How many cold nights are there?

Nineteen degrees Fahrenheit (-7 Celsius) in Philadelphia this morning.

A photographer that I know went out and took some beautiful pictures of downtown Philadelphia overnight; then he complained that he picked the "coldest day of the year" to do it. Of course, he meant the coldest day of the year so far.

But I suspect it won't be the coldest day of the year, for the following reasons:

• the coldest weather in Philadelphia usually comes in late January. (But this is a bit specious; if this morning's low had been 5^oF (-15^o C) I wouldn't be saying that.)


• In the last eleven years, it has been 19 degrees or below on some day after January 3 every year. (There's more complete climate data available out there, but I don't feel like looking at it.)


• Forecasters are forecasting lows around 16 degrees F (-9 C) for tonight.

But that raises an interesting question: what can we expect the coldest temperature all year to be?

The normal low temperature in Philadelphia on any given night in late January is 25 degrees -- that's when the normal low is at its lowest -- but only an idiot would say that this means the average lowest temperature all year is 25 degrees. (In fact, even without looking at the records, I'd be shocked to learn that there has been a year in recorded history in which Philadelphia didn't go below 25 at some point.) One tempting thing to do is to make the following assumptions:

• the coldest day of the year will always fall between, say, January 5 and February 10, a period of 37 days. Call this number t;


• the low temperature on any day in that interval is normally distributed with mean 25 degrees and standard deviation σ


• thus the annual minimum temperature should be at the 1/(t+1) quantile of that normal distribution.

That turns out to be roughly 2 standard deviations below the mean. Unfortunately, I don't know what the standard deviation is!
And there's a much bigger problem here -- I've implicitly assumed that the annual minimum always falls in a certain cold period, and that the temperature on each day is independent of each other day! The second assumption is spectacularly bad. If it's colder than average today, it'll probably be colder than average tomorrow.

Also, I've assumed that the low temperatures on a given day of the year are normally distributed, which probably isn't true...

However, one could use a method like this to estimate, say, the mean number of days below, say, fifteen degrees in a given year. If we know the distribution then we can compute, say, that the probability of it being below fifteen degrees on the night of January 3 is ten percent; adding up these probabilities for every night of the year gives an expected mean number of cold nights; call this μ. But the variance is important here, as well. All I can say instinctively is that the variance is probably greater than that of a Poisson distribution with mean μ (which is what you'd usually use to model "rare events"), since if it's cold today it'll probably be cold tomorrow, and it probably doesn't tell us much about whether next week will be cold; in particular I suspect that within the winter season, there aren't pairs of days for which being cold is inversely correlated.

(Of course, this is all testable, but I don't care that much.)