Why not just average them
Take a set of errors: +10, −10, +20, −20. Their arithmetic mean is exactly zero, which correctly says the errors are unbiased and says nothing at all about how large they are.
A model with those errors is not a perfect model. Averaging the absolute values would work — that is mean absolute error, and it is a real measure — but squaring has two properties the absolute value lacks: it is smooth everywhere, which matters for anything differentiable, and it penalises large errors disproportionately.
So RMS is 15.81 here. Larger than the mean absolute error of 15, because the twenties count for more than their size.
Why 230 volts is not the peak
A UK mains supply is quoted at 230 V and peaks at about 325 V. Neither number is wrong; they measure different things.
The voltage is a sine wave swinging symmetrically about zero, so its arithmetic mean is zero. The RMS is the DC voltage that would deliver the same power into the same resistance — which is the only number that lets you compare AC with DC at all.
For a pure sine wave the peak is exactly √2 times the RMS. 230 × √2 is 325, and every mains figure you have ever seen is an RMS one for this reason.
Crest factor
Peak divided by RMS. It says how spiky a signal is, independently of how large it is.
A pure sine wave is exactly √2 ≈ 1.414. A square wave is exactly 1, since it spends all its time at its peak. A signal that is mostly quiet with occasional spikes can go far above 2.
It matters practically: an amplifier or a power supply has to survive the PEAK while doing the work described by the RMS, so a high crest factor means over-specifying for something that rarely happens.
RMS error, and why it is squared
Root mean square error is the standard way to summarise how far a model’s predictions sit from reality, and the squaring is a deliberate choice rather than a convenience.
It makes one error of 10 count more than ten errors of 1 — 10 against 3.16 — which is right whenever a large miss is worse than several small ones. If it is not right for your problem, mean absolute error is the alternative and it treats them equally.
RMS error is also in the original units, unlike the mean squared error it comes from. That is the practical reason for the final root: an RMSE of 3.2 metres is a sentence anyone can act on.
Standard deviation is this
Standard deviation is the root mean square of the deviations from the mean. Not analogous to it — identical, applied to a shifted set of numbers.
Subtract the mean from every value and the deviations sum to zero by construction, so their arithmetic mean is useless and the same cancellation problem appears. Squaring solves it the same way.
Seeing them as one operation is usually the moment standard deviation stops feeling arbitrary. This page computes both, so the identity is visible on your own numbers. The full spread analysis is on the Standard Deviation Calculator.
Where it sits among the other means
For positive data, the four classical means are always in this order: harmonic ≤ geometric ≤ arithmetic ≤ root mean square.
RMS is the largest, and it is largest by the most when the values are spread out — because squaring exaggerates the big ones before the averaging happens.
So the gap between the RMS and the arithmetic mean is itself a measure of spread. In fact it is exactly related to the standard deviation: RMS² = mean² + SD², which is the same Pythagorean relationship in a different costume.
Sources and methodology
RMS is defined the same way across engineering and statistics; these are the references.
Method. Each value is squared, the squares are averaged, and the root is taken — the three steps are computed and displayed separately rather than collapsed, because the middle one is where the sign disappears. The peak and crest factor are computed from the same samples, and the RMS of the deviations from the mean is computed alongside so the identity with standard deviation is visible rather than asserted. That engine is verified on every change against 61 hand-written assertions, including that the root mean square never falls below the arithmetic mean on any generated series, which is the inequality this measure sits at the top of. The count and the per-case breakdown are published on the formula verification page.
Read the guide
The full spread analysis — variance, quartiles, z-scores, outliers — is on the Standard Deviation Calculator, which uses exactly this operation on the deviations. For quantities that multiply, the Geometric Mean Calculator is the right average.