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 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.

All of this of course also works the other way around.

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={99,10000,20000}

with arithmetic means 1.5 and 10033. Moving 99 from S to R gives means 34 and 15000. 99 is orders of magnitude above 1 and 2, and orders of magnitude below 10000 and 20000. 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. Consider this example:


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

Moving 10 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 more healthy than the people already in the unhealthy set, so adding them raises the average lifespan of that group as well.