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 worst coverage over δ of the two-stage rule for a squared mean, exact, at pretest levels 1%, 5%, 10%, 20%, 30%, 50%, 70%, 80%, 90%, 95%, 99%: 49.55%, 49.72%, 49.84%, 50.38%, 51.82%, 58.84%, 72.02%, 80.61%, 85.98%, 85.98%, 85.98%. The tangent interval alone has a worst of 85.98%, and the second-order interval alone 8.22%. No level puts the two-stage rule above the tangent; from 90% upward the pretest almost never declares flatness and the rule is the tangent.

Against δ: the chance the pretest declares the point flat (|Z| ≤ 1.96), the coverage of the second-order interval among the samples it is used on, and the coverage of the tangent interval among the samples it is used on. At δ = 1 the pretest declares flatness 83.0% of the time and the second-order interval it then reports covers 42.5%; at δ = 2 those are 48.4% and 0.0%. Given flatness, the squared z statistic is below 3.84, and the second-order interval's upper end is that squared statistic itself, so once the squared distance passes it the interval cannot reach the truth.

The ratio of the expected widths of the squared exact interval and the tangent interval for the squared mean, exact. The two are identical whenever the absolute z statistic exceeds 1.96, and otherwise the image is wider by the square of its shortfall below 1.96. The ratio is 1.272 at δ = 0, 1.130 at δ = 1, 1.025 at δ = 2 and 1.000 at δ = 4.

Coverage of three nominal 95% intervals for a saddle at the true mean, counted over 40,000 draws at each distance of the true mean from the stationary point along the first axis: the tangent interval, the two-stage rule that uses the second-order law — half the quadratic form of the z vector in the Hessian — when a 5% chi-square test on two degrees of freedom does not reject stationarity, and the image of the 95% confidence disc for the mean. Tangent: 100.0%, 99.8%, 98.0%, 94.3%, 90.6%, 90.5%, 91.3%, 92.5%, 93.6%. Two-stage: 98.0%, 97.7%, 97.6%, 96.7%, 91.9%, 82.2%, 80.1%, 92.5%, 93.6%. Image: 100.0%, 100.0%, 99.9%, 99.8%, 99.7%, 99.1%, 98.6%, 98.6%, 98.5% — at 0, 0.5, 1, 1.5, 2, 2.5, 3, 4, 6.

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.

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 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.

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.

Approximate tail probability divided by the exact one. At one sigma the ratio is 0.98; at 4 sigma it is 0.104, so a rare event is understated by a factor of 10.

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.

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.

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.

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%.

The content of the band is a random variable. Across 20,000 normal samples of 10 it averages 91.1%, its fifth percentile is 74.7%, and it falls short of 95% on 59.9% of samples. The band that is drawn to show where 95% of the data lies.

At ten observations it is 3.382 sample standard deviations, against the 1.96 an interval for the mean uses. The two only converge in the hundreds: at n = 300 the factor is still 2.106.

Half-widths in sample standard deviations: 0.468 for the mean, 2.14 for one future observation, 2.75 to hold 95% of the population. The first two differ by exactly the square root of n + 1, which is 4.58 here.

The interval between the extremes of n draws holds at least 95% of the population with probability 1 - n p^(n-1) + (n-1) p^n, whatever the population is. Reaching 95% confidence takes 93 observations.

The chance that the smallest and largest of 20 observations hold at least 90% of the population, counted under a normal, an exponential, a uniform and a t on three degrees of freedom. The closed form gives 60.8% and knows nothing about which.

How often the band holds the 95% of the population it promises. On the normal it is 95.1%, as designed. On a Laplace it is 78.4%, and on a uniform it is 99.5% — conservative rather than short, which is the direction nobody predicts.

Both tails should be 2.5%. At 8 observations the interval falls short of the mean on 9.75% of samples and overshoots on 0.31%. At 500 they are 3.31% and 2.05%, and the total is 5.36% — which a coverage table reports as very nearly right.

