# Root Mean Square Calculator

URL: https://speedycalculator.com/calculators/math/root-mean-square/
Last updated: 2026-10-30

> The root mean square (RMS) calculator squares up to 30 values, averages the squares and takes the square root – the quadratic mean used for AC voltages, signal levels and errors.

## Inputs

- Values

## Results

- Root mean square
- Arithmetic mean (for comparison)

## Formula

```
RMS = √((x₁² + x₂² + … + xₙ²) ÷ n)
```

## Explanation

RMS measures the typical size of quantities that can be positive or negative. Mains electricity of 230 V RMS delivers the same power as 230 V DC, although its peak is about 325 V. In statistics, the root mean square error summarises prediction errors.

### How to use the Root Mean Square Calculator

- Type the numbers separated by commas, spaces or new lines.
- Read the result – it updates as you type.

Results update as you type, and everything is calculated in your browser.

### How it works

Each value is squared (so signs do not cancel), the squares are averaged and the square root brings the result back to the original units.

### What each result means

- **Root mean square** = rms([Values])
- **Arithmetic mean (for comparison)** = mean([Values])

## Worked example

Inputs — Values: 3, -4, 5, -2, 6.

Results — **Root mean square = 4.24264069**; Arithmetic mean (for comparison) = 1.6.

## Key terms

- **Quadratic mean**: Another name for the root mean square.

## FAQ

**Why use RMS instead of the mean?**
Positive and negative values cancel in the mean; RMS measures magnitude.

**What is the RMS of a sine wave?**
Its peak divided by √2 ≈ 0.707 × peak.

**Is RMS always at least the mean?**
It is always at least the mean of the absolute values.

**What is RMSE?**
Root mean square error – the RMS of prediction errors.


## Sources

- [OpenStax – Prealgebra 2e](https://openstax.org/details/books/prealgebra-2e)
- [Wolfram MathWorld – Root-mean-square](https://mathworld.wolfram.com/Root-Mean-Square.html)