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

League table — where it appears

A ranking of units — hospitals, schools, surgeons — by an estimate of their performance. Noisy estimates for small units crowd the extremes, and how the estimates are pooled decides which units are flagged at the top and at the bottom.

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

How many of a league table's top ten are small groups, ranked four ways. A hundred groups with sizes from 4 to 400, of which 36% have twenty units or fewer. Small groups make up 36.3% of the true top ten, 62.1% of the top ten by raw means, 13.4% by posterior means and 22.9% by the posterior chance of being in the top ten. The three rankings recover 4.43, 5.38 and 5.47 of the true top ten.

A league table of a hundred

A hundred groups with sizes from 4 to 400, and a top ten to publish. Ranked by their own means, small groups fill 62.1% of the top ten against their 36.3% share of the true top ten. Ranked by posterior means they fill 13.4%. The ranking built from each group's chance of being in the top ten recovers 5.47 of the true ten, the best of three and barely half; and the group ranked first could hold any rank from 1 to 31.

borrowed · Shrinkage The bottom ten of a league table of a hundred when larger groups do better, size–effect slope 0.6. Groups of 4 to 400, true effects rising with log size at a slope of 0.6 population widths. The true bottom ten is 80.6% small groups. Ranked by their own means the bottom ten holds 4.28 of the true ten; pooled to one centre, 3.19, with 42.6% small groups; pooled to a centre that rises with size, 4.35, with 83.0%.

The small groups one centre protects

In a league table of a hundred groups where larger groups do better on average, pooling every group towards one centre overstates the small groups by half a population width — 0.501 — because it pulls them towards an average they are not part of. The top ten barely notices. The bottom ten, the list that flags poor performers, holds 3.19 of the true bottom ten when pooled to one centre and 4.28 when the groups are ranked by their own noisy means. Pooling towards a centre that rises with size restores 4.35 and puts the small groups back at the bottom where 80.6% of the true bottom ten sit.

borrowed · Shrinkage

Named alongside it

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

Hierarchical modelPartial poolingPosterior meanRankingRegression to the meanShrinkageEmpirical BayesSample size

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