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> The normal distribution is not about default humans. It’s about the fact that in many statistical studies, the characteristics of the population tend to clust

by bambax 5y ago

> The normal distribution is not about default humans. It’s about the fact that in many statistical studies, the characteristics of the population tend to cluster around a mean. So if the average height of the human population is 5 foot 7, then most individuals will be around that height. NO!! This is known to be false. The average does not mean "most" measurements match the... mean. And it actually does happen that the normal distribution is often (wrongly) interpreted as defining some kind of standard specimen that does not, in fact, exist. There's a famous story about that exact topic, relating to US pilot seats. Excerpts from an article about it: > Using the size data he had gathered from 4,063 pilots, Daniels calculated the average of the 10 physical dimensions believed to be most relevant for design, including height, chest circumference and sleeve length. These formed the dimensions of the “average pilot,” which Daniels generously defined as someone whose measurements were within the middle 30 per cent of the range of values for each dimension. So, for example, even though the precise average height from the data was five foot nine, he defined the height of the “average pilot” as ranging from five-seven to five-11. Next, Daniels compared each individual pilot, one by one, to the average pilot. > Before he crunched his numbers, the consensus among his fellow air force researchers was that the vast majority of pilots would be within the average range on most dimensions. After all, these pilots had already been pre-selected because they appeared to be average sized. (If you were, say, six foot seven, you would never have been recruited in the first place.) The scientists also expected that a sizable number of pilots would be within the average range on all 10 dimensions. But even Daniels was stunned when he tabulated the actual number. > Zero. > Out of 4,063 pilots, not a single airman fit within the average range on all 10 dimensions. https://www.thestar.com/news/insight/2016/01/16/when-us-air-force-discovered-the-flaw-of-averages.html https://www.thestar.com/news/insight/2016/01/16/when-us-air-...

emergentstate 5y ago
But the anecdote you're relating doesn't really say anything about the ability of a normal distribution to describe a population. It's just a natural consequence of having a high-dimensional multivariate distribution without accounting for correlation or covariance, along with a low sample size. If you have 4,063 pilots and you take the middle 30% of ten (assumed independent) factors, you're looking at 0.3^10 * 4,063 = 0.02 people who actually fall into that bin, on average -- it's more a dimensions game than anything profound. That's not to refute the anecdote's takeaway, but the same reasoning doesn't apply to a unidimensional unimodal distribution.
- bambax 5y ago
  Yeah ok my comment was unclear. My quote from the OP isn't false and the "no" in capitals is over the top and wrong. Yet the airline story does show that the mean (a bag of means of different factors) can be used -- is often used, wrongly -- to construct an average standard (or ideal) specimen. That specimen is then used as a reference, yet it doesn't exist. Also 1/ The normal distribution without a standard deviation tells us little about where a specimen lies in a given population; 2/ A normal distribution is often assumed when next to nothing is known about the actual distribution, and that's an even bigger problem, especially in social sciences.
strangeattractr 5y ago
This is a pretty poor criticism of the normal distribution, for one it doesn't contain a single mention of variance or standard deviation which is as important as the mean. Modelling something with a normal distribution does not say that any member of the sample actually recorded the average value, it simply states that the values cluster around the mean. If the average American household has 2.5 children, it's fairly obvious that there is not a household in the country that has the mean value. I haven't bothered to look at the underlying distribution of children but the principle is the same.

by bambax 5y ago

emergentstate 5y ago

But the anecdote you're relating doesn't really say anything about the ability of a normal distribution to describe a population. It's just a natural consequence of having a high-dimensional multivariate distribution without accounting for correlation or covariance, along with a low sample size. If you have 4,063 pilots and you take the middle 30% of ten (assumed independent) factors, you're looking at 0.3^10 * 4,063 = 0.02 people who actually fall into that bin, on average -- it's more a dimensions game than anything profound. That's not to refute the anecdote's takeaway, but the same reasoning doesn't apply to a unidimensional unimodal distribution.

bambax 5y ago
Yeah ok my comment was unclear. My quote from the OP isn't false and the "no" in capitals is over the top and wrong. Yet the airline story does show that the mean (a bag of means of different factors) can be used -- is often used, wrongly -- to construct an average standard (or ideal) specimen. That specimen is then used as a reference, yet it doesn't exist. Also 1/ The normal distribution without a standard deviation tells us little about where a specimen lies in a given population; 2/ A normal distribution is often assumed when next to nothing is known about the actual distribution, and that's an even bigger problem, especially in social sciences.

strangeattractr 5y ago

This is a pretty poor criticism of the normal distribution, for one it doesn't contain a single mention of variance or standard deviation which is as important as the mean. Modelling something with a normal distribution does not say that any member of the sample actually recorded the average value, it simply states that the values cluster around the mean. If the average American household has 2.5 children, it's fairly obvious that there is not a household in the country that has the mean value. I haven't bothered to look at the underlying distribution of children but the principle is the same.