Given N points \(x_ 1,x_ 2,...,x_ N\) on a unit sphere \(S^{d-1}\) in Euclidean d-space (d\(\geq 3)\), we investigate the \(\alpha\)-sum \(\sum ^{N}_{j=1}| x-x_ j| ^{\alpha},\quad \alpha >1-d,\) of their Euclidean distances from a variable point x on \(S^{d-1}\). We obtain an essentially best possible lower bound for the L 1-norm of the deviation of the \(\alpha\)-sum from its mean value. As an application, we prove similar bounds for the \(\alpha\)-sums \(\sum _{j\neq k}| x_ j- x_ k| ^{\alpha}\) of mutual distances. In order to illustrate the nature of the results, let us give two examples. For \(d=3\) and \(\alpha =- 1\), we prove \[ \| \sum ^{N}_{j=1}| x-x_ j| ^{-1}-N \| _ 1\geq c_ 1\sqrt{N}\quad and\quad \sum _{j\neq k}| x_ j- x_ k| ^{-1}-N(N-1)\geq -c_ 2N\sqrt{N}, \] where \(c_ 1\), \(c_ 2\) denote absolute positive constants.