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

Tweedie's formula — where it appears

The exact expected true score behind a reading measured with normal error: the reading plus the error variance times the slope of the log-density of readings there. It turns regression to the mean into a property of the population's shape, and needs only the readings, not the true scores.

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

The top of a heavy-tailed population keeps its lead; the top of a light-tailed one gives it back. Select the top share on the first reading and read the group again: the share of its mean lead the second reading keeps, by integration over the true score (lines) and counted on 400,000 draws a parent in 20 batches (points, with two standard errors). The normal keeps exactly 0.6 at every selection. At the top half the Laplace keeps 0.541, the t 0.535 and the uniform 0.648 — the heavy tails keep LESS than the correlation. By the top one per cent the order has reversed: 0.761, 0.784 and 0.443. At one in ten thousand the t keeps 0.977 and the uniform 0.346.

A lead that a heavy tail keeps

Four populations whose readings all correlate at exactly 0.6, and whose least-squares slopes all read 0.6. Select the top one per cent on one reading and measure them again: they keep 60% of their lead if the true scores are normal, 76.1% if they are Laplace, 78.4% if they are a t on four degrees of freedom — and 44.3% if they are uniform. The correlation predicts the regression of the extremes for one shape of population only.

paradox · Rtm Twenty studies of a thousand readings estimate the regression of the extremes: the t, four degrees parent. Each thin line is one study of 1000 readings from the t, four degrees parent: Tweedie's formula with the log-density's slope estimated by a degree-5 log-spline, drawn up to that study's largest reading. The thick line is the share kept by integration over the true score, and the dashed line the correlation, 0.6. The top ten readings of the median study begin at 2.45. Over 400 studies the corrected share kept by the top one per cent averages 0.8190, with a spread of 0.0806, against 0.7842 by integration; a Gaussian kernel averages 0.7914 with a spread of 0.0853.

The slope of a density nobody can see

Tweedie's formula corrects a reading by the slope of the readings' own log-density, and a study has its readings. Estimated from a thousand of them, the correction for the top one per cent beats the correlation's linear rule on 84.0% to 98.0% of studies from heavy-tailed populations and on 75.0% to 81.5% from a bounded one — and costs an error of 0.09 to 0.12 where the population is normal and the rule was already exact. At 250 readings the log-spline loses to the rule it replaces, and at 16,000 the same log-spline gets worse on a power tail.

paradox · Rtm How many people a study has to retest before its estimated correction is worth having, normal errors. Root mean squared error, over 400 studies of a thousand readings at each retest size, of the share of the top one per cent's lead a second reading keeps: Tweedie's correction with a kernel log-slope and the error variance estimated from the retested people (solid), and the linear rule at the reliability those people estimate (dashed). Laplace parent: 0.121, 0.108, 0.084, 0.086, 0.084, 0.078 against 0.343, 0.259, 0.201, 0.183, 0.171, 0.164. T, four degrees parent: 0.133, 0.103, 0.086, 0.084, 0.080, 0.078 against 0.358, 0.278, 0.235, 0.212, 0.205, 0.187. Uniform parent: 0.247, 0.192, 0.156, 0.140, 0.118, 0.117 against 0.260, 0.208, 0.185, 0.166, 0.160, 0.157 — at 10, 20, 50, 100, 200, 1000 retested.

A correction built on a retest

Tweedie's correction for regression to the mean needs the error variance, and a study gets it from a few people measured twice. Twenty of them are enough: with normal errors the estimated correction then beats the correlation's rule on 73% to 77% of studies of a thousand readings from a heavy-tailed population, though its variance is uncertain by a third. What twenty cannot reveal is the error's shape. Laplace errors of the same variance make the correction overstate the lead a top group keeps by about 0.17, the plain rule then wins on most studies, and a study needs about two hundred retest pairs before a check on its own differences notices.

paradox · Rtm

Named alongside it

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

Heavy tailMeasurement errorMonte CarloRegression to the meanClosed formSelection biasShrinkageTest–retest reliabilityBias-varianceCorrelationEmpirical BayesKernel density estimate

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