Both bars should read 2.5%. The three symmetric sources — normal, Laplace and a t on five degrees of freedom — are balanced whatever their tails do. The two skewed ones are not: the exponential's misses split 7.9 to 1 and the lognormal's 61 to 1.

Student's t is derived on the assumption that the sample mean and the sample variance are independent. Here their correlation is 0.7023 at 8 observations and 0.7093 at 800 — a constant rather than a small-sample effect.

Forty units split between two groups, with the second group's variance a stated multiple of the first's, and the two population means equal. A 5% test should reject 5% of the time. The pooled test runs from 0.55% to 18.91% across this region; Welch's runs from 4.63% to 5.51%.

The pooled test has 38 degrees of freedom at every split, because it counts them. Welch computes them from the two estimated variances, so they are not integers and they differ from sample to sample — the standard deviation of 8-and-32's is 7.36.

The outer band is left by 5% of genuinely normal samples — which is what a reader is using a band for. The inner one holds each point separately at 95%, which is what software draws, and 45.0% of genuinely normal samples step outside it. The outer is the inner widened by a factor of 1.502.

For each source: how often a quantile plot of the data leaves its pointwise band, and how often the 95% t interval for the mean misses. The two-lump source leaves the band on 100% of samples and its interval covers 94.80%; the t on three degrees of freedom leaves it on 57% and covers 95.73%, the best of the five.

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.

The bound divided by the largest distance between the exact distribution function and the normal one. A fair coin reads 1.19, so the bound is nearly attained; an exponential source reads 8.62 and a gamma of shape 16 reads 23.61.

The bound on the distance from the normal falls as one over the square root of the sample size. It drops below a 2.5% tail at 2,103 draws, below 0.5% at 52,573 and below 0.05% at 5,257,207. Before each of those points it cannot vouch for that tail at all.

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.

The sum cannot fall below -2.24 standard deviations, and the exact tail runs down to zero there. The one-term correction crosses zero at about -1.97 and reports -6.20e-3 at -2.19 — a negative probability.

One to six standard deviations out on an exponential sum. The saddlepoint's relative error at a single draw runs from 1.5e-3 to 1.6e-2, and at thirty from 2.2e-5 to 1.3e-4. The normal's at 30 draws runs from 7.6e-3 to 1.5e+3.

Both with a continuity correction. The saddlepoint's ratio stays within 0.07% of exact across 5 thresholds; the normal's falls to ×0.051 at 5 standard deviations.

The lower tail turns negative 1.97 standard deviations below the mean at 5 draws, 3.13 at a hundred and 9.83 at 100,000. The threshold recedes like the sixth root of the sample size, following the dashed prediction, so there is no sample size at which the region is gone.

The largest threshold before each approximation's ratio to the exact tail first leaves 1 ± 0.1. At a hundred draws: the normal 1.66, one Edgeworth term 3.09, two 4.11, and the saddlepoint past 12, the edge of the range searched. At a hundred thousand draws the normal has reached only 4.66.

On exponential samples the sample skewness has a median of 0.93 at ten observations and 1.85 at three hundred, against a true value of 2. The correction built on it gives a median of ×0.834 of the exact tail at ten, where the true-skewness correction gives ×1.081 and the normal ×0.617.

From 10 observations, the band for one future value has factor 2.371, for ten 3.716, for a hundred 4.942 and for a thousand 6.008. The 95%-content tolerance factor is 3.382 and is passed by m = 10; the Bonferroni stretch of the prediction factor reaches 7.567 at a thousand.

The interval holds one future value 95% of the time. It holds all of the next ten 67.9% of the time, against the 59.9% that 0.95 to the tenth power gives, because the future values share the sample's mean and spread and succeed or fail together.

Each cell should read 2.5 / 2.5. Two identical exponentials at 20 and 20 read 2.04 / 2.22; the worst cell, a wide exponential against a normal at 8 and 32, reads 9.79 / 0.47.

