For the unit circle there are several forms of the Heisenberg-Weyl inequality. The starting point for this paper is the following: Let \(f \in L^2(T)\) have Fourier coefficients \((c_k)\). The frequency variation \(\text{var}_F(f)\) is given by \[ \text{var}_F(f) = \sum_{k=- \infty}^\infty k^2|c_k|^2 - \Biggl( \sum_{k=-\infty}^\infty k|c_k|^2 \Biggr)^2, \] and the angular variance \(\text{var}_A(f)\) by \[ \text{var}_A(f) = {{1-|\tau (f)|^2} \over {|\tau (f)|^2}}, \] where \(\tau (f) = \int_Tz.|f(z)|^2dw(z)\). Then the appropriate inequality is \[ \text{var}_A(f)\text{.var}_F(f) \geq {\textstyle{1 \over 4}}. \] Later it was shown by Prestin and Quak that the constant \({1 \over 4}\) is sharp. The corresponding inequality for the unit sphere \(S^2\) in \(R^3\) was proved by Narcowich and Ward in the form \[ (1-|\tau(f)|_2^2)\text{.var}_F(f) \geq |\tau(f)|_2^2, \] with the analogous definitions of \(\text{var}_F(f)\) and \(\tau (f)\). The purpose of this paper is to prove that this inequality is also sharp and give the corresponding results for \(SO(n)\)-invariant functions on the sphere \(S^n\) in \(R^{n+1}\). The details are technical.