# Why heights are normally distributed

The canonical example of the normal distribution given in textbooks is human heights. Measure the heights of a large sample of adult men and the numbers will follow a normal (Gaussian) distribution. The heights of women also follow a normal distribution. What textbooks never discuss is why heights should be normally distributed.

Why should heights be normally distributed? If height were a simple genetic characteristic, there would be two possibilities: short and tall, like Mendel’s peas that were either wrinkled or smooth but never semi-wrinkled. But height is not a simple characteristic. There are numerous genetic and environmental factors that influence height. When there are many independent factors that contribute to some phenomena, the end result may follow a Gaussian distribution due to the central limit theorem.

The normal distribution is a remarkably good model of heights for some purposes. It may be more interesting to look at where the model breaks down. See my next post, why heights are not normally distributed.

## One thought on “Why heights are normally distributed”

1. Douglas Zare

I’ve heard that speculation that heights are normal over and over, and I still don’t see a reasonable justification of it. I think people repeat it like an urban legend because they want it to be true. All bell curves look similar, just as most ratios aren’t terribly far from the Golden Ratio. I would like to see how well actual data fits.

One source suggested that height is normal because it is a sum of vertical sizes of many bones and we can use the Central Limit Theorem. Ok, but the sizes of those bones are not close to independent, as is well-known to biologists and doctors.

Height is obviously not normally distributed over the whole population, which is why you specified “adult men.” However, even that group is a mixture of groups such as races, ages, people who have experienced diseases and medical conditions and experiences which diminish height versus those who have not, etc. If you want to claim that by some lucky coincidence the result is still well-approximated by a normal distribution, you have to do so by showing evidence.

Perhaps because eating habits have changed, and there is less malnutrition, the average height of Japanese men who are now in their 20s is a few inches greater than the average heights of Japanese men in their 20s 60 years ago. It would be a remarkable coincidence if the heights of Japanese men were normally distributed the whole time from 60 years ago up to now. America had a smaller increase in adult male height over that time period. It would be very hard (actually, I think impossible) for the American adult male population to be normal each year, and for the union of the American and Japanese adult male populations also to be normal each year. I don’t believe it.