Root Mean Square (RMS) Calculator
The quadratic mean — square, average, take the root — the measure behind AC voltage, signal power and the magnitude of an error.
RMS = √(Σx² ÷ n).
How the root mean square (rms) calculator works
RMS = √(Σx² ÷ n). Squaring removes the sign, so it measures magnitude without cancellation: a list averaging zero can still have a large RMS.
That is exactly why it is the standard for alternating signals. Mains at 230 V RMS peaks at 325 V; the RMS figure is the DC voltage that would deliver the same power.
Formula: RMS = √((x₁² + x₂² + … + xₙ²) / n)
Worked examples
| Inputs | Root mean square | Note |
|---|---|---|
| 3, −4, 5, −6 | 4.636809 | RMS = 4.743, mean = −0.5 |
| A symmetric pair | 5 | mean 0, RMS 5 |
| 3, 4 | 3.535534 | RMS = 3.536 |
FAQFrequently asked questions
What is RMS used for?
Alternating signals, where the plain average is zero. Mains voltage, audio levels and vibration are all quoted as RMS.
Why is mains 230 V but peaks at 325 V?
Because 230 V is the RMS value — the DC voltage that would deliver the same heating power. For a sine wave, peak = RMS × √2.
How is RMS related to standard deviation?
The standard deviation is the RMS of the deviations from the mean. When the mean is zero, they are the same number.
Is RMS always at least the mean?
It is always at least the absolute value of the mean, with equality only when every value is identical.
What is RMS error?
The RMS of the residuals — the standard way to quote how far a model's predictions land from the truth, in the units of the data.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.