Further and deeper exploration of paradoxes and challenges of intuition and logic can be found in my recently published book, Probability, Choice and Reason.

The Will Rogers Phenomenon occurs when transferring something from one group into another group raises the average of both groups, even though there has been no change in actual values. The name of the phenomenon is derived from a comment made by comedian Will Rogers that “when the Okies left Oklahoma and moved to California, they raised the average intelligence in both states”.

In moving a data point from one group into another, the Will Rogers phenomenon occurs if the point is below the average of the group it is leaving, but above the average of the one it is joining. In this case, the average of both groups will increase.

To take an example, consider six individuals, the life expectancy of whom is assessed in turn as 5, 15, 25, 35, 45 and 55.

The individuals with an assessed life expectancy of 5 and 15 years respectively have been diagnosed with a particular medical condition. Those with the assessed life expectancies of 25, 35, 45 and 55 have not. So the mean life expectancy of those with the diagnosed condition is 10 years and those without is 40 years.

If diagnostic medical science now improves such that the individual with the 25 year life expectancy is now identified as suffering from the medical condition (previously this diagnosis was missed), then the mean life expectancy within the group diagnosed with the condition increases from 10 years to 15 years (5+15+25, divided by three). Simultaneously, the mean life expectancy of those not diagnosed with the condition rises by 5 years, from 40 years to 45 years (35+ 45+55, divided by three).

So, by moving a data point from one group into the other (undiagnosed into diagnosed), the average of both groups has increased, despite there being no change in actual values. This is because the point is below the average of the group it is leaving (25, compared to a group average of 40), but above the average of the one it is joining (25, compared to a group average of 10).

Exercise

Take the following groups of data, A and B.

A={10, 20, 30, 40}
B={50, 60, 70, 80, 90}

The arithmetic mean of A is 25, and the arithmetic mean of B is 70.

Show how transferring one data point from B to A can increase the mean of both.

Now take the following example:

A={10, 30, 50, 70, 90, 110, 130}
B={60, 80, 100, 120, 140, 160, 180}

By moving the data point 100 from B to A, what happens to the arithmetic mean of A and of B?

To demonstrate the Will Rogers Phenomenon, does the element which is moved have to be the very lowest of its set or does it simply have to lie between the arithmetic means of the two sets?

The Will Rogers Phenomenon. Simple City. Dec. 1, 2012. https://richardelwes.co.uk/2012/12/01/the-will-rogers-phenomenon/

Will Rogers Phenomenon. Stats Mini Blog. Nov. 21, 2014. https://blogs.bmj.com/adc/2014/11/21/statsminiblog-will-rogers-phenomenon/

The “Will Rogers Phenomenon” lets you save lives by doing nothing. https://io9.gizmodo.com/the-will-rogers-phenomenon-lets-you-save-lives-by-doi-1443177486

Will Rogers Phenomenon. In: Paradoxes of Probability and Other Statistical Strangeness. Stephen Woodcock. May 26, 2017. https://quillette.com/2017/05/26/paradoxes-probability-statistical-strangeness/

Will Rogers Phenomenon. Wikipedia. https://en.m.wikipedia.org/wiki/Will_Rogers_phenomenon