We consider mean field fluctuations in globally coupled map xn+1(i) = f(xn(i)) + ϵ( 1 N) ∑ j=1 N xn(j) when the local map f has a flat power-law top, e.g. f(x) = 1 − a|x|β with β > 1. It is shown that in the thermodynamic limit N → ∞ the amplitude of mean field fluctuations is O(ϵ 1 (β−1)) for ϵ ⪡ 1 which is in good agreement with numerical calculations. We also demonstrate the relation between this problem and behaviour of averages as functions of the parameter a in 1D maps. For the latter, we give both theoretical grounds and experimental evidence that the “mean deviation” of an average value behaves as a power of the deviation of the parameter, e.g. ∫|〈x〉(a + δa) − 〈x〉(a)| da ∼ |δa| 1 β.