Correlation
Named by 4 essays across 4 fields — each of them below, with the objects they name alongside it.
A majority wiser than its members
Condorcet's other theorem turns voting round: the voters no longer have preferences but judgements about a single fact, each a little more likely right than wrong. Then a simple majority of many of them is almost certainly right — 6,763 voters who are each right 51% of the time make a majority right 95% of the time. The theorem survives voters worse than a coin, if the average is better. It does not survive voters who share their mistakes, and when their skills differ the right rule weighs votes rather than counting them.
Drawn without putting back
Every concentration bound on this shelf assumes the draws are independent. A real sample is not: a pollster does not ring the same person twice, and every ball taken from an urn changes what is left in it. The dependence runs the helpful way. Wassily Hoeffding proved in 1963 that a sample drawn without replacement is at least as concentrated as one drawn with it, for every convex measure of spread at once — and the variance falls by an exact factor that reaches zero when the whole urn is taken.
Two numbers in one jagged record
A measured record comes with no rule, so its dimension has to be estimated from a finite stretch of samples — and two things go wrong that never trouble a graph built from a formula. The popular estimator depends on the units the record is written in, and the dimension, which describes the record up close, turns out to be independent of its memory, which describes it from far away. For a self-affine path the two are tied by D = 2 − H; for a record, they are two numbers.
Regression is a square completed halfway
A bell in two variables, sliced at a fixed x, is a bell in y — and completing the square in y alone says where it is centred: at ρx, on a line flatter than the ellipse's own axis. That gap is regression to the mean, and the same completion says it runs backwards in time just as well as forwards.
Named alongside it
The objects these essays reach for when they reach for this one.
VarianceBinomial distributionBayes' theoremBox dimensionCompleting the squareConcentration inequalityConditional probabilityCondorcet jury theoremConvexityExpectationFractal dimensionFractional brownian motion