Residual Calculator

Four datasets, one regression summary

Anscombe published the quartet in 1973 to make a point that had to keep being made. All four presets here are his data, unaltered.

They agree on every number a regression report prints. Mean x is exactly 9 in all four and the sample variance of x is exactly 11. Mean y agrees to 7.50, the variance of y to 4.1, the correlation to 0.82, the slope to 0.500, the intercept to 3.00 and R² to 0.67. The residual standard error agrees to 1.24.

They describe four completely different situations. The first is a genuine linear relationship with ordinary scatter. The second is an exact parabola. The third is ten collinear points plus one outlier. The fourth has every x equal to 8 except one.

The agreement is close but not perfect, and the usual claim overstates it. “Identical to three decimal places” is repeated everywhere; measured, the four R² values span 0.000465 and the fourth correlation rounds to 0.817 rather than 0.816. Two decimal places is what actually holds, and the verification suite asserts the spread rather than a digit string.

The residual plot separates them immediately. Load each preset in turn: the first gives a structureless band, the second a clean parabola, the third a single point far from a tight line, the fourth a vertical stack and one isolated point sitting exactly on zero.

Which is the whole argument for plotting. No summary statistic distinguishes these four. One glance at the residuals distinguishes all of them, and it takes about a second.

The most broken dataset scores cleanest

The quartet is usually left there. There is a further result in it that is more uncomfortable, and it is about the diagnostics themselves rather than about R².

Anscombe IV has the smallest maximum standardised residual of the four. 1.5687, against 1.7779, 1.8607 and 2.9999 for the others. The usual rule — flag anything beyond two standard deviations — passes it clean, and flags the third dataset.

It also has the smallest finite Cook’s distance. 0.1367, against 0.4892, 0.8079 and 1.3928. On the standard influence diagnostic it looks like the best-behaved of the four.

It is the one where a single observation decides everything. Ten points sit at x = 8 and one sits at x = 19. That single point determines the entire slope; delete it and every remaining x is identical, so no line exists at all. The tool demonstrates this: remove it and the calculator returns nothing, because there is nothing to return.

The reason the diagnostics miss it is exact rather than approximate. Its leverage is 1.0000, which forces the fitted value to equal the observed value, which makes the residual 0.0000. Standardising divides by √(1 − h) and dividing by zero leaves the standardised residual and Cook’s distance undefined — not large, not small, undefined. Software that reports them as blank or as zero turns an alarm into a clean bill of health.

So leverage is not an advanced extra. It is the diagnostic that catches what residuals structurally cannot, it costs one line of arithmetic, and this tool prints it in the main table rather than behind a toggle for exactly that reason.

Why a residual is divided by s√(1 − h)

“More than two standard deviations” is the usual rule, and the question it skips is: two standard deviations of what?

Residuals do not all have the same variance. The variance of residual i is σ²(1 − hᵢ), so a point with high leverage has a genuinely smaller expected residual — not because it fits better, but because the line was pulled toward it.

Dividing every residual by the same s therefore understates the wrong points. It makes the observations with the most influence look the most obedient, which is the reverse of what a diagnostic should do.

The standardised residual corrects it exactly. eᵢ/(s√(1 − hᵢ)) has unit variance for every point, so a threshold of 2 means the same thing everywhere along the x axis.

The studentised residual goes one step further. It leaves point i out of the variance estimate, so a single extreme value cannot inflate the yardstick it is being measured against. That matters most on small samples, where one bad point can raise s enough to hide itself, and it is why the tool reports both columns rather than choosing.

The two agree until they matter. On well-behaved data the standardised and studentised columns are nearly identical. Where they diverge, the studentised one is larger and the point is masking itself — which is the case the whole construction exists for.

Leverage is decided before you look at y

This is the property that makes leverage a different kind of diagnostic from everything else on the page, and it is routinely missed.

hᵢ = 1/n + (xᵢ − x̄)²/Σ(x − x̄)² contains no y at all. It is a property of where you took your measurements, fixed the moment the design is chosen and unchanged by whatever those measurements turn out to be.

So a high-leverage point is not an outlier. It is a point with more say. It might fit the trend perfectly and still be the reason the trend has the slope it has.

The leverages always sum to exactly 2. Two being the number of parameters in a straight line. The total influence available is fixed, so one point taking more means every other point takes less — which is the cleanest way to see why a single extreme x is so consequential. The suite checks the sum across 400 random datasets.

