Will every population always converge to a “middle” that contains greatest amounts of whatever is being measured with “increasing” number of traits on wither side in order to make a bell curve visual distribution?

Question

Or are there populations where the curve graphs will converge on either end instead of the middle? Is it a fixed rule in Statistics that we “should” always have a bell curve distribution? If not, why does it seem like my data must make a bell curve distribution? Is it a rule in nature that that are greatest amounts of something in a group while slope downwards by number and value of trait towards the gretest middle and from it downwards? What is the special trait about the bell curve that it is underscored so much?

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Answer 1

> What is the special trait about the bell curve that it is underscored so much?

The central limit theorem shows that (under some weak conditions) any variable that is the sum of a large number of independent random variables will be normally distributed. The Galton board – the thing where balls are dropped through a board with pegs in it – is the classic example of this. At each peg, a ball falls randomly to the left or right, and the position that the ball lands in is the sum of all these little movements. So you get the classic bell curve shape. This would still work if you held the board at an angle so the balls were biased towards one direction, or if you messed around with the positions of the pegs, within reason.

Situations like that seem to come up a great deal in nature, at least for quantities we care about, and at least approximately. Another factor is that the normal distribution is mathematically easier to work with than many distributions, so it’s often chosen to model things even if it isn’t a perfect fit.

But there are many, many, many situations where data follow a distribution that doesn’t look at all Gaussian. Anything that is bounded (e.g. people’s ages, which can’t be less than 0), or anything that is discrete (e.g. the number of kids you have, which can’t be fractional), or anything that spans many orders of magnitude (e.g. the lengths of a diverse set of organisms, from whales to bacteria) will clearly not be Gaussian. But sometimes they can be approximated by a normal distribution.