Five pairings of populations at five splits each. Whatever makes the difference skewed — unequal sizes, unequal spreads, opposite skews — the cells fall on one rising curve, with a rank correlation of 0.997.

The gap between the low and high rejection rates, as forty units are split between the groups. The skewness of the difference is zero at 29.6 units in the wider group, where the split is 2.83 to one; the split that minimises the standard error puts 26.7 there.

Both bars should read 2.5%. The t interval misses below the mean on 6.38% of samples. Widened until its total is exactly 5%, it misses below on 4.69% and above on 0.30%. Hall's transformation misses below on 3.31% and above on 1.96%.

The nominal rate is 2.5%. At fifteen observations the t interval's upper limit is exceeded on 8.31% of samples, the widened interval's on 4.93%, the shifted interval's on 7.99% and Hall's on 4.49%.

The t interval's upper limit sits 0.526 standard deviations above the mean on average and is exceeded on 8.31% of samples. A fixed one-sided multiplier tuned to 2.5% needs 0.841. Hall's transformation spends 1.044 and is exceeded on 4.49%.

Medians over two thousand samples. Six standard deviations out the normal reads ×6.8e-4 of the exact tail, the empirical saddlepoint ×0.061, a gamma fitted by the sample's skewness ×0.185, and a fixed exponential model ×1.000. The bars run from the tenth to the ninetieth percentile of the two readings that estimate the tail's shape.

Median reading as a share of the exact tail of a standardised sum of thirty. Six standard deviations out the empirical saddlepoint reads 0.069, 0.274, 0.465, 0.655 at samples of 30, 100, 300, 1000, and the fitted gamma 0.189, 0.513, 0.721, 0.873. The sample's skewness has a median of 1.31, 1.70, 1.85, 1.94 against a true 2.

The empirical saddlepoint needs the mean the threshold asks for to lie below the sample's largest value. For a sum of two, a threshold six standard deviations out is out of reach on 95.3% of exponential samples of thirty; for a sum of five, 32.4%; for a sum of ten, 0.7%.

At a nominal 1% the upper-tail test's actual size is ×1.703 of nominal with the normal critical value, ×0.991 with one Cornish–Fisher term and ×0.998 with two. At 10⁻⁸ they are ×464, ×1.418 and ×0.932.

At a hundred draws one term applied to the probability holds the upper tail within 10% to 3.09 standard deviations, and the same term applied to the critical value holds the test's size within 10% to 5.1. Two terms on the quantile reach 8.63; the normal reaches 1.66. The search stops at twelve.

The exact critical value falls from −1.195 at 10% to −3.013 at 10⁻¹⁰, and can never pass −3.162, minus the square root of ten. The one-term value is lowest, at −2.477, at a level of 1.05e-6, and rises again beyond it — a stricter level giving a laxer test. The two-term value falls more slowly than the exact one and turns back itself near 4 × 10⁻¹⁰.

Power at 5% against five departures. The straightness statistic detects the skewed source 83.3% of the time and light tails 30.8%; kurtosis detects light tails 72.5%; skewness detects light tails 0.2%. Rejecting when any of the five exceeds its own 5% point rejects 13.9% of normal samples; correcting that search to 5% gives it 37.4% power against light tails.

Power at 5%: straightness 30.8%, worst point 18.0%, outer eighths 49.0%, skewness 0.2%, kurtosis 72.5%. Any of the five at 5% each: 76.1%, at a false-alarm rate of 13.9%. The search corrected to 5%: 37.4%. A line-up of twenty read by a viewer running the same search: 38.3%.

