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

Every essay

One idea per essay, grouped by field and ordered so that the earlier ones set up the later ones — but nothing here depends on being read in sequence.

Grouped by field. Each heading is one of 93 fields, which is the coarsest of the five ways through this collection — the others are series, threads, figures and concepts.

The distribution itself 8The tail past the last observation 8Intervals, counted 8Tests, and the second number 8Reversals that are not errors 8The prior, doing visible work 8Regression, and what the summary hides 8A standard error for a model that is wrong 8A variable that moves one thing only 8What conditioning on a variable does 8Corrections, and what each controls 8When the data stops early 8The value that is not there 8Stopping rules 8Groups that borrow 8Decided before the data 8Weighting one sample into another 8When the observations repeat each other 8The spread, and its own uncertainty 8Hierarchy past one number 8Series that move together 8Three series, and a count 8The surface between the corners 8A design chosen rather than looked up 8Splitting the units 7Designs that change while they run 8The reference distribution the design supplies 8Coverage without a distribution 8The observation that has not happened 6The criterion, and what it assumes 6Balancing on what was recorded first 6Comparing two forecasters 8A forecast that is a probability 8A design that assumes less 5More arms than two 6The best of a set, and what the search costs 6What the design is asked to guarantee 6Balancing what has no levels 6Searching among fitted models 5What the procedure may not read 5The shape the covariate enters by 6A search with no fixed point 6Choosing what the rule reads 6The block size as a schedule 6Scoring a search without spending data 6When the set is too large to walk 6A promise about two arms 6Counting what is independent 5When the two are not independent 5The weights the corner needs 5Estimating the dependence, not naming it 5A cut point, at a correlation 7What a block may vary 5The shape a dependence has 5A block, weighted inside itself 6What a dictionary buys and what it costs 5When a fixed width is reached 5Fitted together, or fitted after 5Paying for a search 5What a chain cannot report 5A guarantee that needed a symmetry 6Where a taper's case begins 5A covariance with no parameter 5How long the list is 5Two searches over one sample 5The diagnostic after the trial 5The other half of the dependence 5A block length chosen from the data 5A charge for a covariance's own dimension 5The rate and the size of a disagreement 5Two searches over different features 5A probe chosen rather than picked 5Both halves of the dependence at once 6The block length read on a quantile 5A charge that is not a straight line 7What decides whether a tuning list decides 5Overlap and complementarity, separated 5A probe from what the rule blocks 6The same table at seven correlations 6The interval, studentised 5The interval that holds observations, not a mean 8When the stratified answer and the pooled one disagree 8The analyses that were available and not run 8What a summary of a scatter is a property of 8Shape, and what it does to a two-sample test 8Two tests, a threshold, and the rate they are read against 8What a diagnostic plot is showing 8Past the first term of the normal approximation 7A proportion's interval near the boundary, and the coin 6An interval read beside something else 8What partial pooling does to one group, to the set, and to a ranking 8What a sample-size calculation was given 8What makes it checkable 8

The distribution itself

Sums converge on it, which is the most-quoted theorem in the subject. What the quoting leaves out is the rate: the middle converges quickly and the tail does not, and the tail is where the approximation is actually read. The same theorem carried through a function stops being a normal limit where the function is flat, and carried through a ratio it forbids every interval that is always finite.

8 exponential draws, standardised, against the normal. The source is one-sided and skewed. At n = 8 the standardised sum has skew 0.695, and the theory says 2/sqrt(n) = 0.707 — so the convergence is visible AND its rate is predicted.

Sums of almost anything

The theorem says sums converge on one shape whatever they are sums of, which is remarkable and true. Watching it happen from a one-sided skewed source, with the rate of convergence predicted in advance, is more convincing than watching the shape appear.

7 figures · Clt, part 1 Where the normal approximation converges, and where it does not. Relative error against the exact binomial. At n = 1280 the error at the median is 0.96% and three sigma out it is 25.7% — a factor of 27. The tail is where the approximation is used.

