# Mean, variance and RMS

The mean says where the centre is. The variance says how widely the values spread around it. The RMS says how far, typically, the values lie from zero.

Written for this edition, not translated from Einstein.

## If particles wander both ways, which average shows how far they got?

Four particles move −3, −1, +1 and +3 micrometres. Their average position has not changed, yet they have spread out. Three averages answer three different questions.

$$
\langle x\rangle=\frac{1}{M}\sum_{i=1}^{M}x_i
$$

Add the M signed displacements and divide by M.

$$
\langle x^2\rangle=\frac{1}{M}\sum_{i=1}^{M}x_i^2
$$

Square each displacement before adding and dividing by M.

$$
\operatorname{Var}(x)=\langle x^2\rangle-\langle x\rangle^2
$$

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

RMS means root mean square: average the squares, then take the square root. The standard deviation is the square root of the variance, so it measures spread around the mean, where the RMS measures distance from zero. The two are equal only when the mean is zero. The average of the distances, ignoring sign, is a third statistic and gives a different number.

## Worked example: Four displacements, four different averages

1. For −3, −1, +1, +3 the signed sum is 0, so the mean is 0.
2. The distances, ignoring sign, 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, about 2.236. With a mean of 0, the variance is also 5 and the standard deviation is also about 2.236.
5. A small random sample need not have a signed mean of exactly zero, even when the model's mean is zero.

## Where this lesson stops

Say which average you mean. Squaring then averaging, averaging then squaring, and averaging the distances give three different numbers.

## This lesson builds on

- [A sign records direction](/foundations/bridge-negative-numbers-direction/)
- [Squares and square roots](/foundations/bridge-squaring-square-roots/)