# 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 as 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, 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

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

• xrms = $\sqrt{\frac{a}{b}(x_{1}^{2}+x_{2}^{2}…x_{n}^{2})}$

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 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.

### 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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