For example, some time ago I had a discussion with a friend about strength training. The gist of it was that powerlifters are typically much stronger than the average athlete, but they are also much fewer; because of that, in a typical gym the strongest athlete might not be a powerlifter, but as we get into regional competitions and national competitions, the winner is going to be a powerlifter.
And the explanation, which the friend didn't understand, was "because on the upper tail the difference between means is going to dominate the difference in sizes of the population."
So here's an illustration of what I meant, with pictures and numbers and bad jokes.
First let's make the setup explicit. That's the great power of math and numerical examples, making things explicit. "Powerlifters are typically much stronger than the average athlete" will be operationalized with four assumptions:
A1: There's some composite metric of strength, call it $S$ that we care about and we'll normalize it so that the average gym rat has a mean $\mu(S_{\mathrm{GR}})$ of zero and a variance of $1$.
A2: The distribution of strength within the population of gym rats is Normally distributed.
A3: The distribution of strength in the sub-population of powerlifters is also Normally distributed.
A4: For illustration purposes only, we will assume that powerlifters have a mean $\mu(S_{\mathrm{PL}})$ of 2 and the same variance as the rest of the gym rats.We operationalize "they are also much fewer" with
A5: For illustration, the number of powerlifters is $1\%$ of gym rats.(Powerlifters are gym rats, so the distribution for $S_{\mathrm{GR}}$ includes these $1\%$, balanced by CrossFit people, who bring down the mean strength and IQ in the gym while raising the insurance premiums. Watch Elgintensity to understand.)
The following figure shows the distributions:
When we look at the people in a gym with above-average strength, that is people with $S_{\mathrm{GR}}>0$, we find that one-half of all gym rats have that, and $98
\%$ of all powerlifters have that: $\Pr(S_{\mathrm{GR}}>0) = 0.5$ and $\Pr(S_{\mathrm{PL}}>0) = 0.98$. This is illustrated in the next figure:
Powerlifters are over-represented in the above-average strength, approximately twice as much as in the general population, but they are only about $2\%$ of the total, as their over-representation is multiplied by $1\%$.
As we become more selective, the over-representation goes up. For athletes that are at least one standard deviation above the mean, we have:
with $\Pr(S_{\mathrm{GR}}>1) = 0.16$ and $\Pr(S_{\mathrm{PL}}>1) = 0.84$. Powerlifters are over-represented 5-fold, so about $5\%$ of the total athletes in this category.
When we become more and more selective, for example when we compute the number of gym rats that have at least as much strength as the average powerlifter, $\Pr(S_{\mathrm{GR}}>2)$, we get
with $\Pr(S_{\mathrm{GR}}>2) = 0.023$ and $\Pr(S_{\mathrm{PL}}>2) = 0.5$, a 22-fold over-representation, meaning that of every six athletes in this category, one is a powerlifter. (Yes, one out of six, not one out of five. See if you can figure out why; if not, look at the solution for $S>6$ below and you'll understand. Or not, but that's a different problem.)
And as we look at subsets of stronger and stronger athletes, the over-representation of powerlifters becomes higher and higher: $\Pr(S_{\mathrm{GR}}>3) = 0.00135$ and $\Pr(S_{\mathrm{PL}}>3) = 0.159$, $118$-fold ratio. There will be a few more powerlifters in this group that other gym rats; another way to say that is that powerlifters will be a little bit more than one-half of all gym rats that are at least one standard deviation stronger than the average powerlifter.
The ratios grow exponentially with increasing values for strength (the rare correct use of "exponentially" as they are ratios of Normal distribution tail probabilities; see below).
For $S>4$ the ratio is $718$, for $S>5$ the ratio is $4700$, for $S>6$ the ratio is $32 100$, in other words, there will be one non-powerlifter per group of $322$ gym rats with strength greater than 6 standard deviations above the mean of all gym rats.
This is what the effect of the differences in the tails of Normals always implies: eventually the small size of the better population (powerlifters) will be irrelevant as the higher mean will dominate.
See? That wasn't complicated at all.
-- -- -- --
For the mathematically inclined (strangely themselves over-represented in the set of powerlifters...)
Note that the ratio of probability density functions for the two Normal distributions in the post, for realizations of strength $S = x$ is
\[
\frac{f_{S}(x|\mu_{S}=2)}{f_{S}(x|\mu_{S}=0)}= \frac{e^{-(x-2)^2/2}}{e^{-x^2/2}}= e^{2x-2}
\]
which grows unbounded with $x$; no matter how small the fraction of powerlifters, say $\epsilon$, there's always a minimal $\bar S$ beyond which that ratio becomes greater than $1/\epsilon$ Which means that at some point above $\bar S$ the ratio of the remaining tail itself becomes greater than $1/\epsilon$. (It's very easy to calculate $\bar S$ and I have done so; I'll leave it as an exercise for the dedicated reader...)
Oh, that's the rare occurrence of the correct use of "exponentially," which is usually incorrectly treated as a synonym for "convex."