Straightness alone rejects 5% of normal samples. Adding the worst point, the outer eighths, the skewness and the kurtosis, each at its own 5%, raises the rate to 5.0%, 6.8%, 11.1%, 12.3%, 13.9%. Five independent tests would reach 22.6%; these are correlated, so the search costs less than that and more than one test.

t limit: exceeded 8.51% at a mean margin of 0.523; Hall's transformation: exceeded 4.61% at a mean margin of 1.038; gamma family, shape known: exceeded 2.59% at a mean margin of 0.787; gamma family, shape by likelihood: exceeded 4.04% at a mean margin of 0.746; gamma family, shape by skewness: exceeded 17.51% at a mean margin of 0.421. The maximum-likelihood shape has a median of 1.10 and the skewness-based shape 3.20, against a true shape of 1.

The true shape is 1. The maximum-likelihood estimate has a median of 1.10; the estimate from the sample skewness, 4 over its square, has a median of 3.20 — it reads a shape several times too large, which is a distribution nearer the normal and a limit set too low.

A failure costs 20 standard deviations of margin. At the conventional 97.5% the t limit's loss is 2.143, Hall's 1.922 and the fitted family's 1.530. Each at its own best level: the t limit 1.263 at 99.95%, Hall's 1.862 at 98.50%, the family 1.308 at 99.25%. The best fixed multiple of the standard error costs 1.255.

For the t limit on 15 observations, charged twenty per standard deviation of shortfall plus its margin, the loss at each nominal level as an excess over the best level for each source. The best levels are 97% for the exponential, 96.25% for the gamma, shape 2, 97.75% for the gamma, shape ½, 98.25% for the lognormal. The minimax level, 97.25%, costs at most 0.8% on any of them: 0.1% on the exponential, 0.6% on the gamma, shape 2, 0.2% on the gamma, shape ½, 0.8% on the lognormal. The conventional 97.5% costs up to 1.0%.

t limit, charged by size: exponential 97%, gamma, shape 2 96.25%, gamma, shape ½ 97.75%, lognormal 98.25%; minimax 97.25%. fitted family, charged by size: exponential 92.25%, gamma, shape 2 93.5%, gamma, shape ½ 90%, lognormal 96.5%; minimax 93.5%. t limit, charged per failure: exponential 99.95%, gamma, shape 2 99.9%, gamma, shape ½ 99.99%, lognormal 99.995%; minimax 99.97%. fitted family, charged per failure: exponential 99.25%, gamma, shape 2 99.25%, gamma, shape ½ 98.75%, lognormal 99.9%; minimax 99.65%.

t limit, charged by size, 15 observations: minimax 97.25% costs at most 0.8%, 97.5% up to 1.0%; fitted family, charged by size, 15 observations: minimax 93.5% costs at most 3.4%, 97.5% up to 26.0%; t limit, charged per failure, 15 observations: minimax 99.97% costs at most 6.5%, 97.5% up to 119.9%; fitted family, charged per failure, 15 observations: minimax 99.65% costs at most 15.4%, 97.5% up to 71.8%; t limit, charged by size, 30 observations: minimax 97.25% costs at most 0.9%, 97.5% up to 1.3%; fitted family, charged by size, 30 observations: minimax 95% costs at most 4.1%, 97.5% up to 16.3%; t limit, charged per failure, 30 observations: minimax 99.99% costs at most 6.2%, 97.5% up to 178.2%; fitted family, charged per failure, 30 observations: minimax 99.85% costs at most 17.9%, 97.5% up to 157.0%.

exponential: t at 97.5% 2.075, at 99.9% 1.283, at its best level 1.269; Hall's at its best 1.838. gamma, shape 2: t at 97.5% 1.825, at 99.9% 1.156, at its best level 1.156; Hall's at its best 1.623. gamma, shape ½: t at 97.5% 3.002, at 99.9% 1.715, at its best level 1.540; Hall's at its best 2.283. lognormal: t at 97.5% 3.490, at 99.9% 1.908, at its best level 1.542; Hall's at its best 2.659.

