The band the eye was standing in for

Twenty panels of genuinely normal data answer “compared with what” by example. The alternative is to compute the comparison and draw it as a band, which is what statistical software does, and the band it draws is not the one a reader is using.

Two bands, two meanings #

The pointwise band is exact and easy. The i-th smallest of n draws from a uniform is Beta(i, n + 1 − i), so its 2.5% and 97.5% points map through $\Phi^{-1}$ to limits for the i-th point of a normal quantile plot. Each point leaves its own limits 5% of the time, and the count confirms it: averaged over the interior points, 5.07% across four thousand genuinely normal samples of forty.

That is a correct band and it answers a question nobody asks. Nobody looks at the seventeenth order statistic and wonders whether it alone is where it should be. What a reader does is scan the whole plot for anything outside, which is a simultaneous question, and the pointwise band is the wrong band for it by exactly the amount a multiplicity costs.

A sample of forty has forty chances to step outside. 45.0% of genuinely normal samples take one.

The factor #

The simultaneous band is the pointwise band widened until only 5% of samples leave it anywhere. There is no closed form; the factor is found by counting each sample’s worst departure in units of its own point’s half-band and taking the 95th percentile of those.

At forty observations it is 1.502.

Two readings, and the second is the one that matters for practice.

The exit rate rises with the sample. At a hundred and sixty observations, three genuinely normal samples in five produce a quantile plot with a point outside the drawn band. A reader who treats an excursion as evidence will find evidence more often the more data they have.

The factor rises too, and slowly. From 1.37 at ten observations to 1.61 at a hundred and sixty — it grows like the multiplicity does, which is to say logarithmically, so the correct band is roughly half again as wide across the whole range anybody plots.

That slowness is the useful part. A rule of thumb of “one and a half times the drawn band” is approximately right from ten observations to a hundred and sixty, and applying it costs nothing but the willingness to ignore a point that is only slightly outside.

The same arithmetic as everywhere else #

This is a multiplicity problem and it has all the usual features.

The pointwise band is the uncorrected test, applied n times. The 45% exit rate is the family-wise error rate of forty tests at 5% each, reduced from the 87% independence would give because neighbouring order statistics are strongly correlated — the same correlation that makes a family worth fewer tests than it numbers.

The factor 1.502 is the correction, found by exactly the method that field recommends: build the null distribution of the largest departure and read the threshold off it. No formula is used and none is needed, because the correlation structure of order statistics is complicated and a count carries it exactly.

And the error runs in the familiar direction. An uncorrected multiplicity finds things that are not there, which here means rejecting normality on data that are normal — nine times in twenty at forty observations.

What the band is a band for #

There is a second thing the two bands differ on, and it is easy to miss because both are drawn the same way: what they are a band for.

A pointwise band at the i-th position says where that order statistic would be if the data were normal. It is a statement about a quantile of the sample, and it is exact.

What a reader wants to know is whether the sample came from a normal distribution — a statement about the whole sample. The simultaneous band answers a version of that: 95% of normal samples lie entirely inside it. It does not answer the converse, and the converse is what is usually wanted. A sample that stays inside the band is not thereby normal, because plenty of non-normal distributions produce samples that stay inside a band this wide at forty observations.

That asymmetry is worth stating because it is where a band is most often over-read. Staying inside is not evidence of normality; leaving is weak evidence against it. The band is a one-sided instrument, like every other test, and at the sample sizes where normality is being checked a test has very little power against the departures that matter.

Standardising the sample, and what it does to the band #

One detail of the construction is easy to get wrong and it changes the answer.

The bands above are for a sample from a standard normal — mean 0, spread 1. A real sample has neither, so the plot is of standardised values (x − x̄)/s, and standardising uses two quantities estimated from the same data.

That makes the points less variable than the band allows for, because the standardisation has already removed the sample’s own location and scale — the plotted values are constrained to have mean 0 and spread 1 exactly, so they cannot all wander in the same direction. A band built for unstandardised draws is therefore slightly too wide for a standardised plot, and the simultaneous factor computed here inherits that.

The direction is fortunate and the size is not negligible at small n: at ten observations the constraint removes two degrees of freedom from ten, and at a hundred and sixty it removes two from a hundred and sixty. The measurement on this page uses unstandardised draws throughout, so the exit rates are honest for the question posed and slightly conservative for the plot a reader is looking at.

Being explicit about that is the point rather than a caveat. A band’s construction has to name what it is a band for, and the two constructions — unstandardised draws against a known normal, and standardised residuals against an estimated one — give different bands that software rarely distinguishes.

What the band would have to be to answer the real question #

Both bands are computed under the hypothesis that the data are normal, so both answer a question of the form how unusual is this plot if the model is right. That is a test, and it inherits every property a test has.

In particular it inherits the property that a threshold is not the interesting output. A reader deciding whether to use a t interval does not want to know whether normality is rejected; they want to know whether the departure present is large enough to matter for the interval. Those are different questions and the second one has a different answer at every sample size — a departure that is detectable at four hundred observations is usually irrelevant to the interval there, and one that matters at ten is undetectable.

The band could be drawn for the second question instead. It would be a band containing the quantile plots of every distribution whose t interval covers between 94% and 96% at this sample size, and a plot outside it would be a plot whose interval does not work. That band is computable — the set of distributions is large but it can be sampled — and it is not what anyone draws.

Its shape would be different in an interesting way. It would be narrow in the middle and very wide at the ends, because the interval’s coverage is insensitive to the extreme tails at moderate n and sensitive to skew in the body, which is the opposite of the normality band’s shape.

Where the excursions happen #

The exit rate is not spread evenly along the plot, and knowing where it concentrates changes how a plot should be read.

