This talk concerns a somewhat innocent question motivated by an observation concerning the use of Chebyshev bounds on sample estimates of p in the binomial distribution with parameters n, p. Namely, what moment order produces the best Chebyshev estimate of p? If Sn(p) has a binomial distribution with parameters n, p, then it is readily observed that argmax0 ≤ p ≤ 1ESn2(p) = argmax0 ≤ p ≤ 1np(1-p) = 1/2, and ESn2 = n/4. Rabi Bhattacharya observed (personal communication) that while the second moment Chebyshev sample size for a 95% confidence estimate within
±5 percentage points is n = 2000, the fourth moment yields the substantially reduced polling requirement of n = 775. Why stop at fourth moment? Is the argmax achieved at p = 1/2 for higher order moments and, if so, does it help, and can one easily compute the moment bound ESn2m(1/2)? As captured by the title of this talk, answers to these questions lead to a simple rule of thumb for best choice of moments in terms of an effective sample size for Chebyshev concentration inequalities. This talk is based on joint work with Chris Jennings-Shafer and Dane Skinner.