This paper deals with the problem of bivariate regression where both variates are random variables having a finite number of means distributed along a straight line. A regression statistic is derived which is independent of change in scale so that a prior knowledge of the frequency distribution parameters is not required in order to obtain a unique estimate. The statistic is shown to be consistent. The efficiency of the estimate is discussed and its asymptotic distribution is derived for the case when the random variables are normally distributed. A numerical example is presented which compares the performance of the statistic of this paper with that of other commonly used statistics. In the example it is found that the method of estimation proposed in this paper is more efficient.