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

Discreteness — where it appears

That a statistic can take only certain values, so a test's true level jumps rather than moves smoothly and rarely equals the level it claims. It is why a coverage curve oscillates with the parameter and why an acceptance region computed from a continuous approximation can admit everything or nothing.

Named by 24 essays across 15 fields — each of them below, with the objects they name alongside it.

The expansion that never terminates. The Hermite coefficients of a median split, in magnitude, against the reference j to the power −3/4, anchored at the first one. Every even order is exactly zero because sign is an odd function, and every odd order is not, so no truncation is exact — where a polynomial of degree d is exact at any order past d. Summed, the tail past J falls like 1/√J: sixty orders still leave 6.6% of the variance outside. That statement is what made a cut dictionary's geometry unavailable in closed form, and it is a statement about the function against itself. What it is not is the accuracy of an inner product between two correlated variables, where every term past J carries a factor of ρ^m as well.

A cut is not a polynomial, and it does not have to be

A threshold's expansion never terminates, which is why a balancing dictionary's geometry was closed for powers and taken to draws for cut points. Conditioning on the second variable closes it for both.

splits · Blocking 20,000 p-values from a true null, n = 12. Flat, as it must be: under the null a p-value is uniform on (0,1). The Kolmogorov–Smirnov distance from uniform is 0.0090 (p = 0.81). That flatness is the check that catches an error a single rejection rate would miss.

A p-value that is not flat is not a p-value

Under a true null, p-values are uniform. That is stronger than saying the test rejects 5% of the time, it constrains the whole distribution rather than one point of it, and it catches implementation errors that a rejection rate sails past.

testing · Uniformity The normal approximation's error on a sum of 100 exponential draws, under its Berry–Esseen bound. The distance between the exact distribution function and the normal one peaks at 0.0133, at z = -0.01. The Berry–Esseen bound is 0.1146, 8.62 times the real worst error, and larger than the whole 2.5% tail a two-sided test reads.

A bound written for a coin

The Berry–Esseen theorem guarantees how far a standardised sum can be from the normal, and the guarantee is true. On an exponential source it is 8.62 times the real worst error at every sample size, the worst error sits at the centre rather than in a tail, and at a hundred draws the bound is larger than the 2.5% tail it would be asked to vouch for.

expansion · Rate 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.

discrete · Oscillation What the second search finds, alone and afterwards. For four of the pairs, what the second search removes on its own and what it removes once the first has already run. The gap between the two is the overlap in absolute terms. Where the searches share nothing the two readings are the same: an independent column removes 0.0261 alone and 0.0260 afterwards. Where one contains the other they are 0.1387 and exactly zero. The pair the earlier field measured sits between: a whitening window removes 0.5033 alone and 0.3120 after a break search has run. This is the earlier field's own reading of its pair, on the share scale rather than in log-likelihood units, and it is the number a rule that runs both searches actually has to charge for.

A search that is already the other

A break search shifts every coefficient after a row, so a step column is one of the directions it can move in. Paired with a dictionary of them it reads exactly one, on every draw, and that fixes the top of the scale.

apart · Criterion 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.

intervals · Oscillation Ten groups of 10, pooled on the log-odds scale. Each row is a group. The hollow circle is its own proportion, the filled one is the estimate after pooling, and the vertical rule is the pooled population proportion of 31.3%. One group saw no events at all, and its raw proportion of zero becomes 21.2% — an estimate the group's own data cannot produce and the population's can. The arrows are not the same length, and none of the groups differs in size.

Pooling a proportion

A proportion cannot be shrunk on its own scale — an estimate would leave the interval, and how much information a count carries depends on where it sits. Move to log-odds and the approximation works, at the price of a group that saw nothing having no estimate at all until the correction supplies one.

multilevel · Levels Stationary is not the same as convergent. How far each k-swap walk is from uniform after t steps, started at the least balanced admissible assignment of 410. Every one of these chains has a symmetric proposal and rejects by standing still, so every one of them is doubly stochastic and every one preserves the uniform distribution exactly. Only five of the six get there. Exchanging all six units of each arm is a single proposal — the complement — and the admissible set is closed under complement, so the walk takes it every time and oscillates between two assignments for ever: after 160 steps it has visited 1 state and sits 0.9976 from uniform. Its stationary distribution is a fact about the matrix; its limit does not exist.

Stationary is not convergent

A walk that exchanges every unit in each arm preserves the uniform distribution exactly and never gets near it. Every doubly stochastic matrix has the same stationary distribution; only some of them have a limit.

blocks · Randomisation Sheppard's arcsine, by two routes. Corr(sign X, sign Y) as the covariates' correlation runs from zero to one, drawn twice. One route is a sixty-four-node quadrature of the orthant probability over the correlation — the general construction, which works at any pair of cut points; the other is (2/π) arcsin ρ, which is elementary and works only at the median. They agree to 3.3e-16 at every one of 81 correlations, which is what licenses the quadrature everywhere else. The curve is above the diagonal at small ρ and below it at large: two signs agree with probability ½ + arcsin(ρ)/π, so a correlation of 0.5 gives exactly ⅓ and a correlation of 0.8 gives 0.5903.

