The transformation of random noise present in impedance spectroscopy data by the important Kronig-Kramers integral-transform relations is investigated analytically. It is found that the standard deviation of the transformed noise may be smaller than the input noise under certain conditions. The output noise standard deviation depends critically on the details of the numerical quadrature procedure used for the Kronig-Kramers transformations, so the effects of several different numerical integration routines are investigated. In most cases of interest, it is proved that the standard deviation of the output noise is equal to that of the input noise, in agreement with earlier Monte Carlo results. It has been found possible not only to derive expressions for the limiting standard deviation of the transformed noise for several different integration procedures but also to obtain analytic expressions for their statistical distributions in the limit of an infinite number of discrete integration points. Finally, it is demonstrated that the integration routine may be adjusted to obtain either very small integration errors (in the absence of input noise) or smaller output noise with larger integration error.