Root Mean Square (RMS) Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The quadratic mean is calculated with QM = √(Σvᵢ² ÷ n), the square root of the mean of the squares. For the values 2 and 8, the quadratic mean is about 5.83, versus 5 for the classic arithmetic mean.
Explanation
The root mean square (RMS, or quadratic mean) squares each value before averaging, then takes the square root of the result — a construction that gives more weight to large-amplitude values, positive or negative, than a simple arithmetic mean does. It is the fourth of the classic means covered by this site, after the arithmetic, geometric, and harmonic means: these four means always follow the same order for a given set of not-all-identical values, the quadratic mean being consistently the highest of the four (QM ≥ AM ≥ GM ≥ HM), with strict equality between all only when all values are exactly identical. The quadratic mean is particularly useful for values that can be positive or negative but whose overall amplitude you want to measure rather than their signed average value: this is exactly the principle behind the RMS voltage of an alternating signal, which oscillates symmetrically between positive and negative values, but whose quadratic mean gives a non-zero, physically meaningful value — unlike the arithmetic mean of a perfectly sinusoidal signal, which would always be zero. The same idea appears in the root mean square speed of gas molecules.
Example: the values 2 and 8
Inputs
Values: 2 and 8.
Calculation
QM = √((2² + 8²) ÷ 2) = √((4 + 64) ÷ 2) = √34 ≈ 5.831. Arithmetic mean = (2+8) ÷ 2 = 5.
Result
The quadratic mean of these two values (5.831) is greater than their arithmetic mean (5), a systematic relationship between these two means.
Frequently asked questions
Why is the quadratic mean always greater than or equal to the arithmetic mean?
Because squaring each value amplifies large values proportionally more than small ones, which pulls the final result upward compared with a simple arithmetic mean. This inequality (QM ≥ AM) is a general mathematical property, provable for any set of values, with strict equality only in the special case where all values are exactly identical.
Why use the quadratic mean for an alternating electrical signal?
Because a sinusoidal alternating signal oscillates symmetrically between positive and negative values, which would make its classic arithmetic mean always zero over a full period (the positive and negative values canceling exactly). The quadratic mean first squares each value, which eliminates this sign problem, and gives a non-zero, physically meaningful value: the RMS voltage, directly related to the power actually delivered by the signal.
In what other field is the quadratic mean encountered?
It also appears in statistics as the RMSE (root mean square error) to assess the quality of a prediction model, in physics for the root mean square speed of the molecules of a gas, and more generally in any context where an overall amplitude must be measured without the positive and negative signs of individual deviations canceling each other out.