The arcsine that closes it, and the error that was overstated

Two median splits of a correlated pair agree with probability ½ + arcsin(ρ)/π, exactly. And the truncation the field was avoiding falls geometrically in the correlation, not algebraically in the order.

splits · Routes A budget of 4,000, at 1 and 20 a unit. Every affordable pair, enumerated. The best is 280 cheap units and 186 expensive ones — a ratio of 1.51, against the σᵢ/√cᵢ rule's 1.49. The unit rule, which says buy in the ratio of the spreads, lands at 66:197 and costs 17% more variance for the same money. Both rules are right about their own constraint; only one of them was asked.

The cost of a unit

Change the constraint from units to money and the allocation rule changes with it — from σᵢ to σᵢ/√cᵢ, which can point the other way. An arm that is noisy and expensive gets fewer units than the same arm would if the money were not the thing running out.

allocation · Allocation What a disagreement costs, split on whether it decided anything. The regret from choosing the tuning parameter per candidate, on the draws where the candidates disagreed, split on whether the disagreement changed which candidate the table selects. Over 1200 draws at each list length: when the winner changes the regret is 0.02215, 0.03029, 0.03145; when it does not it is -0.00243, -0.00069, -0.00065 — negative, and small enough that it is inside two standard errors of nothing at every length. The whole of the cost lives in the first column, and the second column is not merely small but slightly the wrong sign: when the table's answer is unaffected, letting each candidate use its own window is a very slightly better rule than making them share one. So a disagreement about the tuning parameter is not a cost. A disagreement that changes the winner is.

The quarrel that changes the winner

A disagreement about the tuning parameter costs 0.031 when it changes which candidate the table selects and −0.0007 when it does not. The distance between the values disagreed about has nothing to do with it.

apiece · Order-selection 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.

discrete · Oscillation The quantity that does not depend on the list. The probability that letting each candidate choose its own tuning parameter changes which candidate the table selects — the product of the two moving shares — against the length of the list, over 1200 draws apiece. It is 14.2%, 11.9%, 12.3%: a spread of 2.2% across a list length that moves the disagreement rate by a factor of 1.52. This is the invariant the whole field turns on. Everything downstream of the winner — the coefficients, the regret, whatever a reader is going to quote — is a function of whether the winner changed, and how often that happens is not something the list controls. A longer list changes how often the candidates quarrel and not how often the quarrel matters.

How often it matters

The disagreement rate rises by half across the list and the share of disagreements that decide anything falls by nearly the same factor. Their product — how often the tuning list changes which candidate wins — sits at an eighth and does not move.

apiece · Order-selection A cut at a quantile, and a cut at a value. Two rules that read identically in a protocol. One splits each covariate at its median; the other splits it at 1 on the covariate's own scale — a dose, a temperature, a clinical threshold. At a correlation of 0.5 the first removes exactly nothing of the interaction between its own two splits, under every marginal here, because a median split is a function of the sign of the latent normal whatever the marginal is. The second removes what the bars show, and it does so on a normal covariate too: the threshold sits at 1.000 on the latent scale rather than at zero, so it is 59.4% odd and 40.6% even. The exact zero was never about the cut; it was about the cut being at the median.

The cut that is not a quantile

A protocol that says split the covariate at a threshold and one that says split it at the median read the same and are different rules. One has an exact guarantee under every marginal and the other has none under any.

skew · Criterion How often "there is no spread between the groups" is reported about data that has one. Every dataset here was generated with a real population spread of 1. The moment estimator is the difference between the observed spread and what noise alone would produce, clamped at zero, and the difference comes out negative often: at eight groups it reports exactly zero on 32.6% of datasets, which is an instruction to pool completely and give all eight groups the same estimate. The rate falls to 4.2% at 48 groups.

When the spread estimates to zero

The usual estimate of a population spread is a difference of two positive quantities, clamped at zero. On a third of eight-group datasets with a real spread in them the difference comes out negative, the estimate is exactly zero, and every group is pooled completely on data that said no such thing.

fullbayes · Pooling Twenty cells of an interval that is exactly 95%, 1,000 replications each. The t interval covers exactly 95% in every cell. Estimated at 1,000 replications its cells read 93.9% to 96.5%, and 2 of the twenty are flagged by their own ±1.96 standard errors.

A coverage table with its own error

Twenty cells estimating the coverage of an interval that is exactly 95%, at a thousand replications each, read from 93.9% to 96.5% — and a table like that flags at least one of its correct cells on 69.9% of honest runs. Ten times the replications does not repair it: at ten thousand the same table still flags one 63.3% of the time.

method · Seeds 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.

discrete · Oscillation The upper tail of 10 exponential draws: normal, two Edgeworth terms, saddlepoint, each against the exact tail. Each curve is an approximation divided by the exact gamma tail, so 1 is exact. Six standard deviations out at n = 10: the normal gives ×0.0000670, one Edgeworth term ×0.00159, two ×0.0167 and the saddlepoint ×1.0007.