For an exponential source the best level runs from 92.500%, 98.500%, 99.750%, 99.950%, 99.990%, 99.995% as a failure costs 2, 5, 10, 20, 50, 100 standard deviations of margin. It is higher for a more skewed source at every penalty but the lowest, and the conventional 97.5% is best only when a failure costs a few margins.

At a hundred future observations the band that holds all of them has factor 4.942; allowing one outside, 4.289; two, 3.948; five, 3.378. The tolerance factors for 99% and 95% content are 4.445 and 3.382.

From the sample's smallest and largest: none of the next 100 outside needs 3,850 observations, at most one 637, at most five 107 and at most ten 50. From the second smallest and second largest, 7,750, 1,204, 187 and 85.

The count is beta-binomial with mean 1.852. No miss at all has probability 0.266 against 0.154 for independent misses at the same rate; five or fewer, 0.951 against 0.989; more than eight, 0.007 against 0.000.

95% of the time, for any continuous population. At most 0: fixed 3,850, learning 3,850; At most 1: fixed 637, learning 512; At most 2: fixed 300, learning 197; At most 3: fixed 190, learning 101; At most 5: fixed 107, learning 36; At most 7: fixed 74, learning 15; At most 10: fixed 50, learning 4. With no flag allowed the two are the same, because a lot with no flag never moves the learning range.

Fixed at the start, the count is beta-binomial with mean 1.852; learning, it is a sum of independent draws with mean 1.315. No flag at all has probability 0.266 for both. Five or fewer: 0.9508 fixed, 0.9979 learning. More than five: 0.0492 against 0.0021.

Fixed at the start, every unit is flagged with probability 0.0185, and 0.0275 once another unit is known to have been flagged. Learning, the k-th unit is flagged with probability 2/(107 + k): 0.0185 for the first, 0.0097 for the hundredth and 0.0018 for the 1,000th, whatever happened before. Over 1,000 units the learning range flags 4.66 on average against 18.52 for the fixed range.

A range from 107 normal observations, a lot of a hundred, the mean stepping up by 2 standard deviations after unit 20; 4,000 lots. After the step the fixed range flags 25.03 units on average and the learning range 3.62. A unit in the last twenty is flagged in 97.8% of lots by the fixed range and 25.5% by the learning range.

Share of lots of a hundred in which more than five units are flagged: step 0, fixed 4.7%, learning 0.3%; step 0.25, fixed 7.3%, learning 0.4%; step 0.5, fixed 15.6%, learning 0.6%; step 0.75, fixed 30.6%, learning 1.1%; step 1, fixed 51.6%, learning 2.2%; step 1.5, fixed 84.6%, learning 7.9%; step 2, fixed 97.2%, learning 17.0%; step 2.5, fixed 99.5%, learning 26.1%; step 3, fixed 99.9%, learning 33.3%. Share with a flag among the last twenty units, at no step and at a step of 3: fixed 29.0% and 99.9%, learning 18.4% and 25.5%.

95% of the time, for any continuous population. At most 1: fixed 637, window 564, learning 512; At most 2: fixed 300, window 236, learning 197; At most 5: fixed 107, window 64, learning 36; At most 10: fixed 50, window 27, learning 4. The window counts are from 20,000 lots of uniform draws; the other two are exact.

4,000 normal lots. Flags per lot: range of the first 107, fixed 1.81, range of everything, from 36 2.62, range of the last 64 3.06, last units it accepted, from 141 2.00. A flag among the last twenty units in 29.3%, 28.2%, 47.3%, 38.9% of lots.

4,000 normal lots, the step after unit 20. Flags in the ten units after it: range of the first 107, fixed 3.13, range of everything, from 36 1.93, range of the last 64 1.89, last units it accepted, from 141 2.99. In all eighty after it: 25.05, 3.92, 4.20, 24.46. A flag among the last twenty in 97.5%, 24.8%, 45.0%, 97.8% of lots.