The tail converges last

The central limit theorem is usually shown as a shape arriving. What the demonstration leaves out is the rate — and the rate is wildly different in the middle and in the tail, which is where every approximation in the subject is actually read.

6 figures · Rate, part 2 Twenty samples of 40, every one of them genuinely normal. Each panel is a quantile-quantile plot of 40 draws from a normal distribution. The worst point in the worst panel sits 0.87 standard deviations off the line. Anything a reader would reject here would be a false alarm.

What normal actually looks like

A single quantile plot of forty normal points wanders enough to look suspicious. Twenty of them, all genuinely normal, show what the noise looks like — and any single panel a reader would have rejected is in there.

4 figures · Qq, part 2 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.

6 figures · Bands, part 1 The squared estimate 1 standard errors from the flat point, exact and linearised. At δ = √n·μ/σ = 1 the exact law of the squared estimate has mean 2.00, variance 6.00 and skewness 2.177; the delta method's normal has mean 1.00, variance 4.00, no skewness, and 30.85% of its mass below zero, where a square cannot go. The Kolmogorov distance between them is 0.3085.

Where the derivative is zero

The delta method reads a standard error off a tangent line, and at a flat point the tangent says the spread is zero. The interval built on it for a squared mean covers 99.991% there and 85.978% one and a half standard errors away, with nearly every miss on the same side — and the law it should have used is a χ², not a normal.

7 figures · Clt, part 2 The two standardised means, and the shape of Fieller's set for their ratio at a = 3, d = 1. Each dot is one pair (zx, zy) drawn around (3, 1). Outside the horizontal band |zy| > 1.96 Fieller's set is a bounded interval, with probability 17.01%; inside the band and outside the disc of radius 1.96 it is everything outside an interval, 75.03%; inside the disc it is the whole line, 7.96%. On 40,000 counted draws Fieller covers ρ = 3.00 95.21% of the time and the delta interval 82.48%.

A ratio whose interval has to be the whole line

The delta interval for a ratio of two means covers 95.61% when the denominator is eight standard errors from zero and 1.10% at a ten-thousandth of one, and ten times as wide it still covers only 3.48%. Linearising is not the fault. Gleser and Hwang proved that every interval that is always finite fails the same way, so an interval that keeps its promise has to be the whole line some of the time.

6 figures · Clt, part 3 The law is the eigenvalues, and nothing else. The mean and the skewness of n(ĝ − g) at the stationary point, measured over 40,000 draws, against the closed forms: half the sum of the Hessian's eigenvalues for the mean, and for the skewness twice root two times the sum of their cubes over the sum of their squares to the power three halves. The worst disagreement anywhere is 0.028. In one variable the second-order law is a single chi-square with one degree of freedom and its sign is the sign of the second derivative; here it is a weighted sum with the Hessian's eigenvalues as weights, so a bowl and a valley differ in both moments and a saddle has both equal to zero.

A flat point with more than one direction

At a stationary point of a function of several means the second-order law is ½ Z′HZ, so the bias is half the Hessian's trace — 2.008 for a bowl, 5.028 for a valley, and −0.006 for a saddle, where the eigenvalues cancel. The saddle's coverage is the worst of the three.

4 figures · Clt, part 4 Testing for flatness at 5% before choosing the interval for a squared mean. Coverage of four nominal 95% intervals for μ², exact, against the true mean's distance δ from the flat point in standard errors. The tangent interval Z² ± 2z|Z| has its worst, 85.98%, at δ = 1.45. The second-order interval read off χ²₁ is right only at δ = 0 and falls to 34.22% at δ = 2. The two-stage rule, which uses the second-order interval when a 5% test does not reject flatness, has its worst, 49.72%, at δ = 1.96. The squared exact interval never falls below 95%.

Testing for a flat point first

Test whether the gradient is zero, and if the test does not reject, read the interval off the second-order law that holds at a stationary point. For a squared mean that two-stage rule covers about 50% at its worst whatever the pretest's level — 49.72% with a 5% test, where the tangent interval it was meant to rescue never falls below 85.98% — because its worst sits exactly where the true distance equals the test's critical value, half the samples are declared flat there, and the second-order interval cannot reach the truth from any of them.

