Will Rogers Phenomenon - Numerical Examples

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.