The climatological forecast for 1 July 2023 for some region is computed as follows, based on the meteorological record for that region and that date:
- The predicted high is the average high for that date from all of the meteorological stations in that area, possibly weighted by some weighting scheme.
- The predicted low is the average low for that date from all of the meteorological stations in that area, again possibly weighted by some weighting scheme.
- The chance of precipitation is the product of the percentage of past first of July records where any weather station in the area measured a significant amount of precipitation multiple by the percentage of weather stations that did observe precipitation on those days.
The question then becomes how skillful is a forecast based on current and recent meteorological data compared to the climatological forecast? If it's a coin toss which of the two will be closer to observation, one might as well use the climatological forecast. This coin toss boundary currently is about where a ten days. The coin toss boundary used to be at seven days. Before then, it was four or five days, before that, just two or three days, then just one day. Before that, it was weather adages such as "red sky at night, sailors' delight," which itself is a bit of a coin toss.
A more qualitative approach is a skill score. For example, suppose the meteorological forecast for the high temperature for some day in the future is $H_f$ (f for forecast) while the climatological forecast is $H_r$ (r for reference). Wait until that day passes and note the actual high temperature $H_p$. The skill score for the meteorological forecast is $$SS_H = \frac{H_f - H_r}{H_p - H_r}$$
There are multiple schemes by which skill scores for multiple scores, multiple metrics, and multiple regions are combined. Weather forecasts are a bit more skillful at predicting high and low temperatures than they are at predicting chance of precipitation. I'm not sure what combination scheme is used in determining that the ten day forecast remains a coin toss compared to the climatological forecast.
References:
Murphy, Allan H., and Edward S. Epstein. "Skill scores and correlation coefficients in model verification." Monthly weather review 117.3 (1989): 572-582.
Wheatcroft, Edward. "Interpreting the skill score form of forecast performance metrics." International Journal of Forecasting 35.2 (2019): 573-579.