Tomorrow’s weather forecast for Carlisle, PA, is:

Windy with showers. High 58F. Winds SW at 20 to 30 mph. Chance of rain 40%.

First of all, 58 degrees on December 28? Sweet! But what does “chance of rain 40%” mean? Does it mean that there is a 40% chance that it will rain **at my house**? Does it mean that there is a 40% chance that it will rain somewhere in the Carlisle area? If we get a few drops that barely wet the ground, does that count as rain? Does it mean that 40% of the region will get rain? Maybe that it will rain for 40% of the day? Most importantly, should I take my kids to the park tomorrow to enjoy the unseasonably warm weather?

If mathematicians know anything, it is that definitions are important. What are these weather organizations trying to tell us when they give us the percent chance of rain? What is the definition of the “percent chance of rain”?

The Wall Street Journal’s Carl Bialik, The Number’s Guy, recently wrote an article called Deciphering a 20% Chance of Rain. In it he answers some of these questions. It turns out that “percent chance of rain” does not mean the same thing for every weather organization.

According to the article, AccuWeather.com says that the percentage is the probability that a measurable amount of rain (defined to be at least 0.01″) falls at the ‘official’ verification site. So they may call it a rainy day even if not a drop fell at my house.

On the other hand, the National Weather Service’s percentage, the “probability of precipitation” (PoP), is calculated as follows (their mathematical notation, not mine).

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.

As an example they say that if there is a 50% chance that the region will get rain and they expect this rain to cover 80% of the region then the PoP is 40%. They add that if the meteorologist is sure that it is going to rain (C=100%), then the PoP **is** the percent of the region that will get rain.

This got me wondering about the psychology of weather forecasts: if we were to look back at all the days with a 40% chance of rain, would 40% of them have rain? I’m not talking about the meteorologists’ predictive abilities—I am wondering if meteorologists are ever pressured by their public relations office to **overstate** the case for rain. In my experience people (me included) have a bad intuitive sense of probabilities. If a meteorologist said that there was a 15% chance of rain, and it rained, then there would probably be a lot of angry people. But 15% is not that small of a percentage—it **should** rain 3 out of every 20 such days. It would be more likely to rain that day than to flip three heads in a row.

(I suppose it could go the other way too—there could be angry people who cancelled an outing based on an 85% chance of rain on a day that it didn’t rain.)

As a partial answer to my question, I found the following table on the Internet. It presents the result of a survey showing how the public (in Juneau, Alaska) interpreted various weather-related qualifiers compared to how the qualifiers are used by the National Weather Service.

Term | Survey Mean | Probability Percentage |
---|---|---|

slight chance | 19.7% | 10%, 20% |

few | 28.0% | 10% |

ending | 31.7% | 80%, 90%, 100% |

isolated | 34.0% | 10% |

scattered | 34.0% | 30%, 40%, 50% |

widely scattered | 34.3% | 20% |

chance | 41.8% | 30%, 40%, 50% |

areas of | 43.1% | 80%, 90%, 100% |

occasional | 50.9% | 80%, 90%, 100% |

developing | 52.9% | 80%, 90%, 100% |

periods of | 56.0% | 80%, 90%, 100% |

likely | 62.5% | 80%, 90%, 100% |

frequent | 66.5% | 80%, 90%, 100% |

numerous | 72.3% | 80%, 90%, 100% |

Sure enough, the terms corresponding to low probabilities (10%, 20%) were perceived as higher percentages by the public. Similarly, the terms corresponding to high probabilities (80%, 90%, 100%) were perceived as lower percentages. I take this to mean that the public does not understand high and low probabilities—despite what our intuition tells us, 80% is no guarantee of rain and a 20% day need not be dry.

So… an unseasonably warm December day with a 40% chance of rain—do I take the kids to the park, or not? According to this chart, there is a “chance of rain” or there may be “scattered” rain showers. Hmmm… I think we’ll go—a little rain can’t hurt anyway.

Odds and ends:

Here’s a quote from my favorite movie about a weatherman: L.A. Story. It is an argument between the goofy meteorologist Harris Telemacher (Steve Martin) and his boss. It takes place in Los Angeles.

– So I pretape the weather and some sailors lost their boats. Big deal! Besides, what kind of as$#0le sailor would trust the “wacky weatherman” anyway?

– This one.

– You lost your boat?

– Yes. You’re fired.

The Numbers Guy wrote a few other articles about weather:

Grading Weather Forecasts

How Much Rain Fell in Marble Falls, Texas?

Should World Series Forecasts Be Charged With an Error?

Here is a link to an interesting website that will tell you which weather service does the best job of predicting weather in your area.

Thank you for clarifying the PoP definitions. I think it’s also important to take into account the time-scale. I’m assuming from the definition posted that if qualifying precipitation occurred at any time during the prediction period then that would be enough. So if Accuweather says there’s a 50% chance of rain on Monday that means that there’s a 50% chance SOMETIME in the 24 hours comprising Monday.

The problem is that percentages add up. If I flip a fair coin in the morning so I have a 50% chance of heads in the morning, and I flip it again in the afternoon, so I have a 50% chance of heads then, my total chance of heads at SOME point becomes .75. (1-(.5 x .5)) If I flip it a third time, in the evening, the chance of ONE of them being heads becomes .875. (1-(.5x.5x.5))

So if Accuweather predicts a 50% chance of rain on Monday, then the chances of rain must be much lower in the individual parts of the day, depending on how many there are. I couldn’t get on the accuweather website for some reason so I went to the weather.com website. But they do HOURLY forecasts with PoP numbers. So using the coin-flip model, above, if you were going on a 5 hour hike and each hour was rated at 50% then the chance of getting rained on at some point during your hike would be 97%.

Is that what they really mean?