Despite the first sentence of this post, I don’t think it’s actually about probabilities. The same questions arise when you have any other sort of number to report.
The answer seems to me to be composed of one kinda-obvious part and one kinda-impossible-to-determine part.
The obvious part is that unless for some reason you’re deliberately deceiving, you should do your best to convey the information you have, which includes both your best estimate of the number and how much you think you know about it (really it’s something like a probability distribution over the possible values of the number) within whatever constraints you have—e.g., on the attention span of whoever you’re reporting the number to, or your insight into your beliefs.
The kinda-impossible part is figuring out how those considerations actually trade off. Usually you will have limited resources, limited insight into what you actually believe, an audience with limited patience, a social context in which numbers with lots of nonzero digits in them are taken as implicit claims to detailed knowledge, etc., and exactly what that means for how you should report the numbers is going to be (1) different each time, as all those factors vary, and (2) very difficult to determine.
If you’re dealing with a fairly technical audience, or one strongly motivated to pay attention to the details of what you say, I think it should be OK to say things like “31.5% +- 2.5%”. Otherwise, I suspect there usually is no way to avoid their understanding being seriously deficient, and you get to choose between saying 31.5% and misleading them about your confidence, and saying 30% and misleading them about your best point estimate.
Despite the first sentence of this post, I don’t think it’s actually about probabilities. The same questions arise when you have any other sort of number to report.
The answer seems to me to be composed of one kinda-obvious part and one kinda-impossible-to-determine part.
The obvious part is that unless for some reason you’re deliberately deceiving, you should do your best to convey the information you have, which includes both your best estimate of the number and how much you think you know about it (really it’s something like a probability distribution over the possible values of the number) within whatever constraints you have—e.g., on the attention span of whoever you’re reporting the number to, or your insight into your beliefs.
The kinda-impossible part is figuring out how those considerations actually trade off. Usually you will have limited resources, limited insight into what you actually believe, an audience with limited patience, a social context in which numbers with lots of nonzero digits in them are taken as implicit claims to detailed knowledge, etc., and exactly what that means for how you should report the numbers is going to be (1) different each time, as all those factors vary, and (2) very difficult to determine.
If you’re dealing with a fairly technical audience, or one strongly motivated to pay attention to the details of what you say, I think it should be OK to say things like “31.5% +- 2.5%”. Otherwise, I suspect there usually is no way to avoid their understanding being seriously deficient, and you get to choose between saying 31.5% and misleading them about your confidence, and saying 30% and misleading them about your best point estimate.