  Given a function $f\in L^2(\mathbb R)$, we consider means and variances
associated to $f$ and its Fourier transform $\hat{f}$, and explore their
relations with the Wigner transform $W(f)$, obtaining a simple new proof of
Shapiro's mean-dispersion principle. Uncertainty principles for orthonormal
sequences in $L^2(\mathbb R)$ involving linear partial differential operators
with polynomial coefficients and the Wigner distribution, or different Cohen
class representations, are obtained, and an extension to the case of Riesz
bases is studied.
