FoxxyBrown1111 said:

Deeply sorry.

My mistake. I take back everything.

Spanky was completely wrong. In reality, the gaps at the top are always the same (maybe it´s different in chess

), even if the talent pool grows.

Why is Bolt dominating (besides clinic

)?

I explained that in my example. At the far left side of a (here talent-)bell curve the line is always flatening, no matter how big the number you look at is. So you´ll always have somebody dominating. Be it in Tennis, Darts or Cycling. So if the gaps are getting tighter in GT´s, it has something (if not all) to do with "easier" grand tours. Something Spanky didn´t got.

You refer to a bell curve, that is, a normal distribution (

http://en.wikipedia.org/wiki/Normal_distribution), of talent, and to the "gaps at the top", which I take to mean the difference between the highest talent in a given sample and the second-highest talent, as in your paper-airplane example, or between the highest talent and the

*k*th highest talent for some

*k* which remains constant as the sample size (talent pool) grows (

*k* might be interpreted as the number of professionals, or the number of riders in a Grand Tour, if we assume for simplicity that the number of professionals or riders in a Grand Tour remains close to constant over time, and that the

*k* most talented samples are the ones who make it as professionals or into Grand Tours).

The expected values of the lowest, second-lowest, and so on up through the highest values in samples from a normal distribution are called the "expected normal order statistics" or "rankits" (

http://en.wikipedia.org/wiki/Rankit), so for the

*k*th rankit (starting with the expected maximum as 1st) of samples of size

*N*, I'll write

*R(k,N)*. The expected value of the maximum value within a sample of size

*N*from a normal distribution is then

*R(1,N)*. The expected value of the gap between the maximum value and the

*k*th-best value is equal to the gap between the expected maximum value and the expected

*k*th-best value, so the expected value of the gap between the top

*k* samples can be represented as

*G(k,N) = R(1,N) - R(k,N)*.

Rankits are not trivial to compute, but there are tables of them, for example, at

http://biomet.oxfordjournals.org/content/48/1-2/151.full.pdf (hmm, it seems that page might require a subscription now, but I was able to download it before! Anyway, it's a table of numbers). From that table, we can compute, for example, the following (to five digits after the decimal point):

G( 2,100) = R(1,100) - R( 2,100) = 2.50759 - 2.14814 = 0.35945

G(10,100) =R(1,100) - R(10,100) = 2.50759 - 1.30615 = 1.20144

G( 2,400) = R(1,400) - R( 2,400) = 2.96818 - 2.65761 = 0.31057

G(10,400) =R(1,400) - R(10,400) = 2.96818 - 1.97871 = 0.98947

So, as the sample size (talent pool) grew from 100 to 400, the expected gap between the best and second-best did go down, from 0.35945 to 0.31057, and so did the gap between the best and tenth-best, from 1.20144 to 0.98947.

Furthermore, in one of the answers to a relevant question at

http://math.stackexchange.com/questions/24743/does-exceptionalism-persist-as-sample-size-gets-large, we can see that Mudholkar, Chaubey, and Tian recently proved that as the sample size approaches infinity, the gap between the expected maximum and expected second-to-maximum values in a sample converges to 0 (we could write

*G(2,infinity)=0*).

So, if we assume a normal distribution of talent, as in the bell curve you referred to, then, as far as I can tell, Spanky is right, and the expected gaps between the top two (or top ten, or whatever) samples shrinks as the sample size grows. Of course, that expectation applies to the average of all sports over all periods of time -- among individual samples, we would still expect some cases of large gaps between the top two (or top ten) sometimes to appear in a given sport at a given time (such as in short-distance sprinting today with Bolt).

Furthermore, the distribution of talent might not be normal, or the selection of professional competitors from among talented humans might not be well-modeled by simply taking the

*k* best; these are simplifications. Are you familiar with any research indicating that, for example, athletic talent follows not a normal distribution but some other distribution, perhaps one which is ES-medium in the sense defined by Mudholkar, Chaubey, and Tian, in which the gaps between the best two would be expected to remain roughly constant as sample size grew, as you asserted? The most similar-sounding thing that comes to me off the top of my head is Stephen Jay Gould's attribution of the demise of the .400 hitter to the reduction in variation as the general standard of play improves (see

http://www.pbs.org/newshour/gergen/november96/gould.htm), which seems more to support Spanky's assertion than yours, but it may not be relevant; it refers to an improvement in general standard, of which an increase in talent pool may be one cause (it improves the professional standard), but is only one special case of, and it also relates only to one particular statistic in one particular sport, so it's not the kind of large-scale (cross-sport, cross-time) analysis I'm asking about.