Root mean square vs standard deviation: what's the difference?

The two formulas, side by side

RMS = √(Σx² / n) SD (population) = √(Σ(x − x̄)² / n)

Both square, both average, both take the root. The only thing that differs is what gets squared: the values themselves, or the values with their mean removed. In physical terms, the RMS of a set of measurements says how big the numbers are; the SD says how far apart they are. A city where every house sits 400 metres above sea level and the heights barely vary has a large "mean height" and a small SD. A signal that swings from +10 to −10 with a mean of zero has a large RMS and says nothing about the spread about its own centre.

The identity that connects them

RMS² = x̄² + σ²

The mean of the squares splits exactly into the square of the mean plus the variance, with nothing left over. Because σ² is never negative, the RMS is always at least the absolute value of the mean, and the gap between the two is the spread.

On the six values 3, −4, 5, −2, 6, −1 (the example the RMS calculator loads): the mean is 7 ÷ 6 = 1.1667, the mean of the squares is 91 ÷ 6 = 15.1667, so the RMS is √15.1667 = 3.8944. The deviations from the mean have a mean of squares of 13.8056, so the population SD is √13.8056 = 3.7156. Check the identity: 1.1667² + 3.7156² = 1.3611 + 13.8056 = 15.1667, the mean of squares. The two quantities are nearly equal here because the mean is small next to the spread; it is the spread, not the level, that dominates.

When they become the same number

When the mean is exactly zero there is nothing to subtract, the (x − x̄) column is just the x column, and the RMS is the population standard deviation. Three situations where that happens:

• AC waveforms. A sine wave spends half its cycle negative, so its mean over a complete cycle is zero. The "RMS value" of the waveform is its SD. That is why engineers who say "RMS voltage" and people who say "standard deviation of the voltage" are often naming the same quantity.
• Regression residuals. A least-squares fitted line is positioned so the residuals sum to zero, so their mean is zero. The root mean square error is the standard deviation of the misses, about the fitted value.
• Noise about a baseline. A detector's background fluctuations are usually quoted as an RMS figure, which is the SD once you have subtracted the baseline.

Which one do you actually want?

Ask which question the number must answer. How big is this quantity, ignoring direction? — RMS: mains voltage, signal amplitude, vibration severity, the typical size of an error in a batch of measurements. How much do these values scatter about their own centre? — standard deviation: scores around a mean, daily returns around their average, replicate readings around their mean. If the level matters (the data has a large mean relative to the spread), the RMS and the SD will differ a lot, and reporting the wrong one reports the level instead of the spread.

A concrete pair from the same data: a factory runs at about 98 units with a scatter of about 2. The SD says 2; the RMS says about 98.01, which answers a completely different question. The standard deviation calculator and the RMS calculator show both on the same numbers.

Why the mix-up happens

Both are written with the same three symbols — square root, sigma, division by n — and both sit in the family of "mean square" quantities: the variance is the mean of squared deviations, the RMS is the mean of squared values, and the two differ by exactly the variance of the data. Textbooks that move between physics and statistics sometimes write σ for the RMS of a signal, which is why a "σ of the current" in an electrical paper is not the same concept as a "σ of the current" in a statistics paper. Read the definition line before comparing numbers from two sources; the formula shape alone does not tell you which centre was subtracted.