Root Mean Square


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Root Mean Square

What is Root Mean Square (RMS)?

Statistically, the root mean square (RMS) is the square root of the mean square, which is the arithmetic mean of the squares of a group of values. RMS is also called a quadratic mean and is a special case of the generalized mean whose exponent is 2. Root mean square is also defined as a varying function based on an integral of the squares of the values which are instantaneous in a cycle.

In other words, the RMS of a group of numbers is the square of the arithmetic mean or the function’s square which defines the continuous waveform.

Root Mean Square Formula

The formula for Root Mean Square is given below to get the RMS value of a set of data values.

For a group of n values involving {x1, x2, x3,…. Xn}, the RMS is given by:

  • xrms :\(\sqrt{{\frac{(x_{1}^{2}+x_{2}^{2}…x_{n}^{2})}{N}}}\)

The formula for a continuous function f(t), defined for the interval T1 ≤ t ≤ T2 is given by:

  • frms = \(\sqrt{\frac{1}{T_{2}-T_{1}}\int_{T_{1}}^{T_{2}}[f\left ( t \right )^{2}dt}]\)

The RMS of a periodic function is always equivalent to the RMS of a function’s single period. The continuous function’s RMS value can be considered approximately by taking the RMS of a sequence of evenly spaced entities. Also, the RMS value of different waveforms can also be calculated without calculus.

How to Calculate the Root Mean Square

Steps to Find the Root mean square for a given set of values are given below:

Step 1: Get the squares of all the values

Step 2: Calculate the average of the obtained squares

Step 3: Finally, take the square root of the average

Try out: Root Mean Square Calculator

Solved Example


Calculate the root mean square (RMS) of the data set: 1, 3, 5, 7, 9


Given set of data values:

1, 3, 5, 7, 9

Step 1: Squares of these values 

12, 32, 52, 72, 92


1, 9, 25, 49, 81

Step 2: Average of the squares

(1 + 9 + 25 + 49 + 81)/5

= 165/5

= 33

Step 3: Take the square root of the average.

RMS = √33 = 5.745 (approx)

Root Mean Square Error (RMSE)

The Root Mean Square Error or RMSE is a frequently applied measure of the differences between numbers (population values and samples) which is predicted by an estimator or a mode. The RMSE describes the sample standard deviation of the differences between the predicted and observed values. Each of these differences is known as residuals when the calculations are done over the data sample that was used to estimate, and known as prediction errors when calculated out of sample. The RMSE aggregates the magnitudes of the errors in predicting different times into a single measure of predictive power.

Root Mean Square Error Formula

The RMSE of a predicted model with respect to the estimated variable xmodel is defined as the square root of the mean squared error.

  • RMSE =\(\sqrt{\frac{\sum_{i=1}^{n}(X_{obs,i}-X_{model,i})^{2}}{n}}\)

Where, xobs is observed values, xmodel is modelled values at time i.

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Frequently Asked Questions – FAQs

What is the root mean square?

The root mean square (RMS or rms) is defined as the square root of the mean square, i.e. the arithmetic mean of the squares of a given set of numbers.

How is RMS calculated?

RMS or Root Mean Square value can be calculated by taking the square root of arithmetic mean of squared observations.

What is the RMSE value?

The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is a frequently used measure of the differences between values (sample or population values) predicted by a model or an estimator and the values observed.

Is a higher or lower RMSE better?

Lower values of RMSE indicate better fit.

What is the value of RMS for the data set 2, 3, 5, 7, 11?

Average of squares of given data values = (4 + 9 + 25 + 49 + 121)/5 = 208/5 = 41.6
RMS = square root(41.6) or √41.6 = 6.45