4,000 normal lots. Flags in the ten units after the return: fixed 0.18, learning 0.17, window 0.66; in the thirty after it, 0.55, 0.44 and 1.67. The fixed range's rate during the excursion runs off the top of the frame.

Forty rows, 3,000 samples a point, read against the band for forty independent normals. With the intercept alone the test flags 82.5%. With 1, 2, 5, 10 columns fixed in advance: 79.9%, 76.1%, 65.4%, 47.0%; with the same number picked by a forward search from twenty candidates unrelated to the outcome: 66.6%, 55.5%, 36.8%, 25.8%.

Lognormal errors with σ = 0.5. Straightness statistic of the fixed fit's residuals 0.1090, skewness 1.283; of the chosen fit's 0.0441 and 0.592. The band's 5% point for forty normals is 0.0552.

Forty rows, five columns chosen from twenty, 3,000 samples a row. Normal errors are flagged 6.2% against the band for forty independent normals, 5.9% against the fixed five-column design's band, 5.1% against the search repeated on normal errors; skewed errors 36.8%, 36.2%, 34.0%; light-tailed errors 8.5%, 7.5%, 7.0%. Five fixed columns flag the skewed errors 65.4% and the light-tailed 34.5%.

Fixed columns, 0 to 20: 83%, 79%, 76%, 73%, 69%, 65%, 62%, 58%, 55%, 52%, 48%, 45%, 42%, 38%, 35%, 33%, 29%, 27%, 25%, 23%, 20%. Chosen fits of 1, 2, 5, 10 columns flag 66.6%, 55.5%, 36.8%, 25.8%, the power of 4.6, 7.8, 13.4, 17.6 fixed columns.

The straightness test, each version read at its own 5% point. On all forty errors: 82.9%. At 2, 5, 10, 20 columns: ordinary residuals 79.1%, 71.4%, 51.2%, 23.0%; BLUS residuals 76.4%, 64.9%, 38.3%, 16.2%; recursive residuals 64.4%, 49.5%, 25.9%, 9.1%; 40 − p errors 80.6%, 76.1%, 69.1%, 49.5%.

Own-error share, averaged: ordinary residuals 0.500 over forty, BLUS residuals 0.498 over 20, recursive residuals 0.294 over 20. The least pure recursive residual carries 0.004 of its own error; the least pure ordinary one, 0.231.

The straightness test on recursive residuals, skewed errors, 3,000 samples each. one far point, as the rows came: 73.9%; one far point, shuffled: 67.3%; one far point, highest leverage first: 75.1%; one far point, highest leverage last: 50.1%; ten columns, as the rows came: 25.9%; ten columns, shuffled: 24.9%; ten columns, highest leverage first: 31.2%; ten columns, highest leverage last: 18.6%. With normal errors the same test rejects between 3.9% and 6.2% in every order.

a line, even x: 5.9%, 5.7%, 4.6%, 6.3%, 5.6%; a line, one far point: 6.2%, 5.6%, 4.0%, 6.4%, 5.6%; five columns: 4.7%, 4.3%, 4.0%, 5.2%, 4.8%; ten columns: 3.9%, 4.3%, 4.9%, 4.7%, 4.5%; twenty columns: 5.3%, 5.3%, 4.6%, 4.8%, 5.3%.

On 20,000 samples from each population the promise held 92.75%, 92.75%, 92.75%, 92.75% of the time — normal, exponential, uniform and Cauchy. The exact beta-binomial value is 92.84%.

At most 1 of 100: the range of 637 normal observations averages ±3.109; the normal band from 637 is ±2.934 and from ten ±4.289; with the population known, ±2.914. At most 5 of 100: the range of 107 normal observations averages ±2.530; the normal band from 107 is ±2.329 and from ten ±3.378; with the population known, ±2.220.

On the forty errors themselves the test flags 82.9%. On the residuals of fits with 2, 5, 10, 20 columns: 79.1%, 71.4%, 51.2%, 23.0%. On 40 − p errors, which is what losing the fitted columns' worth of observations alone would give: 80.6%, 76.1%, 69.1%, 49.5%.

