The Shape the Numbers Can't See

In 1973 the statistician Francis Anscombe published four small tables of numbers. How he made them is not recorded; he made them to carry a single, slightly mischievous lesson — and the lesson has outlived almost everything else he wrote.

Each of his four datasets has eleven points. Run the standard summary statistics on any of them and you get, to the precision he reported, the same answers: the average x is 9, the average y is 7.5, the variance of x is 11 in every set, R² is 0.667 as he printed it (a correlation of about 0.816, a figure he did not print), and the least-squares line of best fit is the same line in all four — ŷ = 3.00 + 0.50x. By every number a 1973 statistics package would print, the four are identical twins.

Then you plot them. Here they are, with their shared regression line drawn through each. Read the table of statistics underneath: the columns are the same. The pictures are not.

The summary statistics are blind to shape. They were never a description of the data — only a few of its moments, and a great many different clouds share the same moments.

II · What the line refuses to show

The four are not random near-misses. Anscombe engineered each to fail a different assumption that the regression line silently makes — so the quartet is really a checklist of the ways a single fitted line can be a lie.

I is the honest one: a roughly linear scatter with ordinary noise, the case the line was made for. II is a perfect downward parabola — the relationship is real and strong, but it is not a line, and the straight fit averages the curve into nonsense (fit a quadratic and the curvature it finds is some four times larger than in I). III is a tidy straight line with one outlier dragging the slope; delete that single point and the eleven become ten almost-collinear points whose true slope is gentler — the line you fitted belongs to no one. IV is the cruelest: ten points stacked at x = 8 and a single point far out at x = 19. The entire slope, the entire correlation, rests on that one high-leverage point. Move it and the line swings freely; the other ten say nothing at all about slope.

Turn on residuals above and the four diverge instantly: in I they scatter, in II they bow into a clean arch, in III one spike towers over a flat field, in IV they collapse onto two vertical stacks. Every one of those signatures is invisible in the table of summary numbers. That was Anscombe's whole point — and it is why he wanted the computer to draw the picture, not only print the digits.

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Take the point in your hand

Reading is one thing; the lesson lands harder when your own hand is on the data. Below is a live scatter you can drag. Every point is grabbable (or tab to one and nudge it with the arrow keys); the mean, the standard deviations, the correlation and the line of best fit are recomputed the instant you move anything. Load Anscombe's fourth set and do what the paragraph above only described — take the lone far point and move it. Watch the whole line swing from it while the other ten sit still and say nothing about the slope.

III · Forty-four years later, a dinosaur

How Anscombe made his quartet is not recorded. In 2017 Justin Matejka and George Fitzmaurice asked a sharper question: given a target picture, can you force a dataset into that shape while holding its summary statistics fixed? Their method is a small, stubborn loop — nudge a point a hair in a random direction, keep the nudge only if it leaves the mean, the standard deviations and the correlation essentially unchanged and moves the cloud a little closer to the target outline (early on, the annealing sometimes accepts a nudge away from it); repeat a few hundred thousand times. The statistics are the conserved quantity; the shape is free.

They started from a dinosaur, the Datasaurus, drawn by Alberto Cairo (he redrew it for them, since the 2016 points had not been kept), and drove it toward twelve target shapes. All thirteen carry the same summary statistics. Here they are. The numbers on the right are recomputed from the 142 points of whichever shape is showing. Press play, or pick a shape, and watch the picture change while the numbers hold still.

The statistics do not budge — to within a hundredth — across a dinosaur, a star, a circle, a bullseye, and nine other shapes. That is the same trick Anscombe pulled, however he pulled it, run by a machine to its logical extreme: the summary is so loose a description that you can hide a dinosaur inside it.

IV · The moral, and its modern shape

Anscombe's lesson is old and simple: plot your data. A handful of summary numbers — the things a regression spits out — are projections, and a projection throws away almost everything. Two datasets agree on their first two moments and their covariance and can still disagree about whether the relationship is a line, a curve, a fluke driven by one point, or nothing at all.

The modern shape of the lesson is less comforting. We have automated the summarising — dashboards, model metrics, a single correlation reported in a paper, a leaderboard score, a learned embedding's cosine — and the Datasaurus is the proof that any such summary has a dinosaur in its preimage: a dataset that scores identically and means something entirely different. The summary is necessary; it is never sufficient. Somewhere upstream, someone still has to look at the shape.

This is the venue's recurring confession, in a new register. The Bias in the Sum, The Migration and The Bias in the Sample each show a grouped statistic lying while every number stays honest. Those lies live in how the groups are handled — pooled, reclassified, selected. This one lives earlier, in the act of summarising at all: before any group is formed, the moments have already let the shape slip away.

V · Sources

Corrected 2026-09-28 (found by a scan of the site's pages for verdicts printed whatever the page computed): the caption under the quartet's table said "the rest match to the precision shown" as fixed text, while the table itself shows var(y) as 4.13, 4.13, 4.12, 4.12 and corr(x,y) as 0.816, 0.816, 0.816, 0.817. The caption is now worded from a row-by-row comparison of the printed values, and says which rows match and which only nearly match; the "identical exactly" line for mean(x) and var(x) is now printed only when all four sets compute to 9 and 11.

Corrected 2026-09-29 (claims pass): the page said that to the precision Anscombe reported the four sets share their variances and a correlation of 0.816. His printout (p. 19) gives no correlation and no variance of y; it gives R² = 0.667 and the sum of squares of x, 110. The correlation agrees to within 0.0003, which is 0.816 for three sets and 0.817 for the fourth, as the table shows; the page had called that three significant figures. It said three times that Anscombe built his quartet by hand; he calls them "four fictitious data sets" and says nothing of how he made them, and Matejka and Fitzmaurice write that "it is not known how Anscombe came up with his datasets." It called the Datasaurus their target; it was their starting shape, redrawn by Cairo for them, and the twelve others were the targets. Found by a claims pass on 2026-09-29.