411 Femora, March Twelfth

Weldon,

The thigh-bones from the Biometric Laboratory came back from measurement on the twelfth of March, and I want to tell you what happened to them, because the happening is the lesson. Four hundred and eleven femora, measured for antero-posterior diameter by the same assistant on the same calipers. I had expected a coefficient of correlation between left and right of roughly 0.95 - symmetry, after all, is what every anatomist asserts. The value came out 0.87. My first instinct was to look for error: a misread caliper, a transposed digit, a batch of bones measured after the assistant’s evening meal. I checked the arithmetic twice and found nothing.

Here is what I want you to hold onto. The null hypothesis - that the two sides vary independently - was never the interesting question. The interesting question was whether the deviation from my expectation was large enough to demand explanation, and for that I needed the sampling distribution of the correlation coefficient, not my anatomical intuitions. The standard error of r at n=411 is small enough that 0.87 against 0.95 is not a rounding matter. It is a real deficit, and it means the two sides of the body are not simply mirror images under measurement error; there is a genuine asymmetry of variation, and the bones were telling us so all along.

The point is this: nothing was discovered until the number was put on trial. The expectation of 0.95 was the null statement. The caliper readings were the evidence. The sampling distribution was the standard of proof, and without it the 0.87 would have been dismissed as dirt on the lens - or worse, accepted as dirt on the lens, which is the same mistake made in the opposite direction. Chance arrangements of facts are not discoveries. Chance arrangements of facts, brought before a defined standard of evidence, sometimes are.

A correction, since you will ask for one. I began this inquiry believing that the great enemy was the naturalist’s vague impression, the untested assertion. I now think the greater enemy is my own prior expectation, dressed in numbers. The 0.95 was as vague as any anatomist’s assertion; I had merely laundered it through arithmetic. The discipline is not calculation. The discipline is stating the null before you look, and being prepared to find that your own confident prediction was the thing under test.

Yours, K.P.