Root Mean Square Calculator — RMS of Any Numbers
Enter any set of numbers (positive or negative, up to any count) to compute their root mean square, along with the sum of squares and arithmetic mean.
√( (x₁² + x₂² + … + xₙ²) / n )
- 1
Count values
n = 5 = 5 - 2
Sum of squares
x₁² + x₂² + … + xₙ² = 90Squares all 5 values then adds them together. - 3
Mean of squares
90 ÷ 5 = 18 - 4
Root Mean Square
√(18) = 4.2426
How does this calculator work?
RMS = √((x₁² + x₂² + … + xₙ²) / n). Square every number, average the squares, then take the square root. The RMS is always ≥ 0 and ≥ |arithmetic mean|. Enter any comma-separated numbers to compute it instantly.
Formula
How this is calculated
The root mean square — also called the quadratic mean — is calculated in three steps: square each value, take the arithmetic mean of those squares, then take the square root of that mean. Formally RMS = √((Σxᵢ²) / n). Because squaring removes the sign, the RMS is always non-negative and is always ≥ the absolute value of the arithmetic mean (by the root-mean-square–arithmetic-mean inequality). This makes it useful when the sign of individual values is irrelevant to the "size" of the quantity.
In electrical engineering the RMS of an AC voltage or current gives the equivalent DC value that delivers the same power. In statistics it appears as the quadratic mean. In signal processing it measures the power content of a waveform. For a pure sine wave with amplitude A, the RMS is A/√2 ≈ 0.707A — the reason mains AC at "230 V" really means 230 V RMS.
The arithmetic mean, in contrast, cancels positive and negative values, so it is zero for a symmetric AC signal. The RMS captures the magnitude regardless of sign, which is why it is preferred for power-related calculations.
Frequently asked questions
The arithmetic mean (average) sums values and divides by n, so positive and negative values cancel. The RMS squares each value first, so signs are removed before averaging — giving a measure of magnitude that is always ≥ 0 and always ≥ |arithmetic mean|.
For a sine wave y = A·sin(θ), the RMS over a full period is A / √2 ≈ 0.707 A. This is why a 230 V RMS mains supply has a peak voltage of 230 × √2 ≈ 325 V.
Only if every value in the set is zero. Otherwise, because each value is squared before averaging, the sum of squares is positive, so the RMS is always positive for any non-zero dataset.
Also known as
TG we-Calculate Editorial Team. (2026). Root Mean Square Calculator — RMS of Any Numbers [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/root-mean-square-calculator
TG we-Calculate Editorial Team. "Root Mean Square Calculator — RMS of Any Numbers." TG we-Calculate. 2026. https://we-calculate.com/calculator/root-mean-square-calculator.
TG we-Calculate Editorial Team, "Root Mean Square Calculator — RMS of Any Numbers," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/root-mean-square-calculator
@misc{wecalculate_root_mean_square_calculator, title = {Root Mean Square Calculator — RMS of Any Numbers}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/root-mean-square-calculator}}, year = {2026}, note = {TG we-Calculate} }
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