We discuss minimum mean squared error and Bayesian estimation of the variance and its common transformations in the setting of normality and homoscedasticity with small samples, for which asymptotics do not apply. We show that permitting some bias can be rewarded by greatly reduced mean squared error. We apply borderline and equilibrium priors. The purpose of these priors is to reduce the onus on the expert or client to specify a single prior distribution that would capture the information available prior to data inspection. Instead, the (parametric) class of all priors considered is partitioned to subsets that result in the preference for different actions. With the family of conjugate inverse gamma priors, this Bayesian approach can be formulated in the frequentist paradigm, describing the prior as being equivalent to additional observations.