5 figures · Clt, part 5

The tail past the last observation

The distribution a sum converges on has one limit; the distribution a maximum converges on has three, and every extreme-value analysis is a claim about which. The claim is also an extrapolation: a hundred-block level read from fifty blocks sits above the largest reading in the record half the time. So each measurement here reports not what the estimate is but how much of it came from data.

More blocks, and the light tail is called wrong more often. The share of records whose three-way family call is right, over 400 records at each block count, at 100 readings a block. A 95% interval for the shape is formed and the record is recorded as calling Fréchet, Weibull or Gumbel according to whether that interval sits above zero, below it, or straddles it. The two signed parents go from 51.5% and 71.0% at 20 blocks to certainty by 100. The light-tailed one goes the other way — 83.0%, 75.5%, 55.0%, 36.5%, 4.8% — because its estimate sits at about -0.1157 whatever the record length, and a longer record only shrinks the interval onto that number.

Three shapes, one limit

A normalised sum has one limit and a normalised maximum has three, indexed by a single number. Twenty blocks put the sign of that number right 97.3% of the time — and naming the family from a light-tailed record gets worse as the record grows, from 83.0% at twenty blocks to 4.8% at five hundred.

6 figures · Extremes, part 1 One tail arrives; the other is still on its way at a million. The Kolmogorov distance between the exact law of a normalised maximum and its Gumbel limit, at six block sizes, for two parents that both have that same limit. Both are closed form: the exact law of a maximum is F(x)^n and no simulation is involved. The exponential parent's distance falls from 0.0280 to 2.707e-7 — a factor of a hundred thousand, which is exactly one over n. The normal parent's falls from 0.0522 only to 0.0091, a factor of 5.74, because its rate is one over log n. At a million readings a block the two differ by a factor of 33556.3.

The maximum converges slowly

The rate at which a normalised maximum reaches its limit law is computable rather than simulable, because the exact law of a maximum is always available. For a normal parent the distance falls like one over the logarithm of the block and is still 0.0091 at a million readings; for an exponential parent, with the same limit, it is 2.707×10⁻⁷.

6 figures · Extremes, part 2 The threshold buys accuracy and spends exceedances. The mean squared error of the estimated shape against the threshold, split into the square of its bias and its spread, over 600 records of 2000 readings from a a normal parent. At the 0.9 quantile 199 exceedances are left, the bias is -0.1708, the spread is 0.0701 and the total error is 0.0341. The bias falls as the threshold rises because the exceedances get closer to being generalised Pareto; the spread rises because there are fewer of them. The sum is smallest at the 0.925 quantile, at 0.0340, of which 80.6% is still bias — so even the best threshold on this grid is one where accuracy, not spread, is the binding constraint.

The threshold is a dial

A peaks-over-threshold analysis has one knob, and raising it buys accuracy with exceedances. For a normal parent the error is smallest at the 0.925 quantile and 80.6% of it is still bias there — and both diagnostics practitioners use to set the knob lose to a fixed 0.90 rule, one by a factor of 1.590 and one by 11.881.

7 figures · Extremes, part 3 The record stops here, and the curve does not. The level exceeded once in T blocks, against T, for a normal parent at 365 readings a block. The truth is closed form — the block maximum's own distribution function is Φ(x) raised to the 365, so the T-block level is Φ⁻¹ of the 365-th root of (1 − 1/T), with nothing fitted in it — and the fitted mean over 800 records of 50 blocks sits on top of it, 4.0186 against 4.0330 at 100 blocks. What moves is not the level but its error, which grows from 0.0956 at 10 blocks to 0.6383 at a thousand while the level itself moves only from 3.4421 to 4.5454. The rule marks the largest reading an average record contains, 4.0062: everything to the right of where it crosses is read from a fit rather than from data.

A level with no data in it

The largest of fifty block maxima is a 51-block event by its own plotting position, so a hundred-block level is read 1.96 times past the longest event the record contains — and it lands above the largest reading on 52.4% of records. The estimate stays nearly unbiased out there; what grows is its error, sixfold from ten blocks to a thousand.

