A 75% Kicker Has No 75% Inside Him. The Number Belongs to the Crowd, Not the Man.
The law made the picture. It never described a dot.
Kickers make about 80% of their 43-yard field goals. That number is solid, dependable, built from hundreds of kicks, and you can bank on it. Now put one man on the field with ten seconds left and ask what he will do. The 80% goes strangely quiet.
It goes quiet because it was never a fact about him. It is a fact about a crowd of kicks. He does not carry 80% around inside him like a battery charge; the crowd has that property and he does not. A law that describes many cases at once is nomothetic, from the Greek for law. The single case — this man, this wind, this Sunday — is idiographic, from the Greek for one’s own. Wilhelm Windelband named the pair in 1894, and psychologists have argued about it ever since.
The law of large numbers, which is the engine behind all those tidy percentages, is honest about this if you read it closely. It promises that the average settles down as the cases pile up. It promises nothing whatever about any single case. It cannot. A single kick is not an average of anything; it goes through or it does not.
On 6 January 2019 Cody Parkey lined up 43 yards out with ten seconds left and Chicago a point behind. He had made a 43-yarder that same season, fourth quarter, same one-point deficit, same holder, same long snapper. In January the ball came off a defender’s hand, hit the upright, hit the crossbar, and stayed out.
Two kicks, as alike as two kicks get. One green dot, one red. The 80% was true the whole time and it told you nothing about either one, because it was never that kind of number.
The reference class, and why the probability keeps moving.
Here is the machinery. Every probability you are ever quoted is really a conditional one: the chance of a thing given a stated set of circumstances. Choose a different set and you get a different number, and neither is wrong. Statisticians call the chosen set the reference class, and choosing it is the whole game.
Watch the number move as the class narrows. Across all field goals, kickers make 84.6%. Narrow to 43-yarders and it falls to 80.3%. Narrow again to the last two minutes of a one-score game — using the 40-to-49-yard band, to keep enough kicks to count — and it falls to 71.1% against 78.2% the rest of the time. Nothing about the kickers changed between those sentences. Only the class did.
Push the narrowing all the way and the trouble becomes obvious. What is the reference class for “43 yards, ten seconds, one point down, playoff game, iced by a timeout, and a rookie’s fingertip in the flight path”? There is exactly one member, and it has already happened. Narrow enough and every event is unique; stay broad enough to have data and you are no longer describing the case you care about. That tension has no clean solution. It is the reference class problem, and being able to name it is most of the defence.
Now the split your instincts have probably already found. A strict frequentist holds that a probability just is a long-run frequency in a repeatable class — so the question “what is the chance this kick goes in?” is not merely hard, it is malformed. There is no long run of that kick. A Bayesian treats probability as a degree of belief about a specific case, which can be stated for something that happens once, and which is supposed to move as evidence arrives. Ask both camps what the 80% means and you get an argument; ask them what to bet and they mostly agree.
So when a win probability lurches from 71% to 4% on one play, nothing has been proved wrong. The number was always “given everything known so far,” and a great deal just became known. A probability that never moved would not be more rigorous. It would only be deaf. See Bayesian Inference for how the moving is done properly.
Exchangeability, the reference class, and the single case.
The formal machinery — the weak and strong laws of large numbers and exactly what they do and do not promise, de Finetti’s exchangeability as the condition that licenses treating one case as a draw from a class, the reference class problem, and the frequentist and subjectivist readings of probability set side by side — is developed in The Sports Page’s companion statistics textbook, a free, open, graduate-level text with the R code to build these examples yourself. Read it free here.
- Every win probability graphic on every broadcast — a nomothetic number, quoted about one idiographic afternoon, and jumping about because the reference class narrows with every snap.
- The playoff prediction work — the best measure in baseball picks a series winner 57.8% of the time. A real law; nearly useless about your October.
- Base Rate (Concept No. 2) — the sibling: a base rate is a nomothetic fact, and forgetting it is the opposite error to the one described here.
- Bayesian Inference — the repair kit: how to move from the crowd’s number to a defensible belief about the one.
- Sample Size (Concept No. 6) — the constraint: how many cases you need before a nomothetic claim is worth making at all.