ABSTRACT Scale equivariant estimators of the common variance σ2, of correlated normal random variables, have mean squared errors (MSE) which depend on the unknown correlations. For this reason, a scale equivariant estimator of σ2 which uniformly minimizes the MSE does not exist. For the equi-correlated case, we have developed three equivariant estimators of σ2: a Bayesian estimator for invariant prior as well as two non-Bayesian estimators. We then generalized these three estimators for the case of several variables with multiple unknown correlations. In addition, we developed a system of confidence intervals which produce the desired coverage probability while being efficient in terms of expected length.