Closed forms all the way up: the sample variance of four observations

Source abstract

The distribution function $F(y)=\Pr(S^2\le y)$ of the sample variance $S^2$ of four independent observations is elementary on its entire support for every polynomial parent density on $[0,1]$ (without loss of generality), and we give it in finite closed form for the uniform, Beta$(2,1)$ and Beta$(2,2)$ parents. The support splits at $y = 1/6$, $2/9$ and $1/4$, with the ceiling at $1/3$, all fixed by geometry: in coordinates aligned with the cube diagonal the event $S^2\le y$ is a ball of radius $\sqrt{3y}$, the cube projects to a rhombic dodecahedron, and these are the levels at which the ball reaches its faces, edges and two kinds of vertices. The divergence theorem reduces every moment of the clipped ball to integrals along circular arcs, so each regime closes in algebraic functions of $\sqrt{y}$ and inverse trigonometric functions. For the uniform parent the only non-obvious constant is $K_g = 12\sqrt2\arctan\sqrt2$, proportional to the mean width of the cuboctahedron; the Beta constants are rational expressions in $π$, $\sqrt2$, $\sqrt3$ and $\arctan\sqrt2$, as in the uniform case. At $n = 4$ the normal-theory $χ^2$ recipe used in acceptance testing misplaces the $99$th percentile of $S^2$ by $50\%$ for the uniform parent. For three observations, polynomial parents also give elementary closed forms (arXiv:2609.25333), but two elementary cases in a row do not guarantee a third, and we make no claim beyond $n = 4$.