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RMS Formula ( Root Mean Square Formula)

March 2nd, 2012

The rms value of a set of n values for (x1, x2, .....xn) is given by

This formula is valid for a set of n discrete values. If you have a continuous variable, you can use Calculus to derive the formula for various common waveforms, for example the sinusoidal waveform.

For example if a is the amplitude of a sinusoidal waveform the vrms is given by

You can use the following calculator to calculate the rms value , if you know the peak to peak value

rms to peak to peak calculator

Probably you cam here searching for the formula for the Root Mean Square or, rms for the short. Before we give out the formula and the explanation, let us find the need of a root mean square.

A company Xing Hua, makes nuts and bolts and supplies it to various companies in US and Europe. The company makes a strict record of all the quality procedure that it adapts.

The nuts and bolts had their dimensions measured with automatic tools after they are finished and recorded in a spread sheet. That is a great !!! Said the CEO when he visited the shop and looked into the quality procedure adopted by them. The QA engineer then calculated the mean of the internal diameter of the nuts and found that the average of the dimension of is very close to the required dimension. The required dimension was 2.050 mm while their average dimension was 2.051 mm. Pretty close.

A few months later the order that came from Japan for an automobile company was rejected. The company went bankrupt.


Had the QA engineer measured the rms average in place or regular average, they could have caught the fault much earlier saving the company from bankruptcy.

You may have now understood the story now. Let us say we want to hit an average dia of 2.50 mm. If one piece has dia of 1.5 mm and second one has a dia of 3.5 mm, the average is still 2.50 mm, but both are useless.

So we define root mean square.

We take the difference. The differences are as follows

For 3.5 mm the difference is +1.0 mm
For 1.5 mm the difference in -1.0 mm

The average is 0
But the root mean square is sqrt ( (1)^2 + (-1)^2) /2) = 1.0

This is 1.0 mm - non trivial value.

So coming back to the rms formula. Here is the formula for the RMS

The rms value of a set of n values for (x1, x2, .....xn) is given by

Other Resources

- If you are, however looking at the rms value corresponding to a waveform, you may like to check vrms to vpeak to peek calculator

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  1. Aser Hasan
    February 27th, 2013 at 03:59 | #1

    Thanks for the nice and detailed illustration

  2. Serge
    June 25th, 2013 at 07:04 | #2

    Instead of "For 1.0 mm the difference is +1.0 mm" shouldn't it be:
    "For 3.5 mm the difference is +1.0 mm" ?

  3. admin
    October 2nd, 2015 at 06:57 | #3

    Serge, you are right. Thanks for pointing that out. The error has been fixed.

  4. February 16th, 2016 at 23:35 | #4


  5. Deepak Suwalka
    February 19th, 2017 at 23:27 | #5

    It's a nice approach to . I really like it. It's really appreciable. Thanks for sharing it.

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