The Will Rogers phenomenon is the apparent paradox obtained when moving an element from one set to another set raises the average values of both sets.

When the Okies left Oklahoma and moved to California, they raised the average intelligence level in both states.

-- attributed to Will Rogers

Numerical Example

Consider the sets R and S

R={1, 2, 3, 4}

S={5, 6, 7, 8, 9}

The arithmetic mean of R is 2.5, and the arithmetic mean of S is 7.

However, if we move 5 from S to R, producing

R={1, 2, 3, 4, 5}

S={6, 7, 8, 9}

then the arithmetic mean of R increases to 3, and the arithmetic mean of S increases to 7.5

Stage migration

One real-world example of the Will Rogers phenomenon is seen in the medical concept of stage migration. In medical stage migration, improved detection of illness leads to the movement of people from the set of healthy people to the set of unhealthy people.

Because these people are not healthy, removing them from the set of healthy people increases the average lifespan of the healthy group. Likewise, the migrated people are more healthy than the people already in the unhealthy set, so adding them raises the average lifespan of that group as well.

Reference

• Feinstein AR, Sosin DM, Wells CK. The Will Rogers phenomenon. Stage migration and new diagnostic techniques as a source of misleading statistics for survival in cancer. N Engl J Med 1985;312:1604-8. Fulltext. PMID 4000199.