The errors are lognormal with σ = 0.5. Their straightness statistic is 0.1259 and their skewness 1.161; the residuals of a 20-column least-squares fit to the same errors give 0.0259 and 0.276.

a line, even x: 5.8%, 5.4%, 4.5%, 5.4%, 5.3%; a line, lognormal x: 5.7%, 5.4%, 4.8%, 5.3%, 5.7%; a line, one far point: 6.0%, 6.1%, 4.8%, 5.8%, 5.2%; five columns: 5.9%, 5.7%, 4.5%, 5.1%, 5.7%; ten columns: 4.6%, 5.1%, 5.4%, 4.4%, 5.0%; twenty columns: 4.8%, 5.1%, 5.3%, 4.9%, 4.8%.

The factor is 3.290 at 20, 3.348 at 40, 3.378 at 100, 3.386 at 200, 3.390 at 400, 3.392 at 1000. The tolerance factor for 95% content with 95% confidence is 3.382.

At the factor 4.289, which allows one of the next hundred outside 95% of the time, the band misses at least once with probability 11.0% and, having missed once, misses again 45.4% of the time. Independent misses at the same average rate, 0.263 per hundred, would give 23.2% and 12.5%.

At a nominal 1% the upper-tail t test's actual size is ×0.153 with Student's quantile, ×2.94 with one Cornish–Fisher term and ×1.51 with Hall's cubic, both using the source's skewness, and ×0.878 with Hall's cubic using each sample's own. At 10⁻⁶ they are under a hundredth of nominal, ×1.04e+4, ×50.0 and ×1.59.

T's own upper critical value rises from 1.065 at 10% to 5.457 at 10⁻⁶. The one-term value is highest, at 1.080, at a level of 8.85e-3, and falls beyond it to −0.115 at 10⁻⁶ — a stricter test given a laxer threshold. Hall's cubic keeps rising, to 2.748; Student's quantile reaches 10.720.

Sorted by the sample's own skewness, from a median of 0.63 in the lowest tenth to 2.66 in the highest, the lower-tail test at 1% rejects ×6.83 to ×1.76 of nominal with Student's quantile, ×6.37 to ×0.48 with one term and ×6.14 to ×0.00 with Hall's cubic, both built from that skewness. Overall: ×3.98, ×2.73 and ×2.09.

At 1% on the lower side, Student's quantile rejects ×6.28 of nominal at ten draws, ×3.98 at thirty and ×1.95 at two hundred. Hall's cubic with the sample's own skewness: ×4.99, ×2.10, ×1.12. With the source's skewness known, the one-term value gives ×1.96 at thirty and Hall's cubic ×0.553.

At a failure rate of about 5.4%, the t limit's failures average 0.117 standard deviations, Hall's 0.125 and the fitted family's 0.119. As each limit is set stricter its failures become rarer and a little smaller, along nearly the same curve for all three.

Exponential source. 15 observations at penalty 20: the family +1.1%, Hall's +10.8%; 30 observations at penalty 20: the family −0.9%, Hall's +5.7%; 60 observations at penalty 20: the family −0.5%, Hall's +2.4%; 15 observations at penalty 50: the family +2.1%, Hall's +26.4%; 30 observations at penalty 50: the family −1.4%, Hall's +13.7%; 60 observations at penalty 50: the family −1.1%, Hall's +5.4%.

The saddlepoint density of the draw's mean and mean square, integrated over the samples a tilt can reach that put the t statistic past its exact critical value, over the exact tail those same samples carry, at 30 exponential draws. On the short side the relative error runs from +0.88% at 10% to +2.47% at 10⁻⁵; on the long side from +0.83% to +2.05% at 10⁻⁶. Nothing in the construction was fitted to these tails; its error stays within 2.5% of the truth at every level shown.