# "The distribution of heights is not Gaussian"

1. Jun 17, 2017

### bluemoonKY

I was browsing old threads at Physics Forums, and I came across some information in this thread from 2008 that got my interest. The thread is titled "Child molester avoids prison because he is short." PF member stickythighs wrote the following: "Since the average American man is 5'10", there are about equal numbers of men in Florida 5'5" and shorter as there are men 6'3" and taller."

In post #25 on the thread, PF Member Gokul43201 responded: "Not true. The distribution of heights is not Gaussian. It's almost Boltzmannian, and 5 inches is way bigger than the standard deviation - so a Gaussian approximation could be quite off when you go that far away from the mean. And it is..."

The distribution of human heights is the classic example that statistics textbooks and other textbooks use to show a Gaussian Distribution. By the way, a Guassian distribution = a Normal Distribution. I admit that the distribution of human heights is not Gaussian at the tails. In reality, there are far more people at 5+ standard deviations both above and below the mean than a graph of a 100% Gaussian Distribution of human heights would show. In other words, a graph of a 100% Guassian Distribution of human heights would show less people at 5 SD from the mean than there would actually be in real life.

However, in the example that stickythighs and gokul were discussing, the comparison was between male heights of 5'5" and 6'3". Human height distribution IS Gaussian when you are so close to the mean as 5'5" and 6'3". Therefore, why did Gokul deny that the human height distribution is Gaussian in the 5'5"-6'3" range?

Why did Gokul say that the distribution of human heights is almost Boltzmannian? Clearly it's not.

Note to moderators: The topic of the thread that I am referencing is about a child molester avoiding prison because he is short. The main topic of the thread that I am referencing is NOT about whether or not the distribution of human height is Gaussian or not. The correct etiquette and protocol for a digression in another area is to create a new thread on the digression, not to hijack the previous thread. There is no thread that I am aware of specifically about whether or not the distribution of human heights is Gaussian. Therefore, I should not be breaking any rules by creating this thread. It's a new topic.

2. Jun 17, 2017

### Staff: Mentor

Have a look at the graph here.

This is a distribution that has a lower limit like zero, and a maximum going out the x axis. Maxwell-Boltzmann distribution.

So, with normal human adults there is a minimum height, and a larger maximum. I'm excluding dwarfism and gigantism due to abnormalities. And there is a skewness to the result; the graph is not symmetric across the mean, it is skewed.

What gokul43210 (no longer active on the forums) said was that it was a poor fit to a Gaussian curve, and close (but not really) a M-B distribution.
His link to what he cites as a model of the distribution is broken.

How you find a model to fit an existing distribution is interesting. @Dale works with this kind of thing. Maybe he can help clarify what you do with 'almost-fits' situations.

There is also this: Limpert, E; Stahel, W; Abbt, M (2001). "Lognormal distributions across the sciences: keys and clues". BioScience. 51 (5): 341–352
which I cannot get to show in a link. It says that a good human height distribution model is lognormal.

Also, it is not valid to claim that a part of the data is distribution A, and another part is distribution B. And then apply statistical methods on each part as if they were separate.
Since I cannot get all the facts, I cannot give you a good answer.

Last edited: Jun 17, 2017
3. Jun 17, 2017

### 256bits

You can reference Lambert Adolphe Jacques Quetelet for bringing statistical studies into the humanities. His conclusion is that the traits of the average man follow a normal distribution. That has been followed ever since in many areas.

You may also want to read this.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2831262/
See figure 3 for the tabulated distribution of 20 year old males (1900 ) - raw data, adjusted and fitted normal distribution.

For other populations, the height curve could well not follow a normal distribution.
ie ages 0 to old age ( 60, 70 ... ) whole population - what's that going to look like??
ie sex - male and female follow two difference curves. Bring them together and one gets a flat-ish top

4. Jun 19, 2017

### Staff: Mentor

If you have a large enough sample then even the tiniest departures from normality become significant. With N on the order of a billion I am sure that it is not normal.

The question isn't really whether or not something is normal, just whether or not the approximation is close enough that you can use the nice simplifying assumption that normality provides.

5. Jun 19, 2017

### Staff: Mentor

Closed threads should not be re-opened without moderator approval. I think this has been answered.