Thinking about the Weather: Probability of Precipitation

It's raining. That's a matter of pressing concern at the moment (9:40 a.m.) because, before night falls, I'd like to get in my 26-mile bicycle ride. So I suddenly find myself with an intense interest in that odd probability figure so often bandied about by your friendly neighborhood meteorologist, "chance of rain." (Note: if you're looking for financial commentary today, sorry -- it's Saturday, and my Derivatives Muse has left the building.)

But what does a 30% chance of rain actually mean? Or, in my case, a 50% chance -- that's the grim outlook today for where I live. Does it mean it will rain 50% of the day? Or that, at any moment I step outside, I have a 50% chance of getting wet? I've studied some meteorology, so I was pretty sure that I knew roughly what it meant. But I still had some questions.

Online I soon found this from "The Straight Dope" from Cecil. His answer was sort of like "Percentages for Dummies" -- a bit superficial and not quite what I was looking for:

When you hear there's a 10 percent chance of rain, that means that out of the last 100 times the weather conditions were just like they are now, it rained 10 times.

Yeah, yeah, I knew that part already. It just sort of recapitulates the essence of probability ("If you poke a bear 100 times, and 10 random times he growls at you, there's a 10 percent chance he'll growl at you if you poke him again.") Cecil's response dodges the intriguing questions, such as does a 50% chance of rain mean that that's the probability it will rain somewhere in town, or rain specifically on my head?

The answer is here (I won't drag out the suspense). This explanation comes courtesy of the National Weather Service, so this is the final authority speaking. Below is the long version of the weather agency's explanation for "probability of precipitation," or PoP. It's bound to make some heads hurt among the numerically challenged:

Mathematically, PoP is defined as follows: PoP = C x A where "C" = the confidence that precipitation will occur somewhere in the forecast area, and where "A" = the percent of the area that will receive measureable precipitation, if it occurs at all.

So... in the case of the forecast above, if the forecaster knows precipitation is sure to occur ( confidence is 100% ), he/she is expressing how much of the area will receive measurable rain. ( PoP = "C" x "A" or "1" times ".4" which equals .4 or 40%.)

But, most of the time, the forecaster is expressing a combination of degree of confidence and areal coverage. If the forecaster is only 50% sure that precipitation will occur, and expects that, if it does occur, it will produce measurable rain over about 80 percent of the area, the PoP (chance of rain) is 40%. ( PoP = .5 x .8 which equals .4 or 40%. )

Got it? In other words, "chance of rain" breaks down into two parts. First, the forecaster looks at a given area -- let's say the metro region of Boston. He crunches his models and concludes that there's a 50% likelihood it will rain somewhere in Boston. Then he analyzes how much of Boston will get wet, if it does rain. If he decides it's, say 40% of the area, then the official "chance of rain" for Boston becomes 20%.

If you don't like percentages, and you certainly don't like them on a combo platter, as I've just served up, here's the easy way to understand that figure: there's a 20% chance that any given point in the Boston area will be rained upon.

Okay, now we're getting somewhere. "Chance of rain" is always tied to a spatial component. After all, it doesn't make much sense to talk about the chance of rain without talking about some specific place that's either getting rained on or not getting rained on. Further, we're talking about any particular point in that area (this raises an interesting question -- how big is our point? The size of a period at the end of a sentence, or a beach ball, or a helicopter landing pad? And what happens to the probability of precipitation if we increase the size of the point? Naturally we would expect the percentage to rise, though only by a very small amount -- there will be a rare rain event where the edge of the beach ball gets wet but not a small dot in the middle.)

"Chance of rain" isn't only linked to a certain geographic area though. It also needs a temporal component. This has interesting implications. Illustration: what is the chance of rain for a given point in that metro area of Boston for the entire year of 2009? Well, one would expect that to be about 99.9999% -- it's practically inconceivable that a given location in Boston won't receive some rain, at some point during a year.

Now, running that idea in reverse, we'd expect the opposite to occur: that if there's say a 90 percent chance it will rain at a given point in Boston during a certain week, the likelihood that it will rain at that point on any particular day would be lower. (40% maybe? or even as low as 20%?)

So "chance of rain" lacks sufficient precision without an accompanying timeframe. We all know this, but ordinarily don't give it much thought. The National Weather Service tells us there's a 40% chance of rain "this afternoon." Few of us would then say, "What does this afternoon mean exactly? 12 to 4? 12 to 5? Because how you define the length of the afternoon affects the resulting probability."

Convenient example: wunderground.com has started giving "hourly" forecasts for chance of rain (actually, they represent three-hour blocks). Today, for where I live, there's a 50% chance of rain. And all of the shorter blocks of time also give 50%. Seems logical at first. But what's wrong with this picture?

Recall what a 50% chance of rain for the day means: sometime, over 24 hours, any given point in a certain region has a 50% chance of experiencing rain. The rain could fall at 5 a.m., or noon, or 11:59 p.m. But it will occur sometime in that 24-hour cycle. But when you start chopping that 24-hour cycle into smaller chunks, the percentage for the shorter periods should fall. So when you have a 50% chance of rain over an entire day (and, to keep things easy, let's say the 50% chance is evenly spread over the day, ruling out fronts swinging through in the morning and exiting by the afternoon), a good meteorologist (and statistician) will realize that you don't by extension have a 50% chance of rain for any given three-hour period during that day.

Hope you enjoyed this little detour from the normal programming. By the way, it's still raining, at 10:46 a.m. My back of the envelope calculations indicate there's a 100% chance I'm not going anywhere for at least an hour ...