I have a bag of unfair coins. Some will come up heads half the time, some less, some more. I don't know if they're wildly spread out, usually stay between, say, 45% and 55%, or what.
I take a good handful of the coins and I flip each of them several times. I'm not systematic, though. So coins don't all have the same number of flips. But at least for each coin I flipped, I know how many heads and how many tails I got.
I want to know just how bad the whole bunch of coins are. Or at least, the best guess I can make from the data I have. If I took the whole bag and flipped every coins millions of times, I could easily calculate the standard deviation of heads probability among the coins. But what's the best I can do from the limited information I actually have?