Say we have a 3-4 team. Their observed (mean) record is 3 / 7 = 0.429, and the binomial standard deviation of that is (sqrt((W+L)*WPct*(1-WPct)))/(W+L) = (sqrt((3+4)*0.429*(1-0.429)))/(3+4) = 0.187. Since we’re regressing halfway to the mean, we’ll use a 0.500 WPct as the Bayesian prior mean, with a standard deviation of 0.15 (aka the standard deviation of true NFL winning percentage talent that we derived in the post).

Bayes’ Theorem states that:

Result_mean = ((prior_mean/prior_stdev^2)+(observed_mean/observed_stdev^2))/((1/prior_stdev^2)+(1/observed_stdev^2))

Plugging in the means and standard deviations we found above, we get:

Result_mean = ((0.5/0.15^2)+(0.429/0.187^2))/((1/0.15^2)+(1/0.187^2))

Which equals… 0.472. Or, exactly the same “true” WPct talent we found via (W + 5.5) / (G + 11).

Pretty cool, right?