We present the first large-scale empirical examination of the relation of molecular chemical potentials, mu(0)(mol) = -1/2 (I-0 + A(0))(mol), to the geometric mean (GM) of atomic electronegativities, (GM) = < 1/2 (I-0 + A(0))(at)>(GM), and demonstrate that mu(0)(mol) not equal -(GM). Out of 210 molecular mu(0)(mol) values considered more than 150 are not even in the range min{mu(0)(at)} < mu(0)(mol) < max{mu(0)(at)} spanned by the mu(0)(at) not equal -chi(0)(at) of the constituent atoms. Thus the chemical potentials of the large majority of our molecules cannot be obtained by any electronegativity equalization scheme, including the "geometric mean equalization principle", 1/2 (I-0 + A(0))(mol) = < 1/2 (I-0 + A(0))(at)>(GM). For this equation the root-mean-square of relative errors amounts to SE = 71%. Our results are at strong variance with Sanderson's electronegativity equalization principle and present a challenge to some popular practice in conceptual density functional theory (DFT). The influences of the "external" potential and charge dependent covalent and ionic binding contributions are discussed and provide the theoretical rationalization for the empirical facts. Support is given to the warnings by Hinze, Bader et al., Allen, and Politzer et al, that equating the chemical potential to the negative of electronegativity may lead to misconceptions.