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Mean Square Error

The mean square error (MSE) of predictions y^^_i for observed values y_i is the arithmetic mean of the squared errors e_i=y_i-y^^_i,

MSE=1/nsum_(i=1)^ne_i^2.

For an estimator theta^^ of a parameter theta, its mean square error is

MSE(theta^^)=E[(theta^^-theta)^2]=Var(theta^^)+Bias(theta^^)^2,

where E[Y] denotes the expectation value of Y. Thus mean square error combines the variance and squared estimator bias. Its square root is the root-mean-square error, which has the same units as the measured quantity.

See also

Estimator, Estimator Bias, Expectation Value, Mean Absolute Error, Mean Absolute Percentage Error, Root-Mean-Square Error, Variance

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References

Hyndman, R. J. and Koehler, A. B. "Another Look at Measures of Forecast Accuracy." Int. J. Forecasting 22, 679-688, 2006. https://doi.org/10.1016/j.ijforecast.2006.03.001.Iyanaga, S. and Kawada, Y. (Eds.). "Statistical Estimation and Statistical Hypothesis Testing." Appendix A, Table 23 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1486-1489, 1980.

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Mean Square Error

Cite this as:

Weisstein, Eric W. "Mean Square Error." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MeanSquareError.html

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