In Katrina, New Orleans and the Limits of Knowledge, Art De Vany explains that some phenomena don’t follow the statistical rules we’re used to:
For New Orleans, Katrina is the storm of the century. Or is it?It turns out there is no storm of the century in the statistical sense. Looking back, one can see what the biggest storm was over the past. But, in forecasting storms one never knows what the biggest event will be or how probable it is.
The 100 year flood plain that many of you will know of because of home purchases, does not exist. That’s right, the 100 year flood plain that represents an attempt to forecast what area will flood from the biggest storm likely to occur in the next century is impossible to calculate. But, it is done all the time and has taken hold as the way to quantify flood risk.
It does not exist because the rain fall time series in a location does not have a mean. How can it not have a mean? It has an average, which is just a calculation looking back at past rain falls and taking the average as the sum of the rain falls divided by the number. But, even though the series has an average (called the sample average) it does not have a mean. It turns out the average is dominated by extreme events; forty percent of the total erosion in a decade is done by a single storm. And, of all the storms we have seen we can never know what the future will hold. Larger ones are probable.
The mean is a parameter of the underlying true probability distribution of rain falls. But, it need not exist and does not exist for rain fall. Why should it? Nature doesn’t do means as often as people like to think, given their exposure to averages in every thing they read. In order for there to be a mean the probabilities of large events have to fall off quickly enough for the integral of the size of the event, x, with the probability density of the event, f(x), to converge.
It turns out that it doesn’t for rain fall and for a lot of other natural phenomena. In other cases, the mean exists but the variance does not. This is true of hurricanes and tornadoes for example. And the movies.