Solution
Variance of the Sample Variance of Dice
Show the problem again
Three fair six-sided dice are rolled independently. Let S² = (1/2) Σ (Xᵢ − X̄)² be the usual sample variance with the n − 1 denominator. Find Var(S²) exactly, as a reduced fraction.
Worked solution
For i.i.d. data with variance σ² and fourth central moment μ₄, Var(S²) = (1/n)[μ₄ − ((n − 3)/(n − 1)) σ⁴]. For n = 3 the correction term vanishes and Var(S²) = μ₄/3. For a fair die, σ² = 35/12 and μ₄ = (1/6) Σ (k − 7/2)⁴ = (1/6)(2)((1/2)⁴ + (3/2)⁴ + (5/2)⁴) = (1/3)(1 + 81 + 625)/16 = 707/48. Hence Var(S²) = 707/144, about 4.91. Complete enumeration of the 216 outcomes confirms it, and also E[S²] = 35/12, as unbiasedness requires. The answer is 707/144.
Source: Original problem written for AxiomIQ; uses the exact fourth-moment formula for Var(S²).