Correlation
Named by 8 essays across 5 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.
Edges that crowd each other out
Pick a spanning tree of a graph uniformly at random, and the presence of one edge makes every other edge less likely — a theorem of Kirchhoff's electricity. Pick a forest instead, or a connected subgraph, and the same is believed and unproved. Checking every pair of edges in every labelled graph up to six vertices, 871,926 pairs for forests alone, finds no exception, and comes as close as 0.9994 to one.
Two neighbours share out their small primes
A number and the next one have no prime in common, and the natural guess is that their largest primes are independent. In the limit that is believed and nearly proved. Below ten million it is visibly false: neighbours are both made of small primes only nine-tenths as often as independence predicts, a run of four only two-thirds as often, and the correlation between their largest primes is negative at distance one and positive at distance six — because a small prime that divides one of two neighbours is a small prime the other cannot have.
Holding one still turns the other over
A quantity can rise with an outcome and still predict it downwards once a second quantity is held fixed. Completing the square in two predictors says exactly when: the pair of correlations has to fall in a wedge whose angle is arcsin of the predictors' own correlation. Averaged over every possible correlation matrix the chance is exactly 1/4 − 1/π², about one coefficient in seven; two coefficients never reverse together, and with two dozen predictors almost two in five do.
The element the judges are worst at
A court can vote element by element or on the verdict, and the comparison so far assumed judges equally good at every element. Real judges are surer about the law than about a contested fact. Make one element hard and both procedures sink towards the majority on that element, but they sink differently: the verdict vote sees only the product of the judges' competences and cannot tell an even court from a lopsided one, while the premise vote is ruled by its weakest majority. And when a judge's errors on two elements come from the same misreading of the facts, the gap between the procedures closes.
Named alongside it
The objects these essays reach for when they reach for this one.
VarianceBinomial distributionCompleting the squareConditional probabilityCondorcet jury theoremIndependenceNormal distributionProjectionQuadratic formSimulationBayes' theoremBox dimension