We study Dedekind sums S(a,b) for arguments a near Farey points of the interval [0, b]. The Petersson-Knopp identity connects each of these Dedekind sums with a set of other Dedekind sums. In the case considered here, this identity has a very specific interpretation, inasmuch as each Dedekind sum occurring in this identity is close to a certain expected value. Conversely, each of these expected values occurs with a certain frequency, a frequency that is consistent with the Petersson-Knopp identity.