Understand the subject
What is root mean square?
Root mean square, or RMS, measures typical magnitude without allowing positive and negative values to cancel.
It is used for alternating signals, signed errors, vector components, and other quantities whose squared magnitude carries meaning.
See the structure
Square removes direction, root restores scale
−3 → 9+3 → 9mean square 9 → square root 3
Worked example
RMS of 3 and 4
RMS = √[(3²+4²)/2] = √(25/2) ≈ 3.5355
Interpret with care
What the result does—and does not—mean
RMS is not the same as standard deviation: RMS measures distance from zero, while standard deviation measures spread around the mean.
Quick guide
How to use this calculator
- Add signed values in labelled rows.
- Inspect the exact mean of their squares.
- Read an exact rational RMS when possible, otherwise an explicitly approximate square root.
Calculation method
Square, average, then take the non-negative root
Each value is squared exactly, so signs do not cancel. Their exact mean square is formed before its principal square root is evaluated.
Worked example
RMS of 3 and 4
The mean square is (9 + 16) / 2 = 25/2.
RMS = √(25/2) ≈ 3.53553390593274
Supported inputs
Precision and limits
List size
Add 1–100 labelled value rows.
Decimal size
Each value may contain up to 200 digits and 100 decimal places.
Exactness
Exact rational results are never replaced by a rounded decimal; approximations are labelled.
Signed inputs
Positive, negative, and zero values are accepted.
Exact roots
The RMS is labelled exact only when the reduced numerator and denominator are both perfect squares.
