another) and its [[Bias of an estimator|bias]] (how far off the average estimated value is from the true value).{{citation needed|reason=This is a known fact, you can look up "An Introduction to Statistical Learning" for a discussion, but it requires a citation|date=May 2021}} For an [[unbiased estimator]], the MSE is the variance of the estimator. Like the variance, MSE has the same units of measurement as the square of the quantity being estimated. In an analogy to [[standard deviation]], taking the square root of MSE yields the ''root-mean-square error'' or ''[[root-mean-square deviation]]'' (RMSE or RMSD), which has the same units as the quantity being estimated; for an unbiased estimator, the RMSE is the square root of the [[variance]], known as the [[standard error]].