What does the weather percentage mean?

No it's not the percentage of area that gets rain

The answer is quite simple actually, the percentage reflects the fraction of simulations that had rain in your area. But what does that mean?

What simulation?

A lot of the mathematics and measurements required for weather forecasting has been around since the 18th century, including calculus, the relevant gas laws and the basics of thermodynamics. The actual Navier-Stokes equation describing the movement of liquids (gases are basically liquids) took a bit longer until 1850s to come around, although it would not be used for forecasting until later!

The main thing missing was data, which is why the first modern methods relied on the telegraph to get weather information, thus allowing you to estimate the weather upwind from any nearby weather station. The first people credited for creating forecasting as a science were Francis Beaufort and Robert FitzRoy (of HMS Beagle fame and also first Governor of New Zealand). The latter also coined the term weather forecast. All this new data allowed for the first weather maps to be made to depict pressure, wind, temperatures and importantly for us rain. All these weather maps also lead to new discoveries and theories, including the discovery of cyclones by William Redfield.

The first person to propose solving the Navier-Stokes equation for weather prediction was Lewis Fry Richardson in 1922. At the time computers were still humans who compute and so rather unfeasible, but Richardson proposed that

After so much hard reasoning, may one play with a fantasy? Imagine a large hall like a theatre, except that the circles and galleries go right round through the space usually occupied by the stage. The walls of this chamber are painted to form a map of the globe. The ceiling represents the north polar regions, England is in the gallery, the tropics in the upper circle, Australia on the dress circle and the Antarctic in the pit.

A myriad computers [people who compute] are at work upon the weather of the part of the map where each sits, but each computer attends only to one equation or part of an equation. The work of each region is coordinated by an official of higher rank. Numerous little "night signs" display the instantaneous values so that neighbouring computers can read them. Each number is thus displayed in three adjacent zones so as to maintain communication to the North and South on the map.

Needless to say, this did not happen. However, as a proof of concept he did retroactively forecast the weather for a single day (20 May 1910) by direct computation. He did all of the computations by hand while serving with the Quaker ambulance in France. His forecast turned out to have failed dramatically, though mostly due to the lack of preprocessing his data.

The first forecasts to be produced by an actual computer was run on the ENIAC in the 1950s, based on simplified approximations of weather equations. This was first put into routine use by Carl-Gustaf Rossby in 1954 in Sweden. Eventually, these efforts along with work of the US Air Force, Navy and Weather Bureau were combined to create the first general circulation climate model at what would later become the National Oceanic and Atmospheric Administration (NOAA), which to this day provides the weather forecasts for all weather apps in the US and also does other very cool stuff!

Why run multiple simulations?

In 1961, Edward Norton Lorenz was using a computer to simulate a basic weather model. To double check his solution he wanted to run the simulation again, this time entering the printed variables from the middle of the simulation. He found that this time he got a completely different answer. It turns out that this was due to rounding in his inputs, the computer he was using worked with 6-digit precision, but only printed 3-digit numbers (0.539435 → 0.539). Although the input difference is tiny, the output was dramatically different. This lead to the discovery of chaos theory. As Lorenz put it:

Chaos: When the present determines the future, but the approximate present does not approximately determine the future.

Although a lot better than in the 1960s, our modern day measuring devices are still not perfectly precise. To deal with this we run the simulations many times, each time perturbing the input data slightly. Then, if say rain is predicted in 60/100 of the simulations we say there is a 60% chance of rain. This method is known as ensemble forecasting. This is a massively oversimplified explanation, glossing over the complexity of data collection and post-processing.

The thing about area

As with most urban myths or misconceptions, there is a kernel of truth. The myth goes as follows: if your chosen app displays 60% chance of rain, that actually means that there is a 100% chance of rain, but it will only rain in 60% of the given area. While this is false, it is also kind of approximately true as a coincidence of mathematics.

Rain is an extremely local phenomenon that is hard to capture with the kinds of resolution that standard weather models use. The smallest grid size is usually used for short term predictions, which is around a square kilometer. Furthermore, if a given rain cloud will actually start to rain within a given hour is unclear.

This uncertainty can be modeled naively with a Poisson point process which randomly scatters points across a given area. If we now imagine a larger city that is made up of multiple square kilometers and we give 60% chance of rain for the whole city, whereas in truth there was a 60% chance of rain for each individual square kilometer, we can indeed expect that roughly 60% of the area of the whole city will receive rain. This is a question of how weather stations accumulate the probabilities of the forecasts they received.

Your weather app is lying to you!

Humans are terrible at intuitively judging probabilities. Although they can get better through practice. Thus consumer weather apps don’t optimize to make you smarter, but to make you happy. People don’t like rain. They really don’t like unexpected rain. So weather stations tend to be slightly more pessimistic when broadcasting forecasts, so that people will be better prepared and more likely to be positively surprised.

Your takeaways should be that actual weather predictions are very good and that it took a lot of time and brainpower to make them so. Weather apps are lobbying government to disallow NOAA to give you weather information directly! Apple also does not show 69F which is mildly funny.