# Cautionary tales for Gaussians/the central limit theorem?

In any undergraduate course on probability, one of the fundamental results that is discussed is the central limit theorem. Apart from the interesting mathematics needed to prove the result (most often introducing moment generating functions, and more subtly the idea that the distribution of a random variable $$X$$ can be characterized by expectations $$E[f(X)]$$ over a rich enough class of test functions $$f$$), the central limit theorem can be a good opportunity to teach students about the so-called universality of the Gaussian distribution.

That being said, I would like my explanation of universality to come with one very important caveat, so that my students do not, like many people, become "unreasonably enthusiastic" about the Gaussian distribution. (Essentially the error I would like them to avoid is what is explained in this answer to the question "Impressive common misleading interpretations in statistics to make students aware of".)

So my question would be as follows:

Question. Are there well-documented examples of people who, likely because they erroneously assumed some distribution to be Gaussian, were led to a catastrophic outcome? (Perhaps in the insurance or financial industries, or other.)

I feel that an interesting real-life example would help drive the point home that, if you want to argue something is Gaussian by invoking the CLT, you better be able to write it down as the sum of a large number of i.i.d. variables divided by square root of their number.

• Spin a pen-pointer laser on the ground, and then shine its beam at the resulting random angle onto a nearby wall, measuring the length L of the beam. Now matter how many of these i.i.d. variables like L you average together, you will never get a Gaussian. – Ben Crowell Sep 30 '20 at 18:37
• @Ben Crowell, wait, isn't this $L=1+X^2$ where it is $X$ itself with the cauchy distribution? Not that I think the $L_i$ are likely to obey central limit theorems either, but the additional wrinkle caught me off-guard. – Vandermonde Oct 1 '20 at 19:09
• it obeys the CLT if you are in a room with four walls. – Michael Bächtold Nov 3 '20 at 8:20

You may want to lookup

Donnelly, C., & Embrechts, P. (2010). The devil is in the tails: actuarial mathematics and the subprime mortgage crisis. Astin Bulletin, 40(1), 1-33.

It confronts the story of the use of the Gaussian Copula in the financial industry. The Gaussian Copula does not allow for co-variance of the included variables at the extremes of the support (in technical terms, the Gaussian Copula does not exhibit "tail-dependence"). This simply means that if we use the Gaussian Copula to model the dependence structure between variables, say, between the value of financial securities, then by construction we have assumed that these values will not co-vary together at the extremes: in other words they will not go down all together... but which is what happened during the 2008 financial crisis.

Of course this did not cause the financial crisis of 2008, but it certainly did not contribute any warning sign either to the professionals and the practitioners. So we can fantasize that maybe, if said warning signs were given by the financial model, some mitigating actions may had been taken.

A real world, time I have encountered this, was doing process variation calculations (e.g. "Cpk") for pharmaceutical analytical or mechanical quality measurements. In fact, our issue was taking people who were doing zero annual analysis (not even trending, averaging, standard deviation, etc.) and moving to doing some control measurements, anything.

But then after installing those, I got the criticism that FDA has fussed (rightfully in some cases) about non-normal variations. So we had to do normality tests, to tie the bow, and say we had gone from a C to a B to an A.

for an overview.

My understanding is there have been a few high profile product recalls where failure to check for normality was an issue (see the intro section of the thesis, I linked to). Honestly we were worse than that...since we didn't even do any SPC. but that's another issue...and just shows that there can be a value in doing some tests, even imperfect. All that said, I ran the normality tests, so we didn't get fussed at, since at that point we were in the gunsights and giving the "went from a C to a B" argument wasn't going to work. Just had to do it at A standards, since our facility was shut down for poor quality, patient injuries, etc. :-(