Statistics How To

Quadratic Mean / Root Mean Square

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. This type of mean gives a greater weight to larger items in the set and is always equal to or greater than the arithmetic mean. It is used for specialized purposes, such as the expression of average stand diameter in forestry. In this case, the quadratic mean of tree stand diameter is closer to the “true” mean of a sample of trees rather than the arithmetic mean. The quadratic mean is also used anywhere where it’s the square of the values that matters, rather than the values themselves. For example, electrical current squared is proportional to power, so if you’re interested in total power (rather than current), this type of mean is a good choice.

Sometimes the quadratic mean is referred to as being “the same as” the standard deviation. 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.

*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

Worked Example

Find the Root Mean Square of 2,4,9,10,and 12.

Step 1: Count the number of items.
N = 5.
Set this number aside for a moment.

Step 2: Square all of the numbers. 22,42,92,102, 122 = 4, 16, 81, 100, 144.

Step 3: Add the numbers from Step 2 up:4 + 16 + 81 + 100 + 144 = 345.

Step 4: Divide Step 3 (the sum) by Step 1 (number of items in the set):
345/5 = 69.

Step 5: Find square root of Step 4. √(69) = 8.31.
That’s it!

Kenney, J. F. and Keeping, E. S. “Root Mean Square.” §4.15 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 59-60, 1962.
Wofram. Root MEan Square. Available at:


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