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Root Mean Square Calculator

Free Root mean square Calculator for arithmetic. Enter values to get step-by-step solutions with formulas and graphs. Free to use with no signup required.

Reviewed by Manoj Kumar, Mathematics Educator

Reviewed by Manoj Kumar, Mathematics Educator

Formula

RMS = โˆš(ฮฃxยฒ / n)

Square every value in the dataset, average the squares, then take the square root of that average. Because squaring removes the sign, RMS reflects the true magnitude of a dataset even when values are both positive and negative โ€” unlike the arithmetic mean, which can average out to zero.

Worked Examples

Example 1: RMS of a small dataset

Problem:Find the RMS of the values 3, 4, and 5.

Solution:Sum of squares = 3ยฒ + 4ยฒ + 5ยฒ = 9 + 16 + 25 = 50. Mean of squares = 50/3 โ‰ˆ 16.667. RMS = โˆš16.667 โ‰ˆ 4.0825.

Result:RMS โ‰ˆ 4.0825 (arithmetic mean = 4.0000)

Example 2: RMS voltage of a sine wave

Problem:A sinusoidal AC voltage has a peak (amplitude) of 170 volts. Find its RMS voltage.

Solution:For a pure sine wave, RMS = peak / โˆš2 = 170 / 1.4142 โ‰ˆ 120.2.

Result:RMS voltage โ‰ˆ 120 volts (matches standard US household mains)

Frequently Asked Questions

What is the root mean square (RMS), and how is it different from the arithmetic mean?

The root mean square is found by squaring every value, averaging those squares, and then taking the square root of that average โ€” RMS = โˆš(ฮฃxยฒ / n). Unlike the arithmetic mean, squaring the values before averaging means negative and positive values can never cancel each other out, and larger-magnitude values are weighted more heavily. The RMS of any dataset with more than one distinct value is always greater than or equal to its arithmetic mean.

References

Reviewed by Manoj Kumar, Mathematics Educator ยท Editorial policy