Thursday, April 09, 2015
Expectation from a random drawing
?Say you have a uniform distribution, like rolling a single dice (*). Whether you throw a 2 or 3 or 6, all are equally likely. The AVERAGE throw will be 3.5. If you happen to throw a 6, was that "2.5" more than expected? Is throwing a 4 only 0.5 more than expected? Or, was 6 simply as expected as anything else, and you wouldn't describe it as anything other than that, and you certainly wouldn't try to differentiate throwing a 6 from throwing a 4?
(*) Die is a stupid word. It's a dice.
Now, instead of a uniform distribution, you have a binomial distribution of observations for a single entity. And you randomly draw from that distribution. Again, would you say that randomly drawing a number from the tail is "higher" than expected than drawing from near the midpoint? Or, is it simply comparing one random draw to another random draw? That we don't need to "compare" each of the drawn samples to the midpoint. It's all random anyway, so, we're bound to have one sample further away from the middle than another sample. None of that tells us anything, once we've established it's random.
So, suppose that the league BABIP is .300, and there is no such thing as a talent at BABIP. If we observe one pitcher at .340 and another at .270, do we need to say that one is +.040 and another is -.030? Or, can we say that compared to the DISTRIBUTION of all possible values, .340 was just one random sample, and .270 is another random sample of the same distribution. And so, since both are part of the same distribution, we don't need to worry about representing them as anything other than a random variation of the same talent. And so, once you account for the fact that the .340 must have benefited from bad luck and the .270 pitcher benefited from good luck, they are both exactly the same thing.
Basically, it doesn't make sense to compare an observation to the average, without accounting for the random variation.
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