A few weeks ago I wrote a column for the Globe and Mail discussing whether Warren Buffett was a genius, or just lucky. He’s been described as a five-sigma event, meaning that his performance falls within five standard deviations from the average. To put it into context, NBA player Manute Bol stood at 7’7″ and this put him at 10 standard deviations from the average height for an American male. Clearly height is not something you can manipulate easily so it demonstrates that, while rare, extreme data points can occur by chance. Warren Buffett’s performance record would be the equivalent of a 6’9” male, rare but not unimaginable. You can read the original column here.

One of the concerns I had with this analogy is that investing performance probably does not follow a normal distribution. Gilles Gignac provides some more in-depth analysis at his stats blog. His analysis would strongly suggest that Buffett’s performance is indeed due to skill. Thanks Gilles, interesting read.

Based on the figures provided, I suspect that height must not nicely fall into a normal distribution, as the calculation results don’t seem to jive with common sense. I’d say that there are at least a couple of hundred 6’9″+ males in the NBA. I think it is also safe to assume that less than 5% of the worlds 6’9″+ males play in the NBA. So this means there is greater than 200/0.05 (i.e. > 4,000) 6’9″+ males in the world. I think it’s safe to say there are many more than this; I tried to use some heuristics that would be hard to dispute.

So with more than 4,000 6’9″+ males in the world, and assuming there is about 3.5B males in the world, the probability of a male being 6’9″+ is > 1:875,000. But a 5-sigma event should only occur with a probability of 1:1,744,278. So even with very conservative assumptions, we can say that either being a 6’9″ male is not a 5-sigma event, or the distribution of height does not fit a normal distribution.

Also, an 8-sigma event should only occur once in every 8.9×10^14 (my spreadsheet ran out of floating point precision beyond 8 sigmas). With a world population of about 6.75B, about half male, this means the probability of a 8-sigma male existing at all would be less than 1:1,500,000. If I extrapolate a bit, the probability of a 10-sigma male existing in the world is likely less than 1 in billions or trillions. Yet we know that 7′ somethings, while not common, aren’t completely unheard of.

I’m just stirring the pot a bit here. I guess this is a long way of saying that heights don’t seem to nicely fit standard distributions, and many other things don’t either. So caution must be used when applying formulae for normal distributions to data sets, without first demonstrating that the data set fits a normal distribution.

So you’re suspect on height following a normal curve and I’m suspect of investor performance following a normal curve and Gilles is suspect as well.

Yep, statistical analysis is great, but I’m guessing that normal distributions are probably widely used when they shouldn’t be. :)

Great comment Returns Reaper!