Root Mean Square Formula: Definition, Calculation & Solved Examples

Root Mean Square or RMS is described as the square root of the arithmetic mean of the squared terms of a data set. Root Mean Square Error or RMSE is a measure of the standard deviation of the difference between the predicted value and the estimated value. Root Mean Square is also known as quadratic mean and is a special case of the generalized mean whose exponent is 2. Root mean square is also described as a varying function based on an integral of the squares of the values which are instantaneous in a cycle. In short, we can say that the Root Mean Square Formula is used to calculate the square root of the arithmetic mean of the square of the function that defines the continuous waveform.

Key Terms: Root Mean Square, Arithmetic Mean, Average, Root Mean Square Error, Quadratic Mean, Square Root, Exponent, Natural Numbers, Continuous Function

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What is Root Mean Square?

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The name Root Mean Square defines itself well enough. The Root Mean Square is the actual square root of the arithmetic mean of the squares of a set of numbers. Root Mean Square Speed or RMS hold a very high significance in Mathematics as well as Thermodynamic Physics. 

Read More: Variance

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Root Mean Square Formula

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In order to calculate the Root Mean Square, two formulas have been provided below. 

• Formula 1

For a group of "n" values, let us consider a Set A of numbers. If A = { x₁, x₂, x₃….x_n }, the Root Mean Square or RMS of the Set A is given by the formula given below. 

X_rms = \(\frac{\sqrt{x_1^2+x_2^2+x_3^2....x_n^2}}{n}\)

where X_rms refers to the Root mean square value of given "n" observations.

• Formula 2

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

where f_rms refers to the Root mean square value of given function f(t).

The Root Mean Square of a periodic function is always equivalent to the Root Mean Square of a function’s single period. The RMS value of a continuous function can be approximated by calculating the RMS of a sequence of evenly spaced entities.

Read More: Nature of Roots of Quadratic Equation

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How to Calculate Root Mean Square?

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Given below are the steps to calculate the root mean square for a given set of values: 

• Step 1: First, calculate the squares of all the values.
• Step 2: Now, calculate the average of the obtained squares.
• Step 3: Finally, the last step is to take the square root of the average.

Formula to Calculate Average

Example: What will be the root mean square (RMS) of the given data set:1, 3, 5, 7, 9?

Solution: Here, the given set of data values is 1, 3, 5, 7, 9

Step 1: First, calculate the square of the data values.

1², 3², 5², 7², 9² = 1, 9, 25, 49, 81

Step 2: Now, we have to find the average of the squared numbers

(1 + 9 + 25 + 49 + 81)/5 = 165/5 = 33

Step 3: The last step is to find the square root of the average calculated.

 √33 = 5.745 (approx)

Thus, the root mean square of 1, 3, 5, 7, 9 is 5.745

Read More: Geometric Mean Formula

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Root Mean Square Error

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One of the major applications of RMS or Root Mean Square is when two sets of data one derived theoretically and the other by performing experiments are compared. So, when performing the comparison of pairwise results, the RMS values of both are taken into consideration to calculate the extent of error from zero. Due to compatibility with other formulae and the mathematical convention, the RMS of the difference between the order pairs of two datasets are used to find the Root Mean Square Error. 

Thus, the Root Mean Square Error can be defined as a measure of the standard deviation of the difference between the predicted value and the estimated value. 

Root Mean Square Error Formula

Read More: Mean and Variance of Random Variable

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Things to Remember

• Root Mean Square is the square root of the arithmetic mean of the individually squared terms of a set.
• Root Mean Square is also called quadratic mean. 
• Root Mean Square is also sometimes denoted as RMS or rms.
• The Root Mean Square Error or RMSE is used to measure the deviation of the experimental values from the theoretical value. 
• Mathematically, RMSE is the Root Mean Square of the differences between the experimental values and the theoretical results.
• The lower is the RMSE value, the better fit it suggests. 

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