Root Mean Square Formula
The root mean square formula gives the square root of the total sum of squares of each data in an observation. Root mean square, abbreviated as RMS is the square root of the arithmetic mean of the squares of a group of values. It is also called quadratic mean. Root mean square value for a function can also be defined for a continuously varying function in terms of an integral of the squares of the instantaneous values during a cycle. Here, the root mean square formula calculates the square root of the arithmetic mean of the square of the function that defines the continuous waveform.
What is Root Mean Square Formula?
Formula 1
For a group of "n" values involving \({x_1, x_2, x_3,…. x_n}\), the root mean square formula is given as,
\( X_{rms} = \sqrt{\frac{x_1^2 + x_2^2 + x_3^2+ ... + x_n^2}{n}}\)
where \(X_{rms}\) = Root mean square value of given "n" observations.
Formula 2
The root mean square formula for a continuous function f(t), defined for the interval T_{1} ≤ t ≤ T_{2} is given as,
\( f_{rms} = \sqrt{\frac{1}{T_2  T_1}} \int_{T_1}^{T_2}[f(t)^2 dt]\)
where \(f_{rms}\) = Root mean square value of given function f(t).
Solved Examples Using Root Mean Square Formula

Example 1:
Calculate the root mean square of the following observations: 6, 5, 4, 2, 7?
Solution:
To find: Root mean square of the given observations.
Using the root mean square formula,
\( X_{rms} = \sqrt{\dfrac{x_1^2 + x_2^2 + x_3^2 +...+ x_n^2}{n}}\)
\( = \sqrt{\dfrac{6^2 + 5^2 + 4^2 + 3^2 + 7^2}{5}} = 5.196\)
Answer: The root mean square of given observations = 5.196.

Example 2:
Find the root mean square value of f(t) = t over the interval 2 ≤ t ≤ 5.
Solution:
To find: The root mean square value of f(t) = t over the interval 2 ≤ t ≤ 5.
Using the root mean square value formula for th given function f(t),
\( f_{rms} = \sqrt{\dfrac{1}{T_2  T_1}} \int_{T_1}^{T_2}[f(t)^2 dt]\)
\( f_{rms} = \sqrt{\dfrac{1}{5  2}} \int_{2}^{5}[t^2 dt]\)
\( \begin{equation*} = \dfrac{1}{3} \left[ \dfrac{t^3}{3} \right]_2^5 = \dfrac{1}{3} \left[ \dfrac{125}{3}  \dfrac{8}{3} \right]= \dfrac{117}{9} = 13 \end{equation*} \)
Answer: Root mean square value of the given function, f(t) = 13.
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