# Mean square error or mean squared error

As a non-native english speaker I was wondering which of the square or squared expression I should use. For instance in mean square error or mean squared error.

According to the Internet, it seems both forms are used indistinctly. Is one expression more square than the other ?

• Just a side note. "Error" in abbreviations like MSE usually actual mean "residuals". Practically, residuals and errors frequently are treated as synonims, MSE=MSR. But theoretically there's important distinction between the two terms – ttnphns Jun 20 '12 at 17:55

The conceptual uses of "square" and "squared" are subtly different, although (almost) interchangeable:

• "Squared" refers to the past action of taking or computing the second power. E.g., $x^2$ is usually read as "x-squared," not "x-square." (The latter is sometimes encountered but I suspect it results from speakers who are accustomed to clipping their phrases or who just haven't heard the terminal dental in "x-squared.")

• "Square" refers to the result of taking the second power. E.g., $x^2$ can be referred to as the "square of x." (The illocution "squared of x" is never used.)

These suggest that a person using a phrase like "mean squared error" is thinking in terms of a computation: take the errors, square them, average those. The phrase "mean square error" has a more conceptual feel to it: average the square errors. The user of this phrase may be thinking in terms of square errors rather than the errors themselves. I believe this shows up especially in theoretical literature where the second form, "square," appears more often (I believe: I haven't systematically checked).

Obviously both are equivalent in function and safely interchangeable in practice. It is interesting, though, that some careful Google queries give substantially different hit counts. Presently,

"mean squared" -square -root -Einstein -Relativity


returns about 367,000 results (notice the necessity of ruling out the phrase "$e=m c^2$" popularly quoted in certain contexts, which demands the use of "squared" instead of "square" when written out), while

"mean square" -squared -root  -Einstein -Relativity


(maintaining analogous exclusions for comparability) returns an order of magnitude more, at 3.47 million results. This (weakly) suggests people favor "mean square" over "mean squared," but don't take this too much to heart: "mean squared" is used in official SAS documentation, for instance.

Nope! Both can be used.. interchangeably :-) it's the same.

Mean squared error sounds better to me but indeed both forms are used (see, e.g., the Wikipedia page).

They are absolutely NOT the same.

mean SQUARE error: square the quantity => calculate the error => calculate the mean

mean SQUARED error: calculate the error => square the result => calculate the mean

• Not so. I've never met your first usage, even as a mistake. If you were right, and I don't think you are, it would be much flagged that a minute difference in wording was associated with such a big difference in meaning: textbook writers would be obliged to explain at length and there would be campaigns to change the terminology. I challenge you to find even one explanation of your definition in the literature. @whuber already gave an excellent answer that remains definitive. – Nick Cox Oct 15 '13 at 18:55