# Root Mean Square Calculator

Created by Anna Szczepanek, PhD
Reviewed by Bogna Szyk and Jack Bowater
Last updated: Jun 05, 2023

Welcome to the root mean square calculator! Use it if you want to know how to calculate the RMS (A.K.A. quadratic mean) of any data set. But what is the root mean square? What is the root mean square formula? Scroll down - besides an RMS calculator, we also give the root mean square equation and explain how to calculate root mean square by hand!

## What is root mean square? Root mean square formula

The root mean square formula for a set of $n$ numbers $x_1, \ldots, x_n$ is the following:

$\sqrt{\frac{x_1^2 + x_2^2 + \ldots + x_n^2}{n}}$

With the help of the summation sign we can rewrite it as:

$\sqrt{\frac 1n \sum_{i = 1}^n x^2_i}$

We see that this expression is the square root of the arithmetic mean of the squares of the values in our set. This is where the name "root-mean-square", which is abbreviated as "RMS", comes from. Sometimes the root mean square is called the "quadratic mean".

## How to calculate the root mean square by hand?

We now know what the root mean square is, so it's time to learn how to calculate RMS by hand.

Let $2, 6, 3, -4, 2, 4, -1, 3, 2, -1$ be our dataset for which we want to apply the root mean square equation to. We need to perform the following steps:

1. The first step is to find the square or each number. The set of squared values is the following:

$4, 36, 9, 16, 4, 16, 1, 9, 4, 1$.

2. Sum those squared numbers:

$4 + 36 + 9 + 16 + 4 + 16 + 1 + 9 + 4 + 1 = 100$.

3. Divide $100$ by the number of values contained in our dataset. We see that there are $10$ of them, so we obtain $100 / 10 = 10$.

4. The last step we need to apply the root mean square equation is to compute the square root of the value obtained above. The quadratic mean of our dataset is $\sqrt 10 \approx 3.16$.

## How to use this RMS calculator?

To use our root mean square calculator, start by entering your values. At first, eight fields are visible, but more fields will appear as you go. You can enter up to 30 numbers.

The root mean square of your numbers will be displayed at the bottom of the RMS calculator. It will update as you enter more and more variables.

## Weighted root mean square

Similar to other means, the root mean square has a weighted variant. It is used when we want some values to contribute more to the average than others (in the default version each value contributes the same, i.e., they all have the same weight).

Suppose that we have a list of numbers $x_1, x_2, \ldots, x_n$, along with an associated list of weights $w_1, w_2, \ldots, w_n$.
The weighted root mean square is given by the formula:

$\sqrt{\frac{w_1x_1^2 + w_2x_2^2 + \ldots + w_nx_n^2}{w_1 + w_2 + \ldots + w_n}}$

## Generalized (power) means

For all you more advanced readers, it may be useful for you to know that the quadratic mean (and most other means, like the arithmetic, geometric, and harmonic are particular cases of a more general concept. Namely, we let $p ≠ 0$ and define the generalized mean with the exponent $p$ of the values $x_1, x_2, \ldots, x_n$ as:

$\sqrt[p]{\frac{x_1^p+ \ldots x_n^p}{n}}$

which can be rewritten as:

$\left(\frac 1n \sum_{i=1}^n x_i^p\right)^{\frac 1p}$

Clearly, for $p = 1$, we have the arithmetic mean. By setting $p = 2$, we obtain the quadratic mean, while $p = 3$ would give the cubic mean, and so on. Also, note that for $p = -1$, we arrive at the harmonic mean:

\begin{align*} \left(\frac 1n \sum_{i=1}^n x_i^{-1}\right)^{-1} & = \frac{n}{\sum_{i=1}^n x_j^{-1}} \\ & = \frac {n}{\sum_{i=1}^n \frac {1}{x_j}} \end{align*}

Finally, it can be proved that the geometric mean is the limit value when $p$ approaches $0$.

## Applications

It is true that the root mean square is less popular than the arithmetic or geometric mean, but it does have several important applications:

• In statistics, the population standard deviation is just the root mean square of the differences between data points and the population expected value. In fact, the root mean square can be used to determine the standard deviation via the following formula:

$x_{\rm RMS}^2 - \bar{x}^2 = \sigma^2$

where:

• $x_{\rm RMS}$ is the root mean square;

• $\bar{x}$ is the population mean; and

• $\sigma^2$ is the squared standard deviation (i.e., variance).

• In physics, the RMS appears, e.g., in electrical engineering, signal processing, and the physics of gas molecules.

Let us just discuss one example.

Imagine we have $N$ identical particles with mass $m$, and the $i$-th particle is moving at speed $v_i$. We know that the kinetic energy of such a particle is equal to $\frac 12 mv_i^2$.

Go to our kinetic energy calculator if you need a refresher on this concept ;)

In total, they have the kinetic energy of

$\frac 12 m (v_1^2 + ... +v_n^2)$.

If we want to find one speed which, with respect to all these particles, would give the same total kinetic energy, we quickly conclude that this speed would be the root mean square of all the speeds $v_1, \ldots, v_n$.

Anna Szczepanek, PhD
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