A procedure for combining averages, each estimating the same parameter, is proposed. If the averages arise from a multivariate normal sample, the procedure yields the maximum likelihood estimate of the parameter. It is shown that providing the number of vector observations is greater than the number of averages being combined, exact confidence intervals can be obtained based either on Student's “t” distribution or a related distribution (unfortunately not tabulated). If the averages arise from unequal sized samples on independent normal distributions with unequal variances, similar results hold subsequent to random matching and data transformation providing the minimum sample size is greater than the number of averages being combined. In both cases, the proposed estimate is almost linearly optimum in the sense that it has variance equal to that of the minimum variance linear unbiased estimate of the parameter except for a multiplicative factor which approaches unity as all sample sizes become large.