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

Mid-p — where it appears

A p-value or interval for a discrete count that counts only half the probability of the observed value. It is the randomised interval with its coin fixed at one half, so its coverage swings above and below the nominal level rather than staying above it.

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

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

Clopper–Pearson, mid-p and the randomised interval: coverage across the proportion, 20 trials. Clopper–Pearson never falls below 95% and runs up to 99.80%. The mid-p interval, which is the randomised interval with its coin fixed at one half, runs from 92.94% to 99.80%. The randomised interval covers 95% at every proportion, to within the 0.043% of the numerical integration over the coin.

The coin that makes it exact

Every interval for a proportion either covers less than 95% somewhere or more than 95% on average, because a count is discrete. One construction covers exactly 95% at every proportion: it adds a uniform random draw to the count. At thirty trials it is 0.9% wider than Wilson's interval and narrower than both exact ones — and two analysts with the same data report different intervals, and one study in forty that sees nothing reports an empty one.

discrete · Oscillation Coverage across the proportion of the randomised interval with the arrangement of the successes as its coin, 20 trials. Between proportions of 0.2 and 0.8 the arrangement coin's coverage stays within 0.121% of 95%, where mid-p's strays by 2.76%. Its worst coverage is 92.60%, at a proportion of 0.140, near the ends of the range where the count leaves few arrangements; mid-p's is 92.94%. Averaged over the proportion, 95.63% against mid-p's 96.11% and Clopper–Pearson's 97.71%.

The coin that is already there

The randomised interval covers exactly 95% at every proportion because it adds a drawn coin to the count, and a drawn coin is why nobody reports it. A sequence of trials already holds one: given the count, the order the successes arrived in is equally likely to be any of its arrangements whatever the proportion is. Used as the coin, it keeps twenty trials' coverage within 0.12 points of 95% from proportions of 0.2 to 0.8, where mid-p strays by 2.8, and it gives the same interval to every analyst. It fails where the count has few arrangements — a count of none has one — and it holds only while the order is read as recorded: an analyst choosing among sixteen orders lifts a 5% test on sixteen trials to 7.68%.

discrete · Oscillation

Named alongside it

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

Clopper–PearsonCoverageDiscretenessRandomised intervalReproducibilityBinomial proportionExchangeabilityFuzzy intervalWilson interval

All concepts