Gaussian and Gamma approximations appear very frequently in communications theory applications. However, in most of the applications, numerical methods are used to illustrate the accuracy of such approximations as there are no simple general purpose analytical tools available to provide a more rigorous assessment of the error. We present a simple and accurate tool to measure the accuracy of Gaussian and Gamma approximations. Specifically, we consider the first order correction terms of the approximations and derive their upper bounds. Our tool provides an accurate approximation of the error term based on the third and the fourth moments of the random variable. To demonstrate the usefulness of our analysis we consider two topical applications based on mmWave channel modeling and distributed massive multiple-input-multiple-output systems. In both cases, the proposed technique provides valuable insights into the accuracy of the commonly used approximations.