An almost painless lesson in practical statistics

 

Upper-left, similar populations but different number totals. Upper-right, same distribution characteristics but shifted-mean. Middle-left, same mean but distribution in red crayon has lower standard-deviation. Middle-right, same process but red-crayon has skirts sorted off. Lower-left, a centered bell-shaped distribution and a random but sorted-to-limits distribution.

There are many times we want to know if two populations are "the same" or "are different".

A computationally simple but mathematically robust way to make that determination is to look at the skirts of the two populations that you are comparing. The method is robust in that it kicks out the correct answer for a wide range of ways that populations might differ from each other.

The method entails collecting a fairly large, random samples from both populations. Mark each part so you can identify which population it came from. Then sort them, rank-order, by the characteristic of interest. Let's say we are sorting 18 year-old humans by height with one population identified as "girls" and the other population identified as "boys".

The classical method of performing the analysis is to start at one end and to start counting the "run" that came from the one population. Suppose the short end looked like:

G-G-G-G-G-B-G-G-G....

Then going to the opposite end of the rank-ordered list and continuing the count but counting "Not-G"

...G-B-G-B-B-B-B-B-B-B-B

The first string yields five "G" in a row before being interrupted with a "B" while the tall-end of the list (counting from the right) yields an additional eight "not-G" in a row before being interrupted by a "G". That gives us a Tuckey End-Count of 13.

But what does that mean?

If the two populations that the samples were drawn from were the same, then each position in the list would have an equal likelihood of being a "G" or a "B".

The first one we pick has no influence because we will accept either a "G" or a "B".

But to continue the count we require that the second individual match the first. So the random chance of the second being the same as the first is 50%. (1/2 * 1/2)

The chance of the third matching the first two is 25%.

The fourth matching is 12.5%

The fifth is 6.25%

But then we get an individual from the other population.

Shifting to the other end, if we are interested in shifted means but similar variance, we are not looking for "not-G".

The first "not-G" on the tall end (end-count of 6) will be 6.25 * 1/2 or 3.125% or (1/2)^ (n-1)

Cutting to the chase, we had an end-count of 5 + 8 or 13. (1/2)^(13-1) pencils out to a random chance of 0.02% of this distribution if the two populations were identical for height characteristics.

Many statisticians who work in agriculture will happily accept an end-count of 6 as "proof" that the two populations are different. Sometimes they will fudge and perform an end-count from each end and accept whichever is longer. Essentially, they are hanging-their-hat on the likelihood of the end-count of six being from identical populations as a one-in-thirty-two likelihood. That is pretty good odds.

Kids as young as first graders can sort themselves by height and count up to six and can identify "Boy" and "Girl". This is not very challenging math.