In an election held in a noisy environment, agents may unintentionally perturb the outcome by communicating faulty preferences. We investigate this setting by introducing a theoretical model of noisy preference aggregation and formally defining the (worst-case) robustness of a voting rule. We use our model to analytically bound the robustness of various prominent rules. The results show that the robustness of voting rules is diverse, with different rules positioned at either end of the spectrum. These results allow selection of voting rules that support preference aggregation in the face of noise.