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,…. X_n}, the RMS is given by:

• \(\begin{array}{l}x_{rms} = \sqrt{{\frac{(x_{1}^{2}+x_{2}^{2}…x_{n}^{2})}{N}}}\end{array} \)

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

• \(\begin{array}{l}f_{rms} =\sqrt{\frac{1}{T_{2}-T_{1}}\int_{T_{1}}^{T_{2}}[f\left ( t \right )^{2}dt}]\end{array} \)

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

Question: 

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

Solution:

Given set of data values:

1, 3, 5, 7, 9

Step 1: Squares of these values 

1², 3², 5², 7², 9²

Or

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 x_model is defined as the square root of the mean squared error.

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

Where, x_obs is observed values, x_model is modelled values at time i.

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