An approximation built at the threshold

The saddlepoint approximation reads the tail of a sum of five exponential draws to within 0.19% six standard deviations out, where the normal is short by a factor of more than sixty thousand. It is within 2.2% out to ten standard deviations on a single draw, where there is nothing to average, and within 1.1% on a binomial whose expected count is one. It works because it is built where the tail is read rather than at the mean.

expansion · Rate Where a walk is cheaper than a hunt. Both costs in the same unit. A rejection sampler evaluates 1/p assignments per independent draw and does not care how large the trial is; a walk evaluates one per step and yields an effective draw every τ steps, and τ is a property of the constraint and the statistic together. They cross at a tolerance of 0.194 standard deviations, where about one assignment in 396 is admissible — far tighter than any trial is designed at. And the walk does not remove the acceptance cost; it pays it once, hunting for somewhere to start.

Draws that repeat each other

A hunt costs 1/p evaluations per independent draw. A walk costs one per step and yields an effective draw every τ steps. Both are counted in the same unit, and the walk is dearer at every tolerance a trial is designed at.

joint · Reference Expected width against coverage, n = 30, p = 0.15. The Wald interval is the shortest and covers 94.2%. Clopper–Pearson covers 98.3% and is 13% wider. Shortness is not a virtue on its own — an interval of zero width is the shortest of all.

The shortest interval is the one that misses

Four intervals for the same data, with their widths and their coverage measured together. The narrowest is the one that fails its stated level, which is exactly why it looks the most appealing.

intervals · Width Which samples Wilson and Clopper–Pearson each cover, n = 50, p = 0.2. Each bar is the probability of one count, shaded by which interval built on that count contains 0.2. Both cover 95.1% of samples, only Wilson 0.0%, only Clopper–Pearson 1.6%, neither 3.3%. The correlation between their hits is 0.810, so on shared draws the variance of their difference is 4.891 times smaller than on independent ones.

The same draws for both methods

Two intervals computed on the same simulated datasets give a difference in coverage whose variance can be 4.891 times smaller than on separate datasets — or, for a pair that covers different samples, 1.164 times larger. Which one a comparison gets is an exact sum over the counts each interval covers, and a standard error that ignores the sharing covers 100.00% for one pair and 93.07% for the other.

method · Seeds Two 95% intervals for 2 of 20, under Jeffreys — Beta(½, ½). The equal-tailed interval runs from 0.0214 to 0.2839 and is 0.2625 wide; the shortest interval runs from 0.0093 to 0.2540 and is 0.2447 wide. Both hold 95% of the posterior, and the shorter one buys its 6.8% by moving its lower endpoint towards the denser side.

The shortest interval, and the one that does not move

Two 95% intervals come out of every posterior and they are not the same set. The shorter one is shorter by 4.86% on average and 22.41% at its best, it covers 86.72% where the other covers 95.68%, and it is not even the shortest once the parameter is written a different way.

bayes · Credible Three ways to reject with two studies, drawn where the two z statistics live. Two one-sided studies, each summarised by its z statistic. Fisher's combination rejects outside a curve that runs parallel to both axes, so one study past z = 2.378 decides it alone; Stouffer's rejects above the straight line z₁ + z₂ = 2.326; Tippett's rejects when either z passes 1.955. Each region holds exactly 5% of the standard bivariate normal — Fisher's in closed form, e^(−c/2)(1 + c/2) at c = 9.488 — and of 100,000 counted null pairs they catch 4.95%, 4.88% and 5.04%. Two alternatives carry the same Stouffer evidence: one study at 2.326 and the other at nothing, where the powers are 62.7%, 50.0% and 65.4%; and both at 1.163, where they are 47.7%, 50.0% and 38.3%.

Two ways to combine p-values

Fisher's and Stouffer's combinations are both exactly right when every null is true, for the single reason that each p-value is flat. Under a real effect they disagree about which evidence counts: with Stouffer held at 50% power across ten studies, Fisher is the more powerful while the signal sits in six or fewer of them and the less powerful from seven.

testing · Uniformity What a variance estimated from K units is worth. The between-unit mean square is a scaled chi-square on K − 1 degrees of freedom, so the estimator's whole distribution is decided by the number of units. At two units its interquartile range spans a factor of 13.03 and its ten-to-ninety range a factor of 171.3, and it comes out exactly zero on 26.7% of studies. The closed form and 3,000 simulated studies agree to 0.051 at every quantile.

A level with two units

A variance estimated from two units is a scaled chi-square on one degree of freedom. Its interquartile range spans a factor of thirteen, its ten-to-ninety range a factor of a hundred and seventy-one, and it comes out exactly zero on 26.7% of studies — so the design effect it decides runs from 1.00 to 7.01 against a truth of 4.69.

multilevel · Levels

Named alongside it

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

CoverageMonte CarloSample sizeBinomial proportionClosed formCovariate balanceExperimental designOrthogonalityBasis functionsClopper–PearsonContinuous covariateCorrelation

All concepts