Dividing by n − 1, n or n + 1

Idea. Dividing by more than $n-1$ adds a little bias but removes more variance. For normal data the best trade-off is at $n+1$.

Step 1: the law of $S$. For normal data, $Q=S/{\sigma }^2\sim {\chi }_m^2$ with $m=n-1=9$. So $\mathbb{E}[Q]=m$ and $\mathbb{E}[Q^2]=\mathrm{Var}(Q)+m^2=2m+m^2$.

Step 2: the MSE. With $c=1/d$, $$\begin{array}{rl}\frac{\text{MSE}}{{\sigma }^4}& =\mathbb{E}[(cQ-1{)}^2]\\ & =c^2(m^2+2m)-2cm+1.\end{array}$$

Step 3: minimise. The derivative in $c$ vanishes at $$c=\frac{m}{m^2+2m}=\frac{1}{m+2},$$ so $d=m+2=n+1=11$.

Comparison. $d=9$ is unbiased, $d=10$ is the maximum likelihood estimator, and $d=11$ has the smallest MSE. In units of ${\sigma }^4$ their MSEs are $\frac{2}{9}\approx 0.222$, $\frac{19}{100}=0.19$ and $\frac{2}{11}\approx 0.182$. Unbiasedness is not the same as accuracy.