Two teams, each with its own average. The coach moves one member from the first team to the second. Surely one team has to get weaker. Lock in your answer, then check it with a game, a hospital simulation and a few explanations.
Move someone who is below their own team's average but above the other team's, and both averages go up. The overall average does not change: nobody got better, only the boundary between the groups moved. The same thing "improves" cancer survival when sharper scans move patients from an early stage to a late one.
The name comes from a joke credited to the American humorist Will Rogers: when the Okies left Oklahoma and moved to California, they raised the average intelligence in both states. Whether he ever said it is unknown; more on that in the facts. First, the puzzle.
At a school contest nine players scored from 1 to 9 points, one score each. The strong team A: 5, 6, 7, 8, 9. The weak team B: 1, 2, 3, 4.
Team A's average score is 35 / 5 = 7, team B's is 10 / 4 = 2.5. All nine together average 45 / 9 = 5.
The coach may move exactly one player from team A to team B. Their score stays the same.
Can one player be moved so that the average score rises in both teams? If so, whom?
"Whatever one team gains, the other loses. Both cannot rise"
That would be true of the total score. But an average also depends on the head count. The player with 5 points is the weakest in team A, so A is stronger without them. And they are stronger than everyone in team B, so B is stronger with them too.
Team A: 5, 6, 7, 8, 9 (average 7). Team B: 1, 2, 3, 4 (average 2.5). After one player moves from A to B, the average score must rise in both teams.
Pick an answer. This is where you will see what happens to both averages after each of the five possible moves.
Hard to decide? Open the hints one at a time: the easy one first, then stronger ones.
An average depends not only on the total score but also on the head count. What happens to a team's average when someone below that average leaves?
Compare each player of team A with A's average (7) and with B's average (2.5). Which of them was pulling their own team down?
A newcomer raises team B's average if they are above that average. Is there anyone in team A who is both below 7 and above 2.5?
Tap a player to move them to the other team. The goal: as many moves in a row as you can, each one raising the average of both teams. Keep an eye on the overall average.
Hint: look at where the player sits on the line relative to the two average ticks.
Here is where the trick happens for real. A hospital splits cancer into an early stage (no metastases found) and a late stage (metastases found). Each patient has a hidden measure of how far the disease has spread, from 0 to 100. The higher it is, the lower the chance of living 5 years: from 100% at 0 down to 0% at 100. The scanner only sees metastases once the spread reaches a threshold. The old scanner sees them from 60, the new one already from 40. The treatment is the same, and so are the patients.
Each dot is a patient (400 of 2,000 shown). A filled dot lived 5 years, a hollow one did not. The vertical line marks the spread from which the scanner sees metastases.
A group's average describes not only its people but also where the line between the groups is drawn. Move the line and the averages change, even if no single person changed.
“In plain words” is hidden on all puzzles.
The 5-point player was dragging team A down: they are below its average. When they leave, A gets stronger. In team B they are the best: above its average. When they arrive, B gets stronger too. That is enough to see the trick. Why nobody actually improved is the next card.
The same nine people, the same scores, the same sum. Only who sits in which team changed. Both teams "got stronger" because the strong team got smaller and even stronger, and the weak one got bigger. The weak team's weight in the overall average grew by exactly enough to cancel both gains.
The new scanner moves patients with spread from 40 to 60 into the late stage. They live 5 years 50% of the time: worse than the rest of the early stage (70%), but better than the late stage (20%). They are the "5-point players". Both stages look better, though no patient lived a single day longer.
We are used to reading "the group's average went up" as "the people in the group got better". But an average depends on who is in the group. When the membership changes, the average changes without anyone improving. So whenever a group's average goes up, ask: did the rule for who counts as a member change?
“Rigorous” is hidden on all puzzles.
Group A has nA people and average a, group B has nB people and average b. A person with score x moves from A to B. This is how the averages change.
Both differences are positive exactly when the score lies between the two averages.
The overall average is the weighted average of the two groups. The sum of scores and the head count do not change, so neither does it.
Both averages rose by 0.5 and the overall one did not move, because group B's weight grew from 4/9 to 5/9. Group B is still weak; it is just bigger now. If x equals one of the averages, that average does not change, so the condition is strict.
