Let p ≥ 3 be a prime, S \subseteq \mathbb{F}_p^2 a nonempty set, and w:\mathbb{F}_p^2 \to R a function with supp w = S. Applying an uncertainty inequality due to András Bíró and the present author, we show that there are at most {1 \over 2}\left| S \right| directions in \mathbb{F}_p^2 such that for every line l in any of these directions, one has
\sum\limits_{z \in l} {w\left( z \right) = {1 \over p}\sum\limits_{z \in \mathbb{F}_p^2} {w\left( z \right),} }
except if S itself is a line and w is constant on S (in which case all, but one direction have the property in question). The bound {1 \over 2}\left| S \right| is sharp.
As an application, we give a new proof of a result of Rédei-Megyesi about the number of directions determined by a set in a finite affine plane.