This paper discusses the problem of ordering n jobs on a single machine with the objective of minimizing the variance of completion times. It is shown that for n> 18 jobs, there exits an optimal solution in which the job with the third largest processing time is always at the second position. Using the techniques of partitioning of variances 1 a general formula is given for the change in variance due to the interchange of two jobs whose positions hav nt been fixed in the sequence. Optimal sequences are obtained for n=6 and n=7. The results are extended to the random case when the processing times are random variables and the objective is to minimize the expected variance of the completion times.