meansquared

Meansquared is a term used to describe the mean of the squares of a set of numbers, also known as the second moment about zero. For a dataset x1, x2, ..., xn, meansquared equals (1/n) sum xi^2.

It is not the same as the square of the mean, which is (mean of xi)^2; indeed E[X^2] and (E[X])^2 differ by the variance: Var(X) = E[X^2] - (E[X])^2.

The square root of meansquared is the root mean square (RMS). If the data have zero mean, RMS equals the standard deviation.

In practice, meansquared is used as a measure of energy or magnitude, in signal processing and statistics. It is also a component of many loss functions, most notably the mean squared error (MSE) used to measure prediction error in regression and machine learning. When evaluating models, MSE is the average of squared residuals; it uses the same quantities as meansquared but in a different context.

In mathematics and data analysis, the term second moment about zero or E[X^2] is sometimes preferred. In common usage, people refer to the mean of squares rather than "meansquared" as a single word; the interpretation depends on the context.