Tables of critical values defining an unbiased test are given for testing the null hypothesis $\sigma^2 = \sigma^2_0$ against the two-sided alternative hypothesis $\sigma^2 \neq \sigma^2_0$ where $\sigma^2$ is the variance of a normal population. Use of the tabulated values leads to the logarithmically shortest confidence limits for $\sigma^k, k > 0$. The critical values have been found to five significant figures for $\alpha = .01, .05, .10$ where $\alpha$ is the size of the critical region and for $\nu = 1(1)20, 24, 30, 40, 60, 120$ where $\nu$ equals the degrees of freedom of the chi-square distribution. A least squares equation is given which may be used to find the critical values when $\nu \geqq 10$ for $\alpha = .01, .05, .10$. Since submitting a revision of the present paper for publication, the article by Tate and Klett  appeared necessitating a second revision. An explanation of the overlap of the present paper with  is included in Section 5. In addition, a brief discussion of , which was called to the writer's attention by the editor, has been added in Section 5.
"Tables for Unbiased Tests on the Variance of a Normal Population." Ann. Math. Statist. 32 (1) 84 - 87, March, 1961. https://doi.org/10.1214/aoms/1177705141