Stochastic errors were found to propagate through the Kramers-Kronig relations in such a manner that the respective standard deivations of the real and imaginary components of complex quantities at any given frequency are equal. The only requirements were that the errors be stationary in the sense of replication at each measurement frequency, that errors be uncorrelated with respect to frequency, that the derivative of the variance with respect to frequency exists, and that the Kramers-Kronig relations be satisfied. Experimental results for elecrochemical and rheological systems are presented which support the conclusion that the real and imaginary components have the same variance. This result and the conclusions reported herein appear to be general and should apply for any physical system in which the real and imaginary components are obtained from the same measurement.