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If a popular website provides an hourly forecast, and it states at 12:00pm there will be a 20% chance of precipitation and 2.5 mm of rain...

What does the "2.5 mm of rain" mean?

(A) ... If it does rain, on average there will be 2.5 mm of rain.

(   20% * 2.5mm   +   80% * 0.0mm   ) = 0.5 mm average rainfall 

(B) ... "2.5 mm of rain" is a "statistical prediction" which means that a huge storm could be coming and that if it does rain there will be about ~12.5 mm of rain for the hour but if the storm misses then there will be 0.0 mm of rain.

(   20% * 12.5mm   +   80% * 0.0mm   ) = 2.5 mm average rainfall

How should a person interpret a weather forecast of 2.5 mm with a 20% probability? There is a big difference between potentially 2.5 mm of rain and 12.5 mm of heavy rain.

In (A), a typical person reading may imagine how deep 2.5 mm is and not be worried as that is just a thin film of water. However if a site reported 12.5 mm, they may imagine puddles and hence be worried. Since a typical person is not mathematically inclined and unlikely to be knowledgeable in conditional probabilities, using (A) is potentially more useful for most people.

In (B), a typical person reading imagines how deep 2.5 mm is and is not worried. That person will be very surprised if the storm hits. A statistical prediction is useful for science/math people and keeps the prediction independent of the PoP %. It is however very misleading for 95-99% of people that happen to be not versed in meteorology. I assume (B) is used but wanted to confirm.

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    $\begingroup$ One reason the US city forecasts give probabilities is that the forecast covers more than a 100 mile diameter area and the likelihood of a uniform precipitation in that large area is low. $\endgroup$ Jul 17 at 17:45
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    $\begingroup$ This question already has an accepted answer, PoP is not related to the amount forecast. Probability is only rain or no rain... the amount is a statistical average across all ensemble members: earthscience.stackexchange.com/questions/70/… $\endgroup$
    – f.thorpe
    Jul 18 at 3:24
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    $\begingroup$ Thank you @f.thorpe, the question by naught101 you provided is the same question as mine based on the question description but with a less specific title such that a proper answer may never be given as can be seen by all the incomplete and oblivious upvoted answers on that page. The question title by naught101, ignoring the question description and based on all the answers on that page is effectively... "what does percentage chance of rain mean?" of which there are multiple unmarked duplicate questions already but none answer naught101's question description or my question. $\endgroup$
    – Yosef
    Jul 18 at 4:39
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    $\begingroup$ Thank you again @f.thorpe, it would really be beneficial for future users as an actual answer with more details and a source to either naught101's question or as an answer to my question. Considering the mismatch in question title and question description of naught101's question, number of page views, number of answers and the age of that question... it would be fair to say that my prediction of never being answered properly is reasonable. naught101's question title really should be edited to be more specific if that is even possible, regardless of whether or not you create an answer. $\endgroup$
    – Yosef
    Jul 18 at 4:44
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    $\begingroup$ The best path forward @f.thorpe would be a new "special community wiki" question, eg "ELI5: How do I read an online weather report?" Whoever posts that question could probably get 100 upvotes * 10 points even on quiet earthscience. The top answer should be comprehensive and address temperature, high temperature, wind speed, chance of rain, rainfall amount etc... all with examples and the gotchas. Meta may need to be asked first to avoid the new question being closed as a duplicate. Then every question like mine, naught101 and others can easily be closed or not asked in the first place. $\endgroup$
    – Yosef
    Jul 18 at 6:14
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A popular website (let us not give any names) may have a a) computer generated forecast. b) computer generated forecast interpreted by a man (meteorologist specialized in short-term or long-term forecasts). In either case, "20% chance of rain" for a particular location is simply the full rain probability for a "prognosis area" times a probability to affect x% of the full prognosis area.

From a document downloaded from weather.gov

This point probability of precipitation is predetermined and arrived at by the forecaster by multiplying two factors: Forecaster certainty that precipitation will form or move into the area X Areal coverage of precipitation that is expected (and then moving the decimal point two places to the left) Using this, here are two examples giving the same statistical result: (1) If the forecaster was 80% certain that rain would develop but only expected to cover 50% of the forecast area, then the forecast would read "a 40% chance of rain" for any given location. (2) If the forecaster expected a widespread area of precipitation with 100% coverage to approach, but he/she was only 40% certain that it would reach the forecast area, this would, as well, result in a "40% chance of rain" at any given location in the forecast area.

Moving to mm-s or in-s. The computer software predicts precipitation rates according to input parameters. 2,5 mm occuring in 2hrs time in Columbus OH means that the software used (GFS, for example) probed initial conditions of current/now wether from Columbus and vecinity of, let us say, 100 miles/160 kms and found that if rain clouds form directly above Columbus, or existing rain clouds in vecinity reach Columbus, 2,5 mm is the most likely pp quantity to be measured at a weather station's rain gauge.

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