  Measuring the correlation (association) between two random variables is one
of the important goals in statistical applications. In the literature, the
covariance between two random variables is a widely used criterion in measuring
the linear association between two random variables. In this paper, first we
propose a covariance based unified measure of variability for a continuous
random variable X and we show that several measures of variability and
uncertainty, such as variance, Gini mean difference, cumulative residual
entropy, etc., can be considered as special cases. Then, we propose a unified
measure of correlation between two continuous random variables X and Y, with
distribution functions (DFs) F and G, based on the covariance between X and
H^{-1}G(Y) (known as the Q-transformation of H on G) where H is a continuous
DF. We show that our proposed measure of association subsumes some of the
existing measures of correlation. Under some mild condition on H, it is shown
the suggested index ranges between [-1,1] where the extremes of the range,
i.e., -1 and 1, are attainable by the Frechet bivariate minimal and maximal
DFs, respectively. A special case of the proposed correlation measure leads to
a variant of Pearson correlation coefficient which, as a measure of strength
and direction of the linear relationship between X and Y, has absolute values
greater than or equal to the Pearson correlation. The results are examined
numerically for some well known bivariate DFs.
