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

Oscillation — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Coverage against sample size, true proportion 0.15. Coverage does not improve monotonically. n = 19 covers 93.8% while the larger n = 20 covers 81.9%. The sample space is discrete, so the endpoints jump as n changes.

    More data is not monotonically better

    Coverage of an interval for a proportion does not improve smoothly as the sample grows. It oscillates, and there are larger samples that cover materially worse than smaller ones — a sample of twenty covers twelve points worse than a sample of nineteen.

    part 2 · intervals
  2. Coverage of the Wilson interval against the expected count, at 10, 30, 100 and 1,000 trials. Read against the expected number of successes the four sample sizes draw the same curve near the boundary. The worst coverage is 83.50% at n = 10, 83.71% at n = 30, 83.79% at n = 100, 83.81% at n = 1000, each at an expected count near 0.177, and the limiting depth is e^(−0.1765) = 83.82%.

    A hole no sample size fills

    Wilson's interval is the recommended repair for a proportion, and away from the boundary it wobbles a point or two around 95%. Near zero it has a hole: at an expected count of 0.1765 its coverage is 83.50% at ten trials, 83.79% at a hundred and 83.81% at a thousand, and it never climbs past e to the minus 0.1765, which is 83.82%. The hole is where the interval built on one success stops containing the truth, and it belongs to the count rather than to the sample size.

    part 3 · discrete
  3. Worst and average coverage of six 95% intervals for a proportion, 30 trials. The worst coverage over every proportion beside the average over a uniform one, with the average expected width. Clopper–Pearson: worst 95.05%, average 97.34%, width 0.299. Blaker: worst 95.00%, average 96.31%, width 0.283. Wilson: worst 83.71%, average 95.24%, width 0.271.

    What a guaranteed minimum costs

    Clopper–Pearson's interval never covers less than 95%, and at thirty trials it averages 97.34% and is 10.4% wider than Wilson's. Blaker's interval keeps the same guarantee, averages 96.31% and is 4.6% wider. The difference is not waste: Clopper–Pearson guarantees each side separately, holding both below 2.5%, and Blaker guarantees only their sum — so at ten trials and a proportion of 0.15 it misses on one side 5.00% of the time.

    part 4 · discrete
  4. 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.

    part 5 · discrete

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