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Quadratic Mean / Root Mean Square

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Descriptive Statistics > Quadratic Mean / Root Mean Square

What is the Quadratic Mean / Root Mean Square?

The quadratic mean (also called the root mean square*) is a type of average.

Sometimes the quadratic mean is referred to as being “the same as” the standard deviation. This isn’t strictly true: standard deviation is actually equal to the quadratic deviations from the mean of the data set. For example, quadratic mean is used in the physical sciences as a synonym for standard deviation when referencing the “square root of the mean squared deviation of a signal from a given baseline or fit”(Wolfram).

The quadratic mean is also called the root mean square because it is the square root of the mean of the squares of the numbers in the set.

*Note: This is different from the root mean square error (RMSE), which is a value used in regression analysis to describe how spread out data is around a regression line.


The quadratic mean is equal to the square root of the mean of the squared values. The formula is:
quadratic mean

An equivalent formula has a summation sign (summation means “to add up”, so it’s telling you here to add all of the squared x-values up):
quadratic-mean 2

Kenney, J. F. and Keeping, E. S. “Root Mean Square.” ยง4.15 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 59-60, 1962.
Wolfram. Root Mean Square. Available at:

Stephanie Glen. "Quadratic Mean / Root Mean Square" From Elementary Statistics for the rest of us!

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