Galton's Ellipse

Galton studied regression to the mean in traits influenced by both heredity and chance. To hear what the data was telling him, Galton had to put it in a form more graphically revealing than a column of numbers. He later recalled, "I began with a sheet of paper, ruled crossways, with a scale across the top to refer to the statures of the sons, and another down the side for the statures of their fathers, and there also I had put a pencil mark at the spot appropriate to the stature of each son and to that of his father."

This method of visualizing the data is the spiritual descendant of René Descartes’s analytic geometry, which asks us to think about points in the plane as pairs of numbers, an x-coordinate and a y-coordinate, joining algebra and geometry in a tight clasp they’ve been locked in ever since.

Each father-son pair has an associated pair of numbers: namely, the height of the father followed by the height of the son. My father is six foot one and so am I—seventy-three inches each—so if we’d been in Galton’s data set we would have been recorded as (73,73). And Galton would have recorded our existence by making a mark on his sheet of paper with x-coordinate 73 and y-coordinate 73. Each parent and child in Galton’s voluminous records required another mark on the paper, until in the end his sheet bore a vast spray of dots, representing the whole range of variation in stature. Galton was using the type of graph we now call a scatterplot.

Scatterplots are spectacularly good at revealing the relationship between two variables; look in just about any contemporary scientific journal and you’ll see a raft of them. The late nineteenth century was a kind of golden age of data visualization. In 1869 Charles Minard made his famous chart showing the dwindling of Napoleon’s army on its path into Russia and its subsequent retreat, often called the greatest data graphic ever made; this, in turn, was a descendant of Florence Nightingale’s coxcomb graph showing in stark visual terms that most of the British soldiers lost in the Crimean War had been killed by infections, not Russians.

The coxcomb and scatterplot play to our cognitive strengths: our brains are sort of bad at looking at columns of numbers, but absolutely ace at locating patterns and information in a two-dimensional field of vision.

In some cases, that’s easy. For instance, suppose that every son and father had equal height, the way my dad and I do. This represents a situation where chance plays no role at all, and your stature is completely determined by your patrimony. Then all the points in our scatterplot would have x and y coordinates equal; in other words, they’d be stuck to the diagonal line whose equation is x = y.

Note that the density of the dots is greater near the middle and less near the extremes; more men are five feet nine inches tall (sixty-nine inches) than are six foot one or five foot four.

Now what about the opposite extreme, where the heights of fathers and sons are totally independent? If the two height distributions also had similar spreads, the scatterplot would look like a roughly circular cloud with no bias toward the diagonal. If you restrict your attention to sons whose fathers were six foot one (seventy-three inches), corresponding to a vertical slice in the right half of the scatterplot, the dots measuring the height of the sons are still centered on five foot nine. We say that the conditional expectation of the son’s height is the same as the unconditional expectation. This is what Galton’s sheet of paper would have looked like if there were no heritable differences at all affecting height. It’s regression to the mean in its most intense form; the sons of tall fathers regress all the way to the mean, ending up no taller than the sons of shorties.

But Galton’s scatterplot didn’t look like either of those two extreme cases. Instead, it was intermediate between them. The average son of a six-foot-one-inch father is shorter than the father on average but still taller than the average man. You are looking at a picture of regression to the mean.

Galton noticed very quickly that his scatterplots, generated by the interplay of heredity and chance, had a geometric structure that was anything but random. They seemed to be enclosed, more or less, by an ellipse, centered on the point where both parents and child were of exactly average height. The tilted elliptical shape of the data is quite clear even in the raw data in the table reproduced in Galton’s 1886 paper “Regression Towards Mediocrity in Hereditary Stature.”

In fact, Galton did more—he carefully drew curves on his scatterplot along which the density of points was roughly constant. Curves of this kind are called isopleths. Galton found an amazing regularity: his isopleths were all ellipses, one contained within the next, each one with the same center. It was like the contour map of a perfectly elliptical mountain, with its peak at the pair of heights most frequently observed in Galton’s sample: average height for both parents and children. The mountain is none other than the three-dimensional version of the bivariate normal distribution.

When parent and child heights are independent and have similar spreads, these contours are circles. When the heights have a perfect linear relationship, the dots lie along a straight line. In between, the ellipses have different orientations and degrees of skinniness. Geometers call an ellipse’s departure from circularity its eccentricity. On comparable scales, the orientation and elongation of the cloud show the strength and direction of the association measured by correlation. A nearly round cloud corresponds to correlation near 0; a skinny cloud along the northeast-southwest axis corresponds to correlation close to 1. A cloud sloping the other way indicates a negative association. The comparison measures association, rather than establishing how much of an individual’s height is caused by heredity.

A proper skeptical attitude now requires you to ask: What if your scatterplot doesn’t look like an ellipse? There’s a pragmatic answer: in practice, the scatterplots of real-life data sets often do array themselves in rough ellipses. Not always, but often enough to make the technique widely applicable. Examples include: the share of voters who voted for John Kerry in 2004 against the share Barack Obama got in 2008 (very skinny ellipse, high correlation, floats above the diagonal); daily stock price changes for Google and GE; average SAT score plotted against tuition for North Carolina colleges; U.S. states arranged by average income and George W. Bush’s share of the 2004 presidential vote (negative correlation). These data sets come from very different sources, but all four scatterplots arrange themselves in the same vaguely elliptical shape that the heights of parents and children did.