Root Mean Square Formula

Last Updated : 10 Jan, 2024

Root mean square is defined as the quadratic mean or a subset of the generalised mean with an exponent of 2. To put it other way, the square root of the entire sum of squares of each data value in an observation is calculated using the root mean square formula. It can be interpreted as a changing function based on an integral of the squares of the values that are instantaneous in a cycle. It is used here to compute the square root of the arithmetic mean of the square of the function that describes the continuous waveform. It is abbreviated as RMS.

Formula

For a data set of n values, that is, {x₁, x₂, x₃,…. x_n}, the root mean square value is given as,

$X_{RMS} = \sqrt{\frac{x^2_1+x^2_2+x^2_3+…+x^2_n}{n}}$

Here, X_RMS is the root mean square value of given n observations of the data set.

For a continuous function f(t), defined in the interval [T₁, T₂], the root mean square value is given as,

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

Here, X_RMS is the root mean square value of the function f(t) such that T₁ ≤ t ≤ T₂.

Sample Problems

Problem 1. Calculate the root mean square of the data set: 2, 7, 3, 5, 1.

Solution:

Using the formula we get,

$X_{RMS} = \sqrt{\frac{x^2_1+x^2_2+x^2_3+…+x^2_n}{n}}\\ = \sqrt{\frac{2^2+7^2+3^2+5^2+1^2}{5}}\\$

= √(88/5)

= √(17.6)

= 4.2

Problem 2. Calculate the root mean square of the data set: 10, 12, 9, 3, 6.

Solution:

Using the formula we get,

$X_{RMS} = \sqrt{\frac{x^2_1+x^2_2+x^2_3+…+x^2_n}{n}}\\ = \sqrt{\frac{10^2+12^2+9^2+3^2+6^2}{5}}\\$

= √(370/5)

= √(74)

= 8.6

Problem 3. Calculate the root mean square of the data set: 5, 7, 2, 4, 3, 9.

Solution:

Using the formula we get,

$X_{RMS} = \sqrt{\frac{x^2_1+x^2_2+x^2_3+…+x^2_n}{n}}\\ = \sqrt{\frac{5^2+7^2+2^2+4^2+3^2+9^2}{6}}\\$

= √(184/6)

= √(30.66)

= 5.53

Problem 4. Calculate the root mean square of the data set: 3, 6, 9, 12, 15, 18, 20.

Solution:

Using the formula we get,

$X_{RMS} = \sqrt{\frac{x^2_1+x^2_2+x^2_3+…+x^2_n}{n}}\\ = \sqrt{\frac{3^2+6^2+9^2+12^2+15^2+18^2+20^2}{7}}\\$

= √(1219/7)

= √(174.14)

= 13.196

Problem 5. Calculate the root mean square of the data set if sum of squares of data set observations is 216 and number of observations is 6.

Solution:

We have,

S = 216

n = 6

Using the formula we get,

R = √(S/n)

= √(216/6)

= √(36)

= 6

Problem 6. Calculate the root mean square of the data set if sum of squares of data set observations is 5832 and number of observations is 18.

Solution:

We have,

S = 5832

n = 18

Using the formula we get,

R = √(S/n)

= √(5832/18)

= √(324)

= 18

Problem 7. Calculate the root mean square value of the continuous function f(t) = t over the interval [4, 7].

Solution:

We have,

f(t) = t and 4 ≤ t ≤ 7.

Using the formula we get,

$X_{RMS}=\sqrt{\frac{1}{T_{2}-T_{1}}\int_{T_{1}}^{T_{2}}[f\left ( t \right )^{2}dt}]\\ =\sqrt{\frac{1}{7-4}\int_{4}^{7}[t^{2}dt}]\\ =\sqrt{\frac{1}{3}[\frac{t^3}{3}]^{7}_4}\\$

= √(343/9 – 64/9)

= √(279/9)

= √(31)

= 5.56