Thanks to everybody who sent in links, story tips, and suggestions. Because of my recent travel and pressures of work, I’m (again) way behind in answering these. I do appreciate you’re taking the effort to send these in, but sometimes it takes me quite a while to get to them. I need a secretary!
First I enjoy your site. I have a technical education (engineering) but was never required to develop an in depth understanding of statistic.
I tend to be a natural skeptic of almost all things. One of my “hobbies” is following “bad science”. It seems that this is more common than most people realize, especially in medical, economic, psychology, and sociology. (All systems that are non-linear and controlled by large numbers of variables.) I think climate falls into this category.
I don’t expect “personal” response to this, but perhaps you could address it on your site someday.
I once read story where a noted hydrologist who was being honored at MIT, was summarizing some of his research and he mentioned this…”precipitation was not a normal distribution”. It has fat tails. (That may be why we always complain that that it raining too much or too little….because rain fall is seldom average.)
My question is this, when a phenomenon is not a normal distribution, and assumed to be so, how could this affect the analysis?
Precipitation does not “have” a normal distribution. Temperature does not “have” a normal distribution. No thing “has” a normal distributed. Thus it always a mistake to say, for example, “precipitation is normally distributed” or a mistake to say “temperature is normally distributed.” Just as it is always wrong to say, “X is normally distributed” where X is some observable thing.
What we really have are actual values of precipitation, actual values of temperature, actual measurements of some X. Now, we can go back in time and collect these actual values, say for precip, and plot these. Some of these values will be low, more will be in some middle range, and a few will be high. A histogram of these values might even looked vaguely “bell-shaped”.
But no matter how close this histogram of actual values resembles the curve of a normal distribution, precipitation is still not normally distributed. Nothing is.
What is proper to say, and must be understood before we can tackle your main question, is that our uncertainty in precipitation is quantified by a normal distribution. Saying instead, and wrongly, that precipitation is normally distribution leads to the mortal sin of reification. This is when we substitute a model for reality, and come to believe the unreality more than in the truth.
Normal distributions can be used to model our uncertainty in precipitation. To the extent these modeled predictions of a normal are accurate they can be useful. But in no sense is the model—this uncertainty model—the reality.
Now it will often be the case when quantifying our uncertainty in some X with a normal that the predictions are not useful, especially for large or small values of the X. For example, the normal model may say there is a 5% chance that X will be larger than Y, where Y is some large number that takes our fancy. But if we look back at these predictions we see that Y or larger occurs (for example) 10% of the time. This means the normal model is under-predicting the chance of large values.
There are other models of uncertainty for X we can use, perhaps an extreme value distribution (EVD). The EVD model may say that there is a 9% chance that X will be larger than Y. Then, to the extent that these predictions matter to you—perhaps you are betting stocks or making other decisions based on these predictions—then you’d rather go with a model which better represents the actual uncertainty. But it would be just as wrong to say that X is EV distributed.
The central limit theorem (there are many versions) says that certain functions of X will in the long run “go” or converge to a normal distribution. Two things. One: it is still only uncertainty in these functions which converges to normal. Two: we recall Keynes who rightly said “in the long run we shall all be dead.”