# Will Rogers phenomenon

The Will Rogers phenomenon is obtained when moving an element from one set to another set raises the average values of both sets. It is based on the following quote, attributed (perhaps incorrectly)[1] to comedian Will Rogers:

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

The effect will occur when both of these conditions are met:

• The element being moved is below average for its current set. Removing it will, by definition, raise the average of the remaining elements.
• The element being moved is above the current average of the set it is entering. Adding it to the new set will, by definition, raise the average.

## Numerical examples

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 5 is moved 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.

Consider this more illustrative example

R={1,2}
S={99, 10,000, 20 000}

with arithmetic means 1.5 for R and 10,033 for S. Moving 99 from S to R gives means 34 and 15,000. 99 is orders of magnitude above 1 and 2, and orders of magnitude below 10,000 and 20,000. It should come as no surprise that the transfer of 99 increases the mean of both R and S.

The element which is moved does not have to be the very lowest of its set; it merely has to have a value that lies between the means of the two sets. Consider this example:

R={1, 3, 5, 7, 9, 11, 13} (mean = 7)
S={6, 8, 10, 12, 14, 16, 18} (mean = 12)

Moving 10, which is larger than R's mean of 7 and smaller than S's mean of 12, from S to R will raise the mean of R from 7 to 7.375, and the mean of S from 12 to 12.333. The effect still occurs, but less dramatically.

## 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 healthier than the people already in the unhealthy set, so adding them raises the average lifespan of that group as well. Both lifespans are statistically lengthened, even if early detection of a cancer does not lead to better treatment: because it is detected earlier, more time is lived in the "unhealthy" set of people. In this form, the paradox can be viewed an instance of the equivocation fallacy. Equivocation occurs when one term is used with multiple meanings in order to mislead the listener into unwarranted comparisons, and life span statistics before and after a stage migration use different meanings of "unhealthy", as the cutoff for detection is different.

## References

• Feinstein AR, Sosin DM, Wells CK (June 1985). "The Will Rogers phenomenon. Stage migration and new diagnostic techniques as a source of misleading statistics for survival in cancer". The New England Journal of Medicine. 312 (25): 1604–8. doi:10.1056/NEJM198506203122504. PMID 4000199.
• Sormani, M. P.; Tintorè, M.; Rovaris, M.; Rovira, A.; Vidal, X.; Bruzzi, P.; Filippi, M.; Montalban, X. (2008). "Will Rogers phenomenon in multiple sclerosis". Annals of Neurology. 64 (4): 428–433. doi:10.1002/ana.21464. PMID 18688811.