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

Frequentist interpretation — where it appears

Reading a probability as a long-run share over the experiments that might have been run, rather than as a degree of belief. It is what makes coverage a property of a rule and what makes a statement about one interval in hand hard to phrase.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Twenty 95% intervals for a proportion that really is 0.35. 2 of the twenty miss the true value. The 95% is a property of the procedure across repetitions — no single interval has a 95% chance of anything, because it either contains 0.35 or it does not.

Twenty intervals and one expected miss

The 95% belongs to the procedure, not to the interval in front of you. Twenty intervals from twenty samples make that visible in a way no definition does, and the one that misses is not a mistake.

intervals · Repetition Counted coverage of two 95% intervals, over 2,000 datasets. Each point is one of the eight groups, at its own standard error. The integrated interval covers 95.2% overall against its stated 95%; the plug-in covers 78.8%, and its shortfall grows with the group's standard error — from 86.1% at se 0.5 to 77.0% at se 2.1. The mean widths are 3.48 and 2.66.

The interval that integrates

A credible interval for one group in a hierarchy has to average over every value the population spread might take. That averaging is what makes it cover — 95.2% against the plug-in's 78.8% — and it costs 31% more width, a heavier tail, and a mixture rather than a normal.

fullbayes · Credible Three promises, and no procedure keeps all three. Average coverage and worst-case coverage for four 95% intervals for a proportion at n = 40, computed exactly. Their expected widths are 0.2418, 0.2417, 0.2472, 0.2641 in the same order. The textbook interval and the score interval have the same expected width to four digits — 0.2418 and 0.2417 — and worst-case coverages of 55.31% and 92.21%. The exact interval never breaks its promise and is 9.3% wider than the score interval to do it. Each of the three columns orders the four procedures differently.

An interval that covers and says nothing

A procedure returning the whole line 95% of the time and the empty set otherwise has coverage exactly 95% at every parameter value. Two real intervals at forty observations have expected widths of 0.2418 and 0.2417 and worst-case coverages of 55.31% and 92.21%.

intervals · Coverage The textbook (Wald) interval's coverage among studies whose count of 40 lands in the tails. Coverage of the nominal 95% textbook (Wald) interval at 40 trials, conditional on the count being at most k or at least 40 − k, computed exactly over the counts for every true proportion. With k = 2 its highest value anywhere is 88.96%; With k = 5 its highest value anywhere is 94.90%; With k = 8 its highest value anywhere is 95.57%. A set whose highest value stays below 95% is a set of counts on which the label is wrong whatever the proportion is.

A count that bets against its interval

A reader who sees only the count can bet against an interval's 95%, and win whatever the proportion is, exactly when some set of counts carries less than 95% at every proportion. Out of forty trials the textbook interval has one — a count of five or fewer, or thirty-five or more, covers at most 94.90% — and the score interval has none: a prior exists under which every one of its forty-one intervals is 96.46% credible, and that prior is the proof that no bet can be found.

intervals · Coverage

Named alongside it

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

CoverageConfidence intervalBinomial proportionClopper–PearsonCredible intervalExact enumerationPriorWald intervalWilson intervalWorst caseClosed formConditional coverage

All concepts