Let be a C
*-algebra, T be a locally compact Hausdorff space equipped with a probability measure P and let (A
t)
tT be a continuous field of operators in such that the function tA
t is norm continuous on T and the function tA
t is integrable. Then the following equality including Bouchner integrals holds
∫TAt-∫TAsdP2dP=∫TAt2dP-∫TAtdP2.This equality is related both to the notion of variance in statistics and to a characterization of inner product spaces. With this operator equality, we present some uniform norm and Schatten p-norm inequalities.