6 figures · Extremes, part 4 The interval every package reports first does not cover. Counted coverage of two 95% intervals for the 100-block return level of a normal parent, against the length of the record they were fitted from, over 300 records at each length. The level they are about is known in closed form, so this is coverage of a number rather than agreement between two estimates. The delta-method interval covers 80.3% at 25 blocks and reaches only 89.0% at 200; the profile-likelihood interval sits between 94.0% and 94.7% throughout. The gap is not a small-sample effect that lengthening the record removes — it narrows by 8.7 points for an eightfold longer record.

Two intervals for one return level

Two 95% intervals read off the same fits of the same records, against a level known in closed form. The symmetric one covers 80.3% at twenty-five blocks and reaches only 89.0% at two hundred — and 99.24% of its misses are the interval sitting entirely below the truth, which is not the endpoint anybody expects to fail.

6 figures · Extremes, part 5 The exceedances arrive together. 300 steps of a max-autoregression with dependence 0.75, drawn on a logarithmic scale because its marginal has no variance. The rule marks the 0.9 quantile: 30 of the 300 readings are above it and they fall into 5 clusters, the largest holding 11. The mean cluster holds 6.000, and its reciprocal — 0.167 — is the runs estimator of the extremal index, whose true value for this process is exactly 1 − 0.75 = 0.25. Every threshold method in the collection assumes exceedances are independent pieces of information; here 30 of them are 5.

The clustering the tail has

Every threshold method counts exceedances as though they were independent pieces of information, and in a dependent series they arrive in clusters. Ignoring that overstates a return level by the reciprocal of the extremal index — ×3.527 counted where the mean cluster holds four — and leaves a reported standard error 2.151 times too small.

7 figures · Extremes, part 6 A run length of 4 makes every exceedance its own cluster. 200 steps of a max-moving-maximum, X(t) = max(0.4·Z(t), 0.3·Z(t−6), 0.3·Z(t−12)) with unit Fréchet innovations Z, whose extremal index is exactly 0.40: one large innovation can put three readings above a threshold, 6 steps apart. The rule marks the 0.9 quantile and 19 readings clear it; 12 of the gaps between consecutive exceedances are exactly 6 steps. With a run length of 4, so that two exceedances 4 or more steps apart start separate clusters, they form 19 clusters, shaded, the largest holding 1. The runs estimator reads 1.000 against 0.40.

The run length a declustering chooses

The runs estimator of an extremal index carries a constant nobody derives. Where a cluster is a run of neighbouring exceedances the constant barely matters; where a cluster's members fall six steps apart, the estimate is 0.9069 at a run length of six and 0.3649 at seven against an index of 0.40, and a run length of four removes under a tenth of the overstatement declustering exists to remove. A rule that reads the run length off the data has the smallest worst error of the three.

7 figures · Extremes, part 7 How much of a return level's uncertainty is the extremal index's, by how far past the record the level is. Delta-method variance of the level from a declustered peaks-over-threshold fit, averaged over 300 records of 4,000 observations. The cluster rate's share at 1.5, 2, 5, 20, 100, 1000 periods: independent, 0.374, 0.253, 0.084, 0.027, 0.012, 0.005; runs of exceedances, θ = 0.6, 0.558, 0.396, 0.141, 0.039, 0.015, 0.006; spaced clusters, θ = 0.4, 0.845, 0.562, 0.222, 0.057, 0.019, 0.007; runs of exceedances, θ = 0.25, 0.787, 0.788, 0.327, 0.083, 0.024, 0.008.

Where the extremal index matters to a return level

A return level from a declustered fit depends on how often clusters arrive, which is the extremal index estimated from the same record. For a level reached once in two periods of a twenty-period record the index's error is a quarter to four fifths of the level's variance, and an interval that leaves it out covers 59.0% when clusters are long; put in, 90.0%. For a level reached once in a hundred periods its share is under two and a half per cent and adding it changes no interval at all — what fails there is the shape, fitted from however many clusters the dependence left: 82.3% coverage from forty-six clusters, 65.7% from nineteen.

