The usual confidence interval for the variance $\sigma^2$ of a normal distribution is a function of the sample variance alone. In this paper, we construct confidence intervals for $\sigma^2$ that also depend on the sample mean. These intervals have the same length as the shortest interval depending only on the sample variance and have uniformly higher probability of coverage. The coverage probabilities of these intervals and others are compared.
"Improved Confidence Intervals for a Normal Variance." Ann. Statist. 18 (2) 972 - 980, June, 1990. https://doi.org/10.1214/aos/1176347636