Comment by wisty

What I mean is, Poisson distributions don't look like normal distributions if lambda is low - there's a long tail.

There's about 500,000 centenarians (thanks bitwize for the spelling) in the world. That's about 0.01%. In a town of 10,000, that's an average (lambda) of 1. About 36% of such towns will have no centenarians, 36% will have one, 18% will have 2, 6% will have 3, 1.5% will have 4, and there's a long tail with 5 (0.3%), 6 (0.05%) or more.

It doesn't seem logical that most towns will have 0 or 1 centenarians, and some will have 5 or more, but it's just the way the numbers work.

If you pick a higher lambda (for example, the number of 50 year olds) it looks like a normal distribution. If some place has 2X the number of 50 year olds, there will be a good reason. If you pick a higher lambda (the number of people who survive incurable cancer) it looks even wackier, and it's very hard to draw conclusions.

What I'm saying is, it's hard to draw conclusions when you are looking at rare events, because there can be so much variation.