For σ2(n−1)S2∼Xdf=n−12 Proof Note that σ2∑i=1n(xi−μ)2=σ2∑i=1n(xi−x)2+σ2n(x−μ)2 Examples With V[χ(m)2]=2m Then, V[σ2(n−1)S2]=2(n−1) ⟹V[S2]σ4(n−1)2=2(n−1)] ⟹V[S2]=n−12σ4