 # Rain percentage : hourly versus daily

@dsj

Just noticed that in my app it is summarising the rain as 10% but if I look at the hourly predictions the maximum is 5% at any time.

Is this a case of rounding up ?

Also I have noticed that on my Turkish site it is advising 10% in the rain summary despite there being 0% for all 24 hours individually.

the 10% is average for the day, think if you add up all the hours and divide you should be there.

@eric no, no, that is the mistake I made yesterday with a post I quickly redrawn, before someone got to comment on it.
I argued that if during the whole day you had a period with 5%, 10%, 15%, 5% rain, that the chance of being dry the whole day would be 0.95 * 0.90 * 0.85 * 0.95 * 100%. = 69% and thus 31% chance of rain. But that didn’t match the value at all. I soon realized that the hourly chances aren’t independent. There could be a small chance for one rainy cloud to pass, but the exact hour wouldn’t be known. You cannot simply add up and divide.

@mrburns it could be that there is a 10% chance that it will rain sometime during the day, but if the exact hour isn’t know, the chance per hour is very small, and rounded down to 0.

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HI @eric

Unfortunately there were 2 hours, both with 5%, average 5%. Not 10%. Sorry

Hi @sunny

Understood, If there are a number of variation during the day I could understand a summary token prediction, but in this case it it either (in the case of the Hinckley site) 2 hours with only 5% noted or in the case of the Turkish site NIL % rain for all 24 hours but still predicts 10% chance.

@mrburns exactly that is what I mean to say. You could even have 2.4% every hour (which rounded down jumps to 0%, as it goes in steps of 5%) and statistically have a chance for the whole day of as much of 44%.
if there is a chance of 2.4% of rain, it means it is 97.6 that it stays dry during that hour. If you have that during the whole day, the chance of staying dry would be 0.976 * 0.976 * 0.976 … (0.976 ^ 24) = 0.56 or 56%. Which means it 44% change of rain. Now that is just very theoretical but shows that daily chance can be way higher (and lower) than based on hourly chances. Also this assumes hourly chances are independent, which they aren’t, which will lower the daily chance.

Eric, That’s not how it works with predictions on a percentage of rain. You can’t add the hourly rates, you can’t average the hourly rates.

@mrburns i just realized that your example of 2 hours each with 5% chance do follow the math as if they were independent. The chance for the first hour to be dry is 0.95. the same for the second hour. The chance for both of them to be dry is 0.95 * 0.95 = 0.90. or 10% that it will rain during the day. Note that, as I described in another message, that this math doesn’t hold up in many cases because the events aren’t independent, so the 10% is only what one would naively expect, with 2 hours of each 5%

HI @sunny
Agree with your understanding but two things jump out:-
1)
Should it be mathematical (where the likelihood is little reliance is given to the days figure) OR an indicator of the maximum perceived chance of rain in an hour over the day. Using the maximum would allow the user to peruse the days likelihood of rain and determine from his / hers viewpoint what is good or bad and when and arrange their day accordingly.

Ultimate decision the user
2)
Still do not understand why if there are 24 hours at NIL displayed how come there is a 10% chance of rain. Is it a display rounding issue?

It would be good if WF could elaborate so we understand and can work with what ever evaluation is displayed

@mrburns

1. to summarize, the daily chance is NOT based on the hourly chance but based on the calculated model, a model that usually runs on very, very big computers. It simply isn’t averaging or taking the max from the hourly observations. There is a huge difference between a rainy cloud passing sometime during the day, and for example the air slowly becoming too saturated so it starts to rain.
2. I tried to explain that mathematically, for independent hourly events, having a chance of only 2.4% will give you a chance of a whopping 44% that it will rain during the day.

As an example if you had 2.4% chance of rain at 1:00, another 2.4% at 5:00, another 2.4% at 18:00 and another 2.4% at 23:00, those would all show up as 0% (because of the rounding to the nearest multiple of 5%), your chance of staying dry the whole day would be 0.976 * 0.976 * 0.976 * 0.976 = 0.907 or almost 91%. So there will be 9% chance of rain. Rounded to the nearest multiple of 5% gives 10%.

Perhaps compare it with the independent events of throwing a dice. The chance of throwing a six somewhere in a series of 24 throws is 99% even though for each throw the chance is only 16.667%.

But as mentioned this mathematics is just wrong, but gives you an idea of what might happen. The real daily chance is based on the complex simulation, not on hourly data.

Since no one really knows, best thing we can do is ask those that made it.

Let me try to find someone tomorrow since it is Sunday and Weatherflow staff is taking a nap (except those on deck)

but I know a bit of statistics/math.

HI @sunny I agree totally with everything you have written. I am an accountant by trade and can make statistics sing and dance. Beauty is in the eyes of the beholder - Its about relevancy, common sense and trusted perception by Joe Public.

Lets see what @eric questioning reveals.

There is some rounding going on, but @sunny is correct in general: The daily probability will almost always be higher than the hourly probability, but it is also not as simple as a textbook probability equation. The details are complicated (and some of them are part of our secret sauce), but we’ve posted an FAQ with a little more detail here:

This is likely the case of a very low hourly probability (1% or 2%) rounding to zero, while the daily probability value rounds to 10%.

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