The average of a function over a set of points is the arithmetic mean of the values of that function at those points. The averaging spectrum is a limit set of average values over periodic pseudo-trajectory. The averaging spectra of a function consist of intervals, each generated by a chain-recurrent set component. The symbolic image allows us to approximate the averaging spectrum. It turns out that the averaging coincides with an averaging over invariant measures, and the relationship between these two concepts is explored. The relationship between averaging, recurrence, and ergodicity is investigated.