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

Quality control — where it appears

The practice of checking that a production process stays within limits set in advance, by measuring samples and comparing them with stated ranges. Its statistical tools are ranges and intervals meant to contain a stated share of future items, not the mean alone.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The band that allows at most r of the next m observations outside it, 95% of the time, from a sample of ten. At a hundred future observations the band that holds all of them has factor 4.942; allowing one outside, 4.289; two, 3.948; five, 3.378. The tolerance factors for 99% and 95% content are 4.445 and 3.382.

A band allowed a few misses

From ten observations, a band holding every one of the next hundred with 95% probability reaches 4.942 sample standard deviations either side of the mean. Allowing one of the hundred outside brings it to 4.289; allowing five, to 3.378 — the 95%-content tolerance factor, 3.382, to within half a hundredth. The tolerance interval turns out to be the limit of a promise about a share of failures. And the misses arrive together: when this band misses once, it misses again 45.4% of the time, where independent misses at the same rate would do so 12.5%.

estimated · Bands How many observations a band from a sample's extremes needs to let at most r of the next 100 outside, 95% of the time, for any population. From the sample's smallest and largest: none of the next 100 outside needs 3,850 observations, at most one 637, at most five 107 and at most ten 50. From the second smallest and second largest, 7,750, 1,204, 187 and 85.

A range allowed a few misses

Without assuming a shape, the band a sample can offer is its own smallest and largest values, and the number of future draws falling outside it is beta-binomial for every continuous population. A promise that at most one of the next hundred lands outside, with 95% probability, then needs 637 observations; none of the hundred, 3,850; at most five, 107. The normal-theory band keeps the one-in-a-hundred promise from ten. On normal data, at the sizes the range needs, it is only 6% wider than the normal band at the same size — the price of dropping the assumption is paid in observations, not in width — and its misses cluster far less than a band from ten: a second follows a first 19.7% of the time, against 45.4%.

estimated · Bands How many starting observations a range needs to flag at most r of the next 100, fixed at the start or learning as the lot is used. 95% of the time, for any continuous population. At most 0: fixed 3,850, learning 3,850; At most 1: fixed 637, learning 512; At most 2: fixed 300, learning 197; At most 3: fixed 190, learning 101; At most 5: fixed 107, learning 36; At most 7: fixed 74, learning 15; At most 10: fixed 50, learning 4. With no flag allowed the two are the same, because a lot with no flag never moves the learning range.

A range that grows with the lot

A band set from the range of everything seen so far flags a unit only when it sets a new record, and for any continuous population those flags are independent, with the k-th unit flagged with probability 2/(n + k). That makes a few-misses promise far cheaper than the fixed range's — at most five of the next hundred from 36 starting observations rather than 107 — and exactly as expensive when no miss is allowed, because a lot with no flag never moves the band. The saving is bought by believing every flag. After a two-standard-deviation step in the mean, the fixed range flags 25.03 of the eighty units that follow and the learning range 3.62; by the last twenty units it sees a unit outside in 25.5% of lots, against 18.4% with no step at all.

estimated · Bands

Named alongside it

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

Sample sizeTolerance intervalBinomial distributionDistribution-freeExchangeabilityOrder statisticPrediction intervalChange pointCoverageEstimated varianceSimultaneous inference

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