Given paired observations, {(xi,yi);i=1,2,…,n} on two variables x and y for a random samples, from some bivariate non-normal population like bivariate gamma, beta-Stacy which are of much use in modeling data obtained in Physical, Social and Life-sciences. This paper considers an improvement of the customary estimator of population variance. A mixture (i.e. a weighted combination) of the customary estimator of the variance and a suitably chosen statistic t is proposed. It is also indicated that under some conditions for a broad range of the values of the mixing constants, the improvement in the sense of a having a smaller mean square error, over the traditional estimator is possible.