Across is the score of the player moved from A to B. Line A crosses zero at 7 (A's average), line B at 2.5 (B's average). Between them both lines are above zero. In team A the players with 5 and 6 points fall there.
| Old scanner (60) | New scanner (40) | |
|---|---|---|
| Early stage | 70% | 80% |
| Late stage | 20% | 30% |
| All patients | 50% | 50% |
| Share in the late stage | 40% | 60% |
Spread is uniform from 0 to 100, the chance of living 5 years is 1 − spread/100. If the scanner sees metastases from a threshold t, the early stage survives on average 100 − t/2 percent, the late stage 50 − t/2, and everyone together always 50. Every step the threshold drops lifts both stages by half a step. Patients with spread 40–60 (20% of all) survive 50% of the time.
“Deeper” is hidden on all puzzles.
In 1985 Alvan Feinstein and two co-authors published "The Will Rogers phenomenon" in the New England Journal of Medicine. They compared lung cancer patients first treated in 1977 with patients from 1953–1964 at the same institutions. The newer group had higher six-month survival in every TNM stage.
The newer group had gone through many new imaging tests, which found metastases that used to be silent. Some patients who would once have been put in a "good" stage were now put in a "bad" one. When the authors split both groups by symptom stages, which the new tests could not change, survival came out similar.
Comparing survival by stage across decades, countries or hospitals is risky when the diagnostics differ. There are three honest ways out. Compare all patients together, like the "all" row of the table. Re-stage the old patients by the new rules, so the line is drawn in the same place. Or split by a feature the new technology does not change, as Feinstein did with symptom stages.
This trap differs from another one doctors know well: early detection lengthens survival "from diagnosis" because the clock simply starts earlier. Here nobody even has to live longer or be found earlier: moving the line between the stages is enough.
First the maths (how many times in a row both averages can rise), then the logic (what really changes when a boundary moves), then practice (how the trick gets into analytics reports).
The same holds for a whole group: move several people at once and both averages rise when the movers' average lies between the two group averages. That is how the "Okies" of the joke could raise the average of both states, even if some of them were bright and some were not.
That no chain is longer than n − 2 was checked by brute force over every split for n from 3 to 11.
Every split into groups has a boundary: a cancer stage, a "top student", "tall", an "active user", a "premium customer". Move the boundary so the borderline cases cross it, and the average can rise on both sides. There is no real improvement: the same people, the same results.
The rule against it is simple: when the averages of all the parts go up, check the whole. If the overall average stays put, only the labelling changed. A related group trap is Simpson's paradox, where the parts and the whole point in opposite directions because the groups carry different weights.
Better diagnostics. More sensitive tests find milder forms of a disease. The mild cases join the sick and improve their numbers, while the healthy look healthier without them.
Ratings and tiers. Redrawing the line between "A-class" and "B-class" hotels, schools or suppliers changes the average scores of both tiers without any change in the hotels themselves.
A product has 1,000 users. The table shows how many sessions a week they have and how much they pay a month on average. Choose from how many sessions a user counts as active.
| Sessions | Users | $ per user | Segment |
|---|
A/B tests are no different: if the new variant changes who counts as "active", "engaged" or as having seen the change, comparing metrics inside those segments between variants can show a gain in both segments with no gain overall. Make the main decision on the metric over all users of each variant, as randomisation assigned them. How to read the test itself is on the page "Two identical pages".
Will Rogers died in 1935. According to Quote Investigator, the joke about the Okies was first credited to him in 1970, when the Los Angeles Times quoted the Oklahoma governor Dewey Bartlett. The line has not been found in Rogers's own writings.
The earliest version found ran in 1939 in the McComb Daily Journal of Mississippi: one man moved from one county to another and "raised the intelligence level of both". The version about Okies and California turns up in 1955.
The New Zealand prime minister Robert Muldoon is credited with a version: New Zealanders who emigrate to Australia raise the IQ of both countries. In 1990 a Sydney Morning Herald journalist wrote that Muldoon told it at a dinner, and added that the joke was very old.
The term "Will Rogers phenomenon" was coined by the Yale clinical epidemiologist Alvan Feinstein in a 1985 paper on lung cancer. He took the joke from a friend who credited it to Rogers. Since then researchers look for "stage migration" in studies of cancer and other diseases.
In 2005 Peter Albertsen and colleagues had an experienced pathologist re-grade the slides of 1,858 men diagnosed with prostate cancer in Connecticut in 1990–1992. The mean Gleason score rose from 5.95 to 6.8. With the new grades, mortality within each score looked 28% lower, though the patients' outcomes were the same.
"Okies" was the name for migrants from Oklahoma and nearby states whom the Dust Bowl and the Great Depression of the 1930s drove west to California. John Steinbeck told their journey in "The Grapes of Wrath" (1939), which won him the Pulitzer Prize.
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