Normal Distribution
nothing is called 95% until it has been counted
Concept

Round trip — where it appears

Computing a quantity one way and recovering it by another, as a check that neither route is wrong. Where the second route is exact it is also an estimator: subtracting its own known error from the first sharpens the answer by a factor the correlation between them fixes.

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

Also named here as verification — the same set of essays touches all of them, so they are one junction rather than several.

Coverage of four nominal 95% intervals, n = 30. Computed exactly by summing over all 31 possible counts, not simulated. The Wald interval drops to 26.0% and is jagged everywhere; Clopper–Pearson never falls below 95% and pays for it in width.

Two routes to every number

A site about probability that only simulates has one route to each answer and no way to tell a right one from a plausible one. Every important number here is computed twice, by arithmetic that shares nothing, and the two are required to agree.

method · Routes Two companions on one simulation, two hundredfold apart. How many times as many draws each companion is worth, on the same 4,000 simulated samples of 40 observations. The coverage of the interval is estimated with the observed count as its companion, whose expectation is 12 exactly; they correlate at 0.2665 and the companion is worth 1.08 times the draws. The expected width is estimated with p̂(1 − p̂) as its companion, whose expectation is 0.20475 exactly; they correlate at 0.9977 because the width is a monotone function of it, and the companion is worth 214 times the draws — 856 thousand simulated samples' worth of precision from four thousand.

The check worth more than the check

The same exactly known companion that verifies a simulation can sharpen it. On one set of four thousand draws, one companion is worth 1.08 times the draws and another is worth 214 times them, and the factor is 1 − ρ² with nothing else in it.

method · Routes

Named alongside it

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

CoverageMonte CarloNumerical methodsVerificationBinomial proportionClosed formCorrelationDegrees of freedomEfficiencyEstimation errorExact enumerationInterval width

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