4 figures · Extremes, part 8

Intervals, counted

An interval that claims 95% is making a checkable statement about a procedure. Build every possible sample and count. The interval taught first fails, the failure is worst where proportions are most often reported, and more data does not monotonically help.

Coverage of four nominal 95% intervals, n = 20. Computed exactly by summing over all 21 possible counts, not simulated. The Wald interval drops to 18.2% and is jagged everywhere; Clopper–Pearson never falls below 95% and pays for it in width.

What the 95% refers to

An interval that claims 95% is making a checkable statement about a procedure, not about the interval in front of you. Build every possible sample and count, and the interval taught first turns out to cover 87.6% of the time.

7 figures · Coverage, part 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.

7 figures · Oscillation, part 2 Twenty 95% intervals for a proportion that really is 0.35. 2 of the twenty miss the true value. The 95% is a property of the procedure across repetitions — no single interval has a 95% chance of anything, because it either contains 0.35 or it does not.

Twenty intervals and one expected miss

The 95% belongs to the procedure, not to the interval in front of you. Twenty intervals from twenty samples make that visible in a way no definition does, and the one that misses is not a mistake.

6 figures · Repetition, part 1 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.

7 figures · Width, part 3 Where the bootstrap works and where it does not. Uniform data on [0, 1]. For the mean the percentile bootstrap covers 93.5%. For the maximum it covers 0.0%, because a resample can never contain a value larger than the largest one observed, so the interval cannot reach above it.

Where the bootstrap lies

Resampling is the most generally useful trick in the subject and it has a failure mode that is easy to state: it cannot see past the data. For a statistic that lives at the edge of the sample, coverage collapses from 95% to almost nothing.

7 figures · Bootstrap, part 3 Student's t on 5 degrees of freedom, against the normal. The two-sided 95% critical value is 2.571 for t(5) and 1.960 for the normal — 31% wider. Using the normal at this sample size makes every interval too short by that much.

The correction for not knowing the spread

The t distribution exists because the standard deviation is estimated rather than known. At eight observations, using the normal instead makes every interval 12% too short — and the coverage that follows can be measured rather than argued about.

5 figures · Student, part 2 Three promises, and no procedure keeps all three. Average coverage and worst-case coverage for four 95% intervals for a proportion at n = 40, computed exactly. Their expected widths are 0.2418, 0.2417, 0.2472, 0.2641 in the same order. The textbook interval and the score interval have the same expected width to four digits — 0.2418 and 0.2417 — and worst-case coverages of 55.31% and 92.21%. The exact interval never breaks its promise and is 9.3% wider than the score interval to do it. Each of the three columns orders the four procedures differently.

An interval that covers and says nothing

A procedure returning the whole line 95% of the time and the empty set otherwise has coverage exactly 95% at every parameter value. Two real intervals at forty observations have expected widths of 0.2418 and 0.2417 and worst-case coverages of 55.31% and 92.21%.

4 figures · Coverage, part 2 The textbook (Wald) interval's coverage among studies whose count of 40 lands in the tails. Coverage of the nominal 95% textbook (Wald) interval at 40 trials, conditional on the count being at most k or at least 40 − k, computed exactly over the counts for every true proportion. With k = 2 its highest value anywhere is 88.96%; With k = 5 its highest value anywhere is 94.90%; With k = 8 its highest value anywhere is 95.57%. A set whose highest value stays below 95% is a set of counts on which the label is wrong whatever the proportion is.

A count that bets against its interval

A reader who sees only the count can bet against an interval's 95%, and win whatever the proportion is, exactly when some set of counts carries less than 95% at every proportion. Out of forty trials the textbook interval has one — a count of five or fewer, or thirty-five or more, covers at most 94.90% — and the score interval has none: a prior exists under which every one of its forty-one intervals is 96.46% credible, and that prior is the proof that no bet can be found.

5 figures · Coverage, part 3

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