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I understand what standard deviation means and what a normal distribution looks like but I don’t understand what normal distribution is used for...

Many typical examples explaining normal distributions goes “X of individuals in a large group follows a normal distribution pattern…” where X can be height/test scores/ etc.

What I don’t understand is HOW can the height or test score end up as a nice symmetric bell curve?

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First it is worth noting that while we often model a certain data type as being normally distributed, this does not necessarily mean that we believe it exactly follows a normal distribution. Take your height example. Many researchers (myself included) believe that modeling height by a normal distribution does a fairly good job of describing the distribution of heights. But clearly it's not exactly normal: you can't have negative height, but all normals with positive variance have a positive probability of being negative. Despite this, the distribution of heights is fairly normal.

So why is it that many data types will follow an approximately normal distribution? To me, this can be motivated by central limit theorems (in an undergraduate course, you probably heard of *the* central limit theorem, but there's more than one...). Under a variety of conditions, averages approach a normal distribution.

Why does this explain why various phenomena are approximately normal? Well, suppose the outcome you are measuring is actually the combination of a whole lot of different small effects. In the case of height, there are a lot of genes that each contribute to your height, along with your diet as a youth, etc. So you could think of the final measure of height as a combination of a large number of factors. Even as an average, if you will. As long as a small number of these factors do not completely dictate height (or similarly, these factors aren't too correlated), then there's a central limit theorem that will suggest that this measurement should be approximately normal.

Keep in mind, this is merely heuristic reasoning and I wouldn't use such logic to decide that a given outcome must be approximately normal. But in my own musings, it's the been my personal explanation why an approximately normal distribution might naturally occur.

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  • $\begingroup$ If it's not exactly normal, why don't scientists just use the exact model? Also, how do we know distribution of heights is fairly normal? If I plot out the graph of students in different classes at my school, all classes' height graphs should look different, no? $\endgroup$
    – Glide
    Commented Oct 11, 2015 at 1:40
  • $\begingroup$ "All models are wrong, but some are useful" -George Box. So heights might not be exactly normal, but they can be close enough for useful approximations. And since there's no distribution that's known to do better... $\endgroup$
    – Cliff AB
    Commented Oct 11, 2015 at 1:42
  • $\begingroup$ In regards to plotting out the height of students in different classes, they would be slightly different, this is standard sampling error which is exactly as we expect. The normality assumption states that you would assume if you plotted all the heights, you would expect a bell shaped histogram, not that you expect each class to look exactly the same. $\endgroup$
    – Cliff AB
    Commented Oct 11, 2015 at 1:46
  • $\begingroup$ If I was being a twit, I'd guess that the reason that normal distributions are assumed so often is that everyone's favorite estimators (mean and standard deviation) are maximum likelihood estimators and sufficient, too, and all our favorite tests work on data with normal assumptions. Somewhat kinder, the central limit theorem makes the tests and confidence intervals based on normal assumptions work reasonably well when you have large sample sizes even if your population is not too horribly non-normal. $\endgroup$
    – AlaskaRon
    Commented Oct 11, 2015 at 3:31
  • $\begingroup$ Truth be known, almost every large dataset that I've worked with deviates from normality, but, being a large dataset, we often don't care because of the CLT. On the other hand, if it were a small dataset, we wouldn't have the power (probably) to show that it isn't normal and ignorance is bliss sometimes... $\endgroup$
    – AlaskaRon
    Commented Oct 11, 2015 at 3:34

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