Mean leverage is therefore always 2/n, and the usual flags are two or three times that. This tool uses 4/n, which is the more common convention in the influence literature and is the same as twice the mean.

The practical consequence is about design, not analysis. If you get to choose where to measure, spreading x reduces the standard error of the slope — but concentrating all the spread in one or two extreme points buys precision that rests entirely on those points being right. Balanced designs keep maximum leverage low on purpose.

What each residual pattern is telling you

A residual plot has a small vocabulary, and each pattern points at a specific defect with a specific fix.

A curve means the shape is wrong. Anscombe II is the pure case: the residuals trace a clean parabola — negative at both ends and positive across the middle, in the order −−+++++++−− — while the linear R² of 0.666 describes a relationship that is very nearly exactly quadratic. A quadratic fit there scores 0.999999.

A fan means the variance is not constant. The widening-spread preset fits well by R² and violates the assumption every standard error and p-value depends on. The coefficients stay unbiased; the confidence intervals become wrong, usually too narrow at the high end. A log transform of y or weighted least squares is the response.

A run of same-signed residuals means something is missing. Consecutive positives followed by consecutive negatives on time-ordered data is autocorrelation, and it inflates significance badly — effectively you have fewer independent observations than rows.

One point far from a tight band is an outlier in y. Anscombe III, where ten points are exactly collinear and one is not. Worth investigating as a data error, and worth reporting either way.

A vertical stack with one point off to the side is Anscombe IV, and it is the pattern the numbers will not tell you about. The residual plot shows it instantly, which is the argument for looking.

A structureless band is what correct looks like. Not a small band — the width is the residual standard error and can be large on genuinely noisy data — but a band with no shape, no trend and no fan.

Cook’s distance, and why influence is not permission to delete

Cook’s distance answers the question the other diagnostics circle: if this point were not here, how different would the answer be?

It is defined by deletion. Drop point i, refit, and measure how far every fitted value moved, scaled by the residual variance. The algebraic shortcut avoids n refits, and the verification suite checks the shortcut against an actual delete-and-refit at every point, so the number is the definition rather than an approximation of it.

It combines the two things that make a point matter. An unusual y, through the standardised residual, and an unusual x, through leverage. A point can have a huge residual and almost no influence if it sits at the centre of the x range, and almost no residual and total influence if it does not.

The threshold of 1 is a convention. So is 4/n, which flags more. Neither is a test, and neither has a null distribution behind it — they are prompts to look at the observation, and that is all they claim to be.

High influence is not evidence that a point is wrong. It is evidence that your conclusion depends on it. The honest responses are to check the record, to report the fit with and without it, or to collect more data near that x. Deleting an inconvenient point because a diagnostic flagged it is the diagnostic being used backwards.

And a clean Cook’s distance proves less than it appears to. Anscombe IV’s largest finite value is 0.1367, comfortably below every threshold, on a dataset that one observation controls completely. Influence measures answer “what changes if I remove this point”, and when removing the point destroys the model entirely they have nothing to compare against.

Sources and methodology

References for the diagnostics and for the dataset that motivates them.

• Anscombe, “Graphs in Statistical Analysis”, The American Statistician 27(1), 1973 — the quartet, exactly as the presets carry it
• Cook, “Detection of Influential Observation in Linear Regression”, Technometrics 19(1), 1977
• Hoaglin & Welsch, “The Hat Matrix in Regression and ANOVA”, The American Statistician 32(1), 1978 — leverage, and why it sums to the parameter count
• NIST/SEMATECH e-Handbook — residual analysis and what each plot pattern indicates
• R² Calculator — the statistic that cannot distinguish any two of Anscombe’s four
• How we calculate — methodology

Method. Leverage is the hat matrix diagonal, and the suite recomputes it from (X′X)⁻¹ built independently rather than from the same expression the engine uses. Cook’s distance is checked against its definition by deleting each point in turn, refitting, and measuring how far every fitted value moved — agreement below 10⁻⁹ at every point. Standardised residuals divide by s√(1 − h) and studentised residuals exclude the point from its own variance estimate, both left undefined rather than clamped where leverage reaches 1. The Anscombe values are the published ones, and the suite pins what actually agrees: mean x of exactly 9 and variance of exactly 11 in all four, slopes within 0.0004, R² within 0.0005 — two decimal places rather than the three usually claimed — and the fourth dataset’s maximum standardised residual of 1.5687 being the smallest of the four. That engine is verified on every change against 188 assertions. The count and the per-case breakdown are published on the formula verification page.

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