Originally Posted by Fat Antar
Also, could someone with a background in statistics could tell me how to calculate "margin of error" for this kind of data?
You could view these battles with one specific match-up as a set of tests you are running, and specific outcomes can be chosen as either "success" or "failure" (at countering the pokemon, that is). If you do this, it would be quite easy to give you a decent estimate of the success-rate + a margin of error (depending to the level you significance you choose)with a "test on a percentage"; However, this isn't perfect of course since forcing something out is "more success" than outright killing it, while stuff like double switches /double kills have to be neglected. But it'll be most likely be usable.
The other thing you could do would assign numbers to every outcome( for example 6 for "killing" , 4 for "opponent forced out" 3 for "opponent switched out" and vice versa -6 for " killed" usw). Then, you could run a "t-test" on all battles with a specific match-up and it will give you estimate of the average outcome + a margin of error(still depending to the level you significance you choose). Than, you have to decide how to use the number you got (for example in clinical studies, you would choose a critical point, and then look whether your average outcome minus the margin is higher than it or not; If yes, it's considered "good enough", if not, not)
This, on the other hand, can be very biased depending on how you assign the numbers and how you choose to process these numbers, so I wouldn't propose this way - you could end up with everything being a counter to everything, stuff like that. Though, if you are determined at spending much time with it and trying out different ways of assigning & processing, it COULD get better than the other test.
On a sidenote:
On lesser used pokemon, the margin of error will most probably be so big that everything you do is most likely rubbish. As a rule of thumb, you should use these statistics only if n*p*(1-p)>9 applies, with "n" being the number of battles against one specific pokemon and "p" being the estimated success rate. So, if you estimate the success-rate at 75%, n should be greater than around 50 and so on.
I could either try to describe these specific test statistics to you or do it myself(if I find the time, though), but describing is somewhat difficult because I'm no native speaker and writing about mathematics is especially difficult in a language you didn't learn it in since you don't know which words are commonly used in respective professional circles.