Root mean square calculator

The formula

RMS = √(Σx² / n)

Three operations, in the reverse of the order the name lists them: square every value, take the mean of those squares, then take the square root. The result is in the same units as the data, and it is never negative. Some texts call it the quadratic mean, which puts it in a family with the arithmetic, geometric and harmonic means; the name is the same quantity.

A worked example

Data: 3, −4, 5, −2, 6, −1. Six values, three of them negative.

The sum of squares is 91. Divide by n = 6 for the mean of squares, 15.1667. Take the square root: RMS = 3.8944.

Compare that with the arithmetic mean. The values sum to 7, so the mean is 7 ÷ 6 = 1.1667 — barely a third of the RMS. Nothing in the data is small; the mean is small because the negatives cancelled most of the positives. The RMS reports how big the numbers are, and the mean reports where their centre sits. Those are different questions.

Why square, then root

Squaring makes every term positive, so the sign of a value stops mattering and a −4 contributes exactly as much as a +4. That is the whole point when the quantity in question swings both ways around zero — an alternating current, a vibration, an error that is sometimes too high and sometimes too low. Averaging such a thing directly gives a number near zero, which is true and useless.

The square root at the end undoes the change of units that squaring caused. The mean of the squares of a set of voltages is in volts squared; its root is back in volts, and can be compared with the original readings.

There is a physical reason for choosing squares over absolute values, which would remove signs just as well. Power is proportional to the square of amplitude: the heat produced in a resistor is V²/R, the energy in a wave goes as the square of its height. So the mean of the squares is the average power, up to a constant, and the RMS is the steady value that would deliver that power. The RMS of an AC voltage is the DC voltage that would make the same resistor equally hot. That is why mains supplies are quoted as 230 V or 120 V — those are RMS figures, and the waveform peaks at √2 times them, about 325 V and 170 V respectively.

RMS and the arithmetic mean

The RMS is never smaller than the absolute value of the mean, and equals it only when every value in the list is the same. The identity behind that fact is the most useful thing on this page:

RMS² = x̄² + σ²

where σ is the population standard deviation. The mean of the squares splits cleanly into the square of the mean plus the variance. Since a variance cannot be negative, RMS² ≥ x̄², and the two are equal exactly when σ = 0. In the example, 1.1667² + 3.7156² = 1.3611 + 13.8056 = 15.1667, which is the mean of squares found above. The calculator reports that check on every dataset you enter.

RMS and standard deviation

The population standard deviation is itself a root mean square — the RMS of the deviations from the mean. Write it out and the shape is identical:

σ = √(Σ(x − x̄)² / n)

The only difference is that the standard deviation subtracts the mean first. RMS measures the values about zero; SD measures them about their own centre. For the example, the RMS of the raw values is 3.8944 and the RMS of their deviations — the population SD — is 3.7156, slightly smaller because shifting the origin from 0 to 1.1667 brought the values a little closer on balance.

When the mean is zero there is nothing to subtract, and the RMS and the population SD are the same number. That happens more often than it sounds: any symmetrical waveform, any set of residuals from a least-squares fit, any noise measured about its baseline. Engineers who say "RMS noise" and statisticians who say "standard deviation of the noise" are describing the same quantity. The variance is the mean square of the deviations, so it sits one square root below the SD in exactly the way the mean of squares sits below the RMS; what the standard deviation means covers the interpretation side.

Root mean square error

In regression, the RMS of the residuals — observed minus predicted — is the root mean square error, RMSE. It is the standard summary of how far a model's predictions typically miss, in the units of the thing being predicted. A model that is off by +1.5, −2, +0.5, −1 and +1 has a mean error of exactly zero and an RMSE of 1.3038; the zero says the model is not biased, and the 1.3038 says how much it wobbles.

Because it squares before averaging, RMSE penalises a few large misses more heavily than many small ones. That is usually wanted — one prediction that is wildly wrong matters more than several slightly wrong — but it does mean RMSE is sensitive to outliers in a way the mean absolute error is not. The linear regression calculator reports the residuals so you can compute either.

The sine-wave factor

For a pure sine wave the ratio of RMS to peak is fixed: 1/√2 ≈ 0.7071. A signal that peaks at 10 V has an RMS of 7.071 V; a supply with an RMS of 230 V peaks at 230 × √2 ≈ 325 V. The factor comes from the mean of sin² over a full cycle being exactly one half. It is specific to sine waves: a square wave has an RMS equal to its peak, a triangle wave has peak/√3, and an arbitrary signal has to be squared and averaged directly, which is what this calculator does with any list of samples you paste in.

Common mistakes

• Averaging first, then squaring. The square of the mean is not the mean of the squares — for the example, 1.1667² = 1.3611 against 15.1667. The order in the name is the order to undo, not to do: square, then mean, then root.
• Mixing units. Every value must be in the same unit before squaring, or the squares are incomparable. Convert millivolts to volts, or metres to millimetres, before you start.
• Applying n − 1. The RMS describes the numbers in front of you and divides by n. The n − 1 correction belongs to the sample standard deviation, which estimates a population; it has no place in an RMS.
• Using the sine-wave factor on a signal that is not a sine wave. Multiplying the peak by 0.7071 is only right for one waveform. For anything else, sample it and compute the RMS properly.