The pointwise band is widest at the ends, because the extreme order statistics are the most variable. But it is proportionally least protective there too: the ends are where a normal sample’s points wander furthest, and the ends are what catches a reader’s eye. A single point past the end of the band is the commonest excursion and the least informative one.

The middle is the opposite. A run of points bowing consistently away from the line in the middle of the plot is a much stronger signal than one extreme point outside the band — because the middle points are tightly constrained and because a run is a joint departure that a per-point band was never going to capture.

So the reading rule the band suggests is the wrong one. A quantile plot is read by shape rather than by excursion, and the shape a reader should be looking for is a consistent bow, a kink or an S, none of which is what a per-point band tests for. The twenty-panel device trains exactly that, which is why it remains better than a band despite being less precise.

What a simultaneous band costs #

Widening by 50% is not free, and it is worth saying what is given up.

Power against a single outlier. One observation genuinely from a different distribution has to be half again as far out to be flagged. That is the price of not flagging the nine-in-twenty normal samples, and it is the ordinary multiplicity trade.

Nothing at all against a shape departure. A band of either width is a per-point device, and a systematic bow is a departure that keeps every point inside while being obviously wrong. Widening does not make that worse because the narrow band was not detecting it either.

So the honest summary of the trade is narrow. The simultaneous band is the right correction for the question the pointwise band was answering badly; neither band answers the question a reader has, which is about shape; and the device that does answer it is a line-up.

Reading a plot with neither band #

Most quantile plots are drawn without any band at all, and the practical question is what to do with one of those. The measurements above give three rules and they are short.

Ignore a single point past the end. At forty observations the extreme points are the most variable in the plot, and a lone excursion at either end is the single commonest thing a normal sample does. Requiring two or more consecutive points outside removes nearly all of the false alarms and costs very little against a real heavy tail, which produces several.

Look for a run of the same sign in the middle. The interior order statistics are tightly constrained and highly correlated, so a systematic departure there is both unlikely under normality and hard to produce by chance in one point.

Read the shape and name it. A bow one way is skew; a bow both ways is heavy or light tails; a kink is a mixture; a staircase is discreteness. Each of those is a description the plot supports and a band does not test for, and naming which one it is does more for a reader than a verdict on whether it is normal.

These are worth stating because the alternative reading — count the points outside — is the reading a band invites, and is the one the 45% makes unusable.

What is claimed here, and what is not #

Three statements, and the first is the calibration for the other two.

Each point leaves its own band about 5% of the time, over the interior points and to within a percentage point. That is what says the pointwise band is the band it claims to be — one drawn from the wrong distribution would miss here, and every comparison on this page would then be a comparison with something else.

The widened band is left by 5% of samples, to within 1.2 percentage points. The factor is found by counting and then confirmed by counting, which is not circular: it is found as a quantile of one statistic and confirmed as an exceedance rate, and taking the wrong quantile would satisfy the first and not the second.

The widening is substantial, a factor above 1.2. A factor near 1 would mean the multiplicity had somehow vanished, and the whole page would be about nothing.

The reading that does not survive is the pointwise band taken as the band a sample should stay inside. The standard a 95% band is held to is that 95% of genuinely normal samples stay inside it, and 55.0% do. That is not a band that is slightly too narrow; it is a band answering a different question, drawn where the answer to this one belongs.

The same correction, in the other diagnostic plots #

A quantile plot is not the only per-point display a reader scans for excursions, and the arithmetic transfers to every one of them without modification.

A residual plot against fitted values. Draw a band at ±2 residual standard errors and a correct model of forty points puts something outside it about the same 45% of the time, for the same reason. The convention of ±2 is a pointwise 95% band with no correction in it at all.

A control chart. Three-sigma limits on a process are explicitly a multiplicity device: three sigma rather than two is a correction for the number of points a chart will show, and the false-alarm rate of one point in 370 is chosen so that a chart with a point per hour raises about one a fortnight. That is the same calculation done properly, in the one field that does it.

A scale-location plot, a leverage plot, a partial residual plot. All per-point, all scanned for excursions, none with any correction.

The control chart is the interesting case because it is the counterexample. That field decided its multiplicity explicitly, chose a limit from it, and states the resulting false-alarm rate as a design parameter — and the rest of diagnostic plotting draws the uncorrected version and reads it as if it were corrected. The difference is not sophistication; it is that a control chart’s false alarms cost money and a diagnostic plot’s cost an unnecessary transformation.

Still open: a band for the shape rather than the points #

Everything above is a per-point device corrected for multiplicity. The reading that matters is about shape, and a band is a poor instrument for it in a way no widening repairs.

The pieces of something better exist. A departure from the line can be summarised by a functional — the integrated squared distance, the maximum of a smoothed deviation, the correlation between the deviation and the quantile — and each of those has a null distribution that can be built by the same counting used here. A plot could then carry a single number saying how unusual its shape is, with an honest reference distribution.

What is not obvious is which functional, and the choice is the whole of the problem: each one is powerful against some departures and blind to others, and a reader looking at a plot is running an unspecified search over all of them. That is the garden of forking paths in visual form, and a number computed from one nominated functional would understate the multiplicity of the eye’s own search by an amount nobody has measured.

The line-up sidesteps that, and it is worth seeing why. A reader shown twenty panels and asked which one is the data is running the same unspecified search on all twenty, so whatever functional their eye is using is applied uniformly — and the probability of picking the real one by chance is exactly 1 in 20 regardless of what that functional is. The device is distribution-free in the reader, which is a strange and rather beautiful property, and it is the reason a procedure that looks less rigorous than a computed band has a rigour a band cannot get.