The convergence that runs just as well backwards

Dark navy slide, masthead “The Stature Register · 205 families · 928 adult children”, headlined “The extremes are being bred back to the middle”. A subhead reads “Every mid-parent height class in the 1886 hereditary-stature survey, joined to the median height of its own adult children. One shared scale in inches, every class shown, all 928 children counted.” A slope chart has two vertical columns: mid-parent height on the left, the median height of their adult children on the right. Nine gold and blue lines run between them: 72.5 to 72.2 (19 children), 71.5 to 69.9 (43), 70.5 to 69.5 (68), 69.5 to 68.9 (183), 68.5 to 68.2 (219), 67.5 to 67.6 (211), 66.5 to 67.2 (78), 65.5 to 66.7 (66) and 64.5 to 65.8 (23). Gold lines mark families above the middle and blue ones families below; a dashed rule crosses the chart at the level of mediocrity, 68¼ inches, and every line leans toward it. Three stat cards down the right side read −1.6 in given back in one generation by the 71½-inch families, 8.0 → 6.4 in span of the nine height classes parents then children, and ⅔ of its parents’ distance from the middle is all a generation keeps. A gold note box adds “Run the two-thirds rule forward. A family sitting 4¼ inches above the middle today is 0.6 inches above it in five generations. The tall are not being replaced — they are being averaged away.” A footnote credits Francis Galton, “Regression towards Mediocrity in Hereditary Stature”, Journal of the Anthropological Institute 15 (1886), Table I, row medians exactly as published, all female heights multiplied by 1.08. — Showing the misleading chart

The same data on a cream background, kicked “The same table · read the other way” and headlined “The convergence runs just as well backwards”. A subhead reads “Galton’s Table I has margins in both directions, and both of them point inward. Group the families by the parents and the children sit closer to the middle. Group the same 928 children by their own height and it is the parents who sit closer to the middle.” Two slope charts sit side by side on the same inch scale, each with a dashed rule at the level of mediocrity, 68¼ inches. The left panel, “Group by the parents → forwards in time”, runs nine mid-parent classes from 72.5 down to 64.5 across to their children’s medians, 72.2 down to 65.8; brackets mark a span of 8.0 inches on the left and 6.4 on the right. Its note reads “Nine mid-parent classes, 8.0 inches apart end to end. Their children’s medians cover 6.4 — 80% of the spread. Galton’s law: the child keeps ⅔ of the parents’ deviation.” The right panel, “Group by the children ← backwards in time”, runs ten adult-child classes from 73.2 down to 64.2 across to their mid-parents’ medians, 70.0 down to 66.3, which bunch tightly around the dashed rule; brackets mark a span of 9.0 inches on the left and 3.7 on the right. Its note reads “Ten child classes, 9.0 inches apart. Their parents’ medians cover 3.7 — 41% of the spread. Galton read his own table this way too: the mid-parentage keeps ⅓.” A card headed “Why it shrinks whichever way you look” explains that a measured height is a lasting family tendency plus a bit of luck — which two parents happened to marry, how the growth years went, whether the shoes came off — so sorting a column by its total sorts partly on the luck, and luck does not appear in the other column, which is why ⅔ forwards and ⅓ backwards come out of one table; if either number were a trend, the population would be closing on the middle in both directions of time at once. A second card, “The same slide, fifty years later”, recounts Horace Secrist’s The Triumph of Mediocrity in Business (1933) and quotes Harold Hotelling’s review in JASA: “the seeming convergence is a statistical fallacy, resulting from the method of grouping” — the diagrams “really prove nothing more than that the ratios in question have a tendency to wander about.” A footnote credits Table I of Galton 1886, notes the ⅔ and ⅓ are Galton’s own figures and that least squares through his nine row medians gives 0.69 and through his ten column medians 0.30, cites Hotelling’s review, and says both drawings are our own. — Showing the misleading chart

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A slide built from Galton’s 1886 stature table: sort 205 families by the parents’ height and every class of children sits closer to the average than its parents did. Read as a trend, the extremes are vanishing. The same table has margins in the other direction too — sort the same 928 children by their own height and it is the parents who sit closer to the average.

