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

Simultaneous inference — where it appears

Statements required to hold together, such as a band that must contain every point or every future value at once. Holding them jointly needs more width than holding each alone, and shared estimates make the misses cluster.

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

Two 95% bands for a quantile plot of 40 points. The outer band is left by 5% of genuinely normal samples — which is what a reader is using a band for. The inner one holds each point separately at 95%, which is what software draws, and 45.0% of genuinely normal samples step outside it. The outer is the inner widened by a factor of 1.502.

The band the eye was standing in for

The confidence band software draws on a quantile plot holds each point at 95%, and a genuinely normal sample of forty has forty chances to leave it — so 45.0% of them do. The band a reader is actually using is that one widened by a factor of 1.502, and nothing draws it.

lineup · Qq The factor that holds all of the next m observations with 95% probability, from a sample of 10. From 10 observations, the band for one future value has factor 2.371, for ten 3.716, for a hundred 4.942 and for a thousand 6.008. The 95%-content tolerance factor is 3.382 and is passed by m = 10; the Bonferroni stretch of the prediction factor reaches 7.567 at a thousand.

All of the next ten

A warranty, a batch release or a monitoring rule promises something about every one of the next ten observations, not about one. From a sample of ten, the band that holds all ten with 95% probability reaches 3.716 sample standard deviations either side of the mean — already wider than the 3.382 of a tolerance interval for 95% of the population — and it keeps widening: 4.942 for a hundred, 6.008 for a thousand, with no ceiling. A 95% prediction interval, read as the answer, holds all ten 67.9% of the time: more than 0.95 to the tenth power, because the ten succeed and fail together.

estimated · Bands

Named alongside it

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

BonferroniConfidence bandEstimated varianceExtreme valueMultiple comparisonsNormal distributionNormalityOrder statisticPrediction intervalQ–Q plotSample sizeSampling variation

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