We identify and correct excess dispersion in the leading eigenvector of a sample covariance matrix, when the number of variables vastly exceeds the number of observations. Our correction is data-driven, and it materially diminishes the substantial impact of estimation error on weights and risk forecasts of minimum variance portfolios. We quantify that impact with a novel metric, the optimization bias, which has a positive lower bound prior to correction and tends to zero almost surely after correction. The sample eigenvalues are used to correct excess dispersion in the leading eigenvector. However, the sample eigenvalues have no direct bearing on large minimum variance portfolios: correcting the sample eigenvalues to their population counterparts does nothing to diminish the optimization bias.