In other words, you assume that every PA has the exact same underlying probability of a walk.

However, if Grilli has a man on 2B and Ryan Howard up to bat, the true probability of a walk increases and if he has bases loaded with AJ Pierzynski up to bat, the true probability of a walk decreases.

When the true probability is fluxuating, your estimate for the binomial variance given X number of PA is going to vastly underestimate the expected variance over X number of PA. Thus, using the formula you used to show that his 2011 and 2012 are significantly different from his 40%+ walk rate pre-2010 is incorrect.

]]>Good article, I think Grilli if healthy can hold down the spot for the Bucs. He had a better season than Hanrahan anyway.

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