Given an expected returns vector and a covariance matrix, one can perform a joint draw and measure the average cross-sectional variation as the standard deviation across returns for a particular joint draw.
Demonstrating the same idea using empirical/historical data, the cross-sectional variation is simply the standard deviation across returns at a point in time. For some intuition, here's a chart plotting cross-sectional dispersion vs. the VIX from a paper by Gorman, Sapra, and Weigand:
Since many shops have a well-designed covariance matrix, rather than looking at the empirical metric to measure dispersion which is noisy and time-varying, I'd rather produce the dispersion metric from an already existing covariance matrix.
What is the analytical relationship between a given covariance matrix and expected returns vector (e.g., a multivariate normal distribution) and the expectation of the cross-sectional dispersion?