Matpic
mathematics, in pictures
Concept

Variance

The average of the squared distances of a quantity's values from its mean, whose square root is the standard deviation.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

-6-4-202460.000.100.200.300.400.50valueprobability1.5σ2σ3σthe meanmean 0.300, standard deviation 1.847; beyond 1.5 of them lies 0.2000 against a bound of 0.4444, beyond 2 ofthem lies 0.1000 against a bound of 0.2500, beyond 3 of them lies 0.0000 against a bound of 0.1111the bound knows only the variance — not the shape, not the number of values, not whether the distribution issymmetric — which is why it is so far from tight here and cannot be improved in general

How far from the average a thing can be

Knowing only an average and a spread — nothing about the shape, nothing about the number of outcomes, nothing about symmetry — the chance of landing three standard deviations out is at most one in nine. And there is a distribution that lands there exactly that often, so the bound cannot be improved.

probability · concentration -1-0.50.510.01.02.03.0the average, minus the meann = 4n = 16n = 64-3-2-11230.00.10.20.30.4the sum, minus n means, over √nn = 4n = 16n = 64the same exact distributions, drawn twice as densities: divided by n the spread falls 0.500 → 0.250 →0.125, divided by √n it is 1.000 every timeso the left curves climb — 0.75 → 1.57 → 3.18 at the peak — and the right ones settle on 0.399, which is theheight of the bell curve

The average settles and the wobble does not

Two theorems are usually met a page apart and sound as though one is a sharper version of the other. They are the same sums looked at through two different magnifying glasses: divide by the number of them and everything collapses to a point, divide by its square root and a shape appears.

probability · central limit

Named alongside it

The objects these essays reach for when they reach for this one.

Convergence rateExpectationNormal distributionCentral limit theoremConcentration inequalityConvergenceExtremal exampleHeavy tailsIndependenceLaw of large numbersRandom walkScaling

All concepts