Francis Galton, “Regression towards Mediocrity in Hereditary Stature”, Journal of the Anthropological Institute of Great Britain and Ireland 15 (1886): 246–263, Table I — 928 adult children of 205 mid-parents, 1886Added August 8, 2026

01The claim

The extremes are being bred back to the middle. Take every mid-parent height class in the 1886 hereditary-stature survey — 205 families, 928 grown children, one shared scale in inches — and join each class to the median height of its own children. Every line leans inward. The 71½-inch families gave back 1.6 inches in a single generation; the 64½-inch families climbed 1.3. Nine classes that spanned 8.0 inches produced children whose medians span 6.4. Galton’s own law does the arithmetic: a generation keeps two-thirds of its parents’ distance from the middle, so a family 4¼ inches above it today is 0.6 inches above it five generations from now. The tall are not being replaced — they are being averaged away.

02The trick

Nothing on the slide is wrong. The nine pairs are the row medians printed in Galton’s Table I, the axis is a single honest scale in inches, no class has been dropped, and the two-thirds rule really is Galton’s. The mistake is one word: the slide reads the inward lean as something happening to the population over time. It isn’t. It is a property of how the nine groups were picked. A mid-parent’s measured height is a lasting family tendency plus a bit of luck — which two parents happened to marry, how the growth years went, whether the shoes came off, which tenth of an inch the tape was read to. Sort the column by its total and you sort partly on that luck, so the 72½-inch class is, on average, a little luckier than it is tall. The luck does not travel to the next column, and the children come out closer to the middle. The proof that this is arithmetic rather than heredity is printed in the very same table, one row lower. Galton gave his columns medians as well as his rows, and he says outright that the table “read in a different way, namely, in vertical columns instead of in horizontal lines” shows that the most probable mid-parentage of a man deviates only one-third as much as the man does. Group by the children and their parents are the ones bunched around the average: ten child classes spanning 9.0 inches came from mid-parents whose medians span 3.7, a tighter squeeze than the forward direction manages. If the inward lean were a trend, this population would be closing on the middle in both directions of time at once. (This exhibit is our own demonstration, drawn in the house style of a modern comparison slide; Galton published the table and his own diagrams, not this.)

03The fix

Read the table both ways and the drama becomes a definition. Forwards, the child keeps ⅔ of the mid-parent’s deviation from 68¼ inches; backwards, the mid-parentage keeps ⅓ of the child’s — Galton’s own two figures, and a least-squares line through his nine row medians gives 0.69 while one through his ten column medians gives 0.30. Both point inward because both groups were picked at their extremes, and a group picked that way always sits further out than the group it is compared against. Galton asked the question this slide answers wrongly, and answered it himself a few pages later: how is it that each generation still contains the same number of men per thousand between any two heights you care to name, when the tall are so rarely descended from the equally tall? Because the process holds two opposite actions in balance, one concentrative and one dispersive — regression pulling families in, the scatter of children around their own family’s centre pushing back out — “like a spring against a weight”, in stable equilibrium. His Table IV is headed “Process through which the Distribution of Statures, in Successive Generations of the same People, remains UNCHANGED”. Nothing is closing in. The practical test is the one the table supplies for free — regroup on the second measurement and look at the first. If it converges in that direction too, nothing is moving, and you are looking at your own selection rule. Fifty years later Horace Secrist missed that test across two hundred charts in The Triumph of Mediocrity in Business (1933), sorting firms by their 1920 results and following them to 1930 as they all drifted average-ward; Harold Hotelling’s review in the Journal of the American Statistical Association named it in one line — “the seeming convergence is a statistical fallacy, resulting from the method of grouping” — and noted the diagrams “really prove nothing more than that the ratios in question have a tendency to wander about.” The same line retires a great many charts about the worst-performing hospitals improving, the top funds falling back, and the rookie who cooled off in year two.