  A possible drawback of the ordinary correlation coefficient $\rho$ for two
real random variables $X$ and $Y$ is that zero correlation does not imply
independence. In this paper we introduce a new correlation coefficient $\rho^*$
which assumes values between zero and one, equalling zero iff the two variables
are independent and equalling one iff the two variables are linearly related.
The coefficients $\rho^*$ and $\rho^2$ are shown to be closely related
algebraically, and they coincide for distributions on a $2\times 2$ contingency
table. We derive an orthogonal decomposition of $\rho^*$ as a positively
weighted sum of squared ordinary correlations between certain marginal
eigenfunctions. Estimation of $\rho^*$ and its component correlations and their
asymptotic distributions are discussed, and we develop visual tools for
assessing the nature of a possible association in a bivariate data set. The
paper includes consideration of grade (rank) versions of $\rho^*$ as well as
the use of $\rho^*$ for contingency table analysis. As a special case a new
generalization of the Cram{\'e}r-von Mises test to $K$ ordered samples is
obtained.
