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

Bands — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The normal density at sigma = 1.00. The bands hold 68.27%, 95.45%, 99.73% of the mass. Those figures are integrals of the curve drawn, not the memorised 68-95-99.7.

    The shape, and where its mass is

    68, 95, 99.7 is recited more often than any other set of numbers in the subject. They are integrals of a specific curve, they are worth computing rather than remembering, and the third one is the one people misuse.

    part 1 · normal
  2. What x-bar plus or minus 2 sample standard deviations holds, at n = 10. The content of the band is a random variable. Across 20,000 normal samples of 10 it averages 91.1%, its fifth percentile is 74.7%, and it falls short of 95% on 59.9% of samples. The band that is drawn to show where 95% of the data lies.

    Two standard deviations of what

    The 95.45% inside two standard deviations is a fact about a curve whose centre and width are given. Drawn from ten observations, the same band holds 91.1% on average and less than 95% on 59.9% of samples — and the average is the reading that hides it.

    part 2 · estimated
  3. 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.

    part 3 · estimated
  4. How many observations the smallest and largest of them need. The interval between the extremes of n draws holds at least 95% of the population with probability 1 - n p^(n-1) + (n-1) p^n, whatever the population is. Reaching 95% confidence takes 93 observations.

    Ninety-three observations, and nothing assumed

    The interval between the smallest and largest of a sample holds a share of the population whose distribution does not depend on the population — Beta(n − 1, 2), for anything continuous. Buying the 95/95 that normality buys at ten observations costs 93 of them, and that number is the exchange rate between an assumption and data.

    part 4 · estimated
  5. 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.

    part 5 · estimated
  6. 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%.

    part 6 · estimated
  7. 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%.

    part 7 · estimated
  8. 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.

    part 8 · estimated
  9. 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.

    part 9 · estimated

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