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Foundation · Explanatory preview

Mean, variance and RMS

The mean tracks the centre; the variance tracks spread around the centre; RMS measures distance from the chosen zero.

Original explanatory text; editorial review pending. This is not a translated source passage.

Which average can reveal wandering that cancels in the signed mean?

⟨x⟩=1M∑i=1Mxi\langle x\rangle=\frac{1}{M}\sum_{i=1}^{M}x_i

Add the M signed displacements and divide by M.

⟨x2⟩=1M∑i=1Mxi2\langle x^2\rangle=\frac{1}{M}\sum_{i=1}^{M}x_i^2

Square each displacement before adding and dividing by M.

Var⁡(x)=⟨x2⟩−⟨x⟩2\operatorname{Var}(x)=\langle x^2\rangle-\langle x\rangle^2

Variance is the mean square minus the square of the mean.

RMS means root mean square: take the square root after averaging the squares. RMS and standard deviation coincide only when the mean is zero. The average absolute displacement is yet another statistic.

One worked example

  1. For −3, −1, +1, +3, the signed sum is 0, so the mean is 0.
  2. The absolute values are 3, 1, 1, 3; their mean is 2.
  3. The squares are 9, 1, 1, 9; their mean is 5.
  4. The RMS is the square root of 5, approximately 2.236.
  5. A small random sample need not have a signed mean of exactly zero even when the model mean is zero.

A stopping point: Specify whether you square before averaging, average before squaring, or take absolute values.

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