Anscombe's Quartet and Summary-Statistic Blindness

Imagine being handed only a spreadsheet of summary numbers for four different backtests — mean daily return, standard deviation, correlation to a benchmark, and a fitted regression line — and being asked to judge which strategy is "the same" as which. If all four summaries match almost exactly, the natural conclusion is that the underlying return series are essentially interchangeable. Anscombe's quartet exists to show that conclusion can be badly wrong.

In 1973, statistician Francis Anscombe constructed four small $(x,y)$ datasets, each with 11 points, that share almost identical mean of $x$, mean of $y$, variance of $x$, variance of $y$, correlation coefficient ($r \approx 0.816$), and best-fit regression line ($y \approx 3 + 0.5x$) — to two or three decimal places. Yet plotted side by side, the four datasets look nothing alike: the first is a genuine linear scatter with noise; the second is a smooth, clearly curved (non-linear) relationship that a straight regression line badly misdescribes; the third is a perfect straight line except for one point knocked far off, which alone determines the fitted correlation and slope; and the fourth is a vertical column of identical $x$-values plus a single outlier that manufactures the entire "correlation" from just one point.

The dataset was built by reverse-engineering points to hit the target statistics, and it makes the point precisely because every one of the four scenarios is common in real return data — genuine linear signal, true non-linearity misfit by a linear model, one influential outlier driving a regression, and one lever point creating a spurious correlation — yet a table of moments alone cannot tell them apart.