How to calculate root mean square
The root mean square (RMS) is the square root of the mean of the squares: RMS = √(Σx² ÷ n). Square every value, average the squares, then take the square root. For 1, 2, 3, 4, 5 the squares are 1, 4, 9, 16, 25 summing to 55. The mean square is 55 ÷ 5 = 11, and the RMS is √11 ≈ 3.32. The final step is a square root, taken after the values have been squared and averaged.
How to use this calculator
Enter your numbers in the value fields and leave unused fields blank. The result shows the RMS along with the mean square and the count. For the ordinary arithmetic mean instead, see the mean, median, mode calculator.
- Signs are irrelevant. Entering −3 gives the same contribution as 3, since both square to 9.
- Every filled field counts, so blank rows are ignored rather than treated as zeros.
- Divide by n, not n − 1. RMS uses the plain average of the squares.
Where RMS is used
RMS is widely used in physics and engineering — for example the effective value of an alternating current or voltage, and the magnitude of varying signals. Because it squares values, both positive and negative numbers contribute, so RMS is never negative. It also relates to the spread measure in the signal to noise ratio calculator.
Frequently asked questions
- How do I calculate the root mean square?
- Square each value, average the squares, then take the square root. For 1, 2, 3, 4, 5 the RMS is √11 ≈ 3.32.
- How is RMS different from the mean?
- The mean averages the values directly; the RMS averages their squares first, so larger magnitudes count more and the result is never negative.
- Can RMS be negative?
- No. Squaring removes signs and the square root of a non-negative number is non-negative, so RMS is always zero or positive.
- Where is RMS used?
- In electrical engineering for AC voltage and current, in signal processing, and anywhere an effective magnitude of varying values is needed.
- Is RMS the same as standard deviation?
- Only when the mean is zero. Standard deviation is the RMS of the deviations from the mean, so for data centred on zero the two coincide; otherwise RMS is always the larger of the two.