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

Prediction interval — where it appears

An interval meant to contain a future observation or a future estimate, rather than a fixed parameter. It is wider than a confidence interval by the future value's own sampling error, and wider again when it must hold several future values at once.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

Where a replication's estimate lands against a 95% interval, replication the same size. The chance that a 95% interval contains a replication's estimate is 95.00% when the original landed on the truth, 82.99% one standard error away and 48.40% two away. Averaged over where originals land it is 83.42%, and 5.00% of originals capture a replication less than half the time.

Five times in six

A 95% interval is read as a 95% chance that a replication's estimate will land inside it. With the spread known and a replication of the same size, the chance is 83.42% — five times in six — because both estimates are uncertain. An original that landed two standard errors from the truth captures a replication 48.40% of the time; a replication a tenth the size lands inside 44.54% of the time; and among significant originals from studies with 17% power, 66.94%.

alongside · Repetition Three bands called 95%, at n = 20. Half-widths in sample standard deviations: 0.468 for the mean, 2.14 for one future observation, 2.75 to hold 95% of the population. The first two differ by exactly the square root of n + 1, which is 4.58 here.

A tenth as wide, and both of them right

The interval for a mean and the interval for one future observation are both labelled 95%, and at a hundred observations one is 10.05 times the other — exactly the square root of n + 1. Read the narrow one as the wide one and it covers a new value 15.7% of the time.

estimated · Bands 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 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 Coverage under a drifting scale, with the quantile taken over fewer and more recent scores. The noise scale grows linearly across the sample to 1 + v times its starting value; 4,000 draws at each v. Worst coverage over the sweep: all 199 scores 85.63%, the last 99 90.88%, the last 39 93.42%, the last 19 93.53%. At v = 2.12, where the best of three detectors first fires 80% of the time, they cover 88.11%, 91.86%, 94.02%, 94.45%. Every window covers 95% when nothing drifts.

A quantile over the recent scores

A noise scale that grows across the sample takes a split conformal interval from 95% to 85.63%, and the best detector cannot see most of that loss. Take the quantile over the last thirty-nine calibration scores instead of all 199 and the worst coverage across the whole sweep is 93.42%, for 3.7% more width when nothing has drifted. The same window covers 91.30% where all 199 cover 93.53% under strongly correlated errors — and switching to it only when the detector fires is worse than either, because the detector fires on correlation too.

conformal · Exchangeability How many units a range needs to let at most r of the next hundred outside, fixed, learning or forgetting. 95% of the time, for any continuous population. At most 1: fixed 637, window 564, learning 512; At most 2: fixed 300, window 236, learning 197; At most 5: fixed 107, window 64, learning 36; At most 10: fixed 50, window 27, learning 4. The window counts are from 20,000 lots of uniform draws; the other two are exact.

A range that forgets

A band set from the range of the last sixty-four units flags each unit of a stable lot with probability exactly 2/65, for ever, and its flags arrive as if independent — so a promise of at most five outside in the next hundred costs sixty-four units, between the learning range's thirty-six and the fixed range's hundred and seven. What it was supposed to buy, a flag that lasts as long as a step and then stops, it cannot give: it absorbs a two-sigma step as fast as the learning range, 1.89 flags in ten units against 1.93, because it learns from what it flags. Refusing to learn from a flag makes it a fixed range that costs 141.

estimated · Bands

Named alongside it

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

Tolerance intervalCoverageSample sizeEstimated varianceExchangeabilityQuality controlBinomial distributionConfidence intervalDistribution-freeOrder statisticSimultaneous inferenceStatistical power

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