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

Reproducibility — where it appears

The property that repeating an analysis on the same data gives the same result. Seeded simulations have it by construction, and a randomised interval deliberately gives it up in exchange for exact coverage.

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

Twenty samples of 40, every one of them genuinely normal. Each panel is a quantile-quantile plot of 40 draws from a normal distribution. The worst point in the worst panel sits 0.87 standard deviations off the line. Anything a reader would reject here would be a false alarm.

The seed is part of the figure

Every other site in this fleet draws from a deterministic rule, so a figure either is or is not what it claims. Here the figures are samples, and a sample can be right by luck. That changes what a figure has to carry.

method · Seeds 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 Coverage against the expected count of the interval whose coin is read from the arrival times. The randomised interval for a Poisson mean, with its coin read from where in the window the events fell, covers exactly 97.5% at every expected count between 0.025 and ln 40 = 3.689 and exactly 95% past it, when a count of zero is given the exact interval's [0, ln 40]; its worst is 95.00%. Given mid-p's [0, ln 20] instead, it falls to 92.50% just past 3.00. Mid-p itself reaches 91.66% at 3.03; the exact interval never falls below 95.62% and averages well above 95%.

The coin a clock supplies

A count observed over a window comes with the times its events arrived, and given the count those times are uniform whatever the rate is — so they are a coin with a continuum of faces at every count from one up. Used to randomise the interval for a Poisson mean, they leave exactly one count without a coin, the count of zero, and what that count is given decides everything: mid-p's limit of ln 20 leaves a hole at 92.5%, and the exact interval's ln 40 is the shortest limit that leaves none, after which the interval covers exactly 97.5% below an expected count of 3.69 and exactly 95% above it.

discrete · Oscillation

Named alongside it

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

CoverageClopper–PearsonDiscretenessMid-pRandomised intervalExchangeabilityWilson intervalBinomial proportionConservative intervalFuzzy intervalInterval widthPoisson limit

All concepts