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

Q–Q plot — where it appears

A plot of a sample's sorted values against the values a reference distribution would put at the same positions. A straight line means the shapes agree; the departures from it are read by eye, and what counts as ordinary wandering has to be supplied from somewhere.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

Two 95% bands for a quantile plot of 40 points. 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.

The band the eye was standing in for

The confidence band software draws on a quantile plot holds each point at 95%, and a genuinely normal sample of forty has forty chances to leave it — so 45.0% of them do. The band a reader is actually using is that one widened by a factor of 1.502, and nothing draws it.

lineup · Qq What the plot says, and what the interval does, at n = 40. 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 plot is about the wrong quantity

A t interval needs the sampling distribution of the mean to be normal, not the data. A two-lump source leaves its quantile band on 100% of samples of forty and its interval covers 94.80%; a t on three degrees of freedom leaves it on 57% and covers 95.73%, the best of five sources.

lineup · Qq 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.

normal · Qq What a design does to the residuals of a correct model. Every residual has standard deviation sigma times the square root of one minus its leverage. On this design the leverages run from 0.045 to 0.663, so the residual spreads differ by a factor of 1.68 — and the model is exactly right. The high-leverage point's residual averages 0.46 of the fitted spread where a typical point's averages 0.79.

Residuals are not the errors

A residual's standard deviation is σ√(1 − hᵢᵢ), so a design whose leverages run from 0.045 to 0.663 produces residuals whose spreads differ by a factor of 1.68 with the model exactly right. On the samples where the high-leverage point really did have the largest error, a raw residual plot shows it as the largest on 0.0% of them.

lineup · Qq How often each one-number summary of a quantile plot detects five departures from normality, forty observations. 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.

A number for the shape

Five one-number summaries of a quantile plot's departure from its line, each a correct 5% test on forty normal observations, and no two agree about what matters. Straightness catches a skewed source 83% of the time and light tails 31%; kurtosis catches light tails 72% and skewness 0.2%. Reading all five and reporting whichever looks bad rejects 13.9% of genuinely normal samples. Correcting that search to 5% costs almost half the power against the departure it would have caught best — and a line-up of twenty panels gets the same correction for nothing.

lineup · Qq How often a quantile plot's shape test sees a skewed error distribution, on the errors and on residuals of fits of more columns. 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%.

Residuals that look too normal

The quantile plot most often read is of a regression's residuals, and residuals look more normal than the errors behind them. Each residual is a mix of all the errors, and the mix moves towards a bell as the fit takes more columns. On forty rows with skewed errors, the straightness test flags 82.9% of samples of the errors themselves, 79.1% of the residuals of a line, 71.4% with five columns, 51.2% with ten and 23.0% with twenty — far below the 49.5% that twenty errors would give, so the fit costs more than the observations it uses. Light tails vanish faster: 73.0% on the errors, 4.7% on twenty-column residuals. The band's level holds at every design, and one point with leverage 0.73 costs almost nothing; what costs is the number of columns.

lineup · Qq Whether residuals built to be independent see a skewed error better than ordinary ones. 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%.

Residuals made independent

Recursive residuals — each row predicted from the fit to the rows before it — are exactly independent normals under the null for any design, so a quantile plot of them has an exact band. They see a skewed error worse than the ordinary residuals do, not better: 25.9% against 51.2% at ten columns on forty rows, 9.1% against 23.0% at twenty. BLUS residuals, the independent set closest to the errors, reach 38.3% and 16.2%. Every such set is a rotation of the ordinary residuals, so none holds information they lack, and independence is bought by mixing some residuals almost entirely out of other rows' errors — the least pure recursive residual at twenty columns is 0.4% its own.

lineup · Qq How often a quantile plot sees a skewed error when the fitted columns were chosen from the data. 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%.

Residuals of a model the data chose

Five columns picked by a forward search from twenty candidates that have nothing to do with the outcome leave residuals that show a skewed error 36.8% of the time, where five columns fixed in advance show it 65.4% — the power of a fit of 13.4 fixed columns. Against light-tailed errors the chosen fit's plot is blind, 8.5% against 34.5%. Simulating the search on normal errors gives a band that holds 5%, which the ordinary band already nearly did; the level was never what broke. A reader told the final model has five columns is reading a plot with the power of thirteen.

lineup · Qq

Named alongside it

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

NormalityVisual inferenceModel diagnosticsStatistical powerLeverageResidualSampling variationHat matrixMultiple comparisonsResidual plotSkewnessCentral limit theorem

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