If noise is uncorrelated during repeated measurements of the same physical variable, averaging these measurements improves the accuracy of estimating the variable. When two values of a variable are measured separately, the smallest separation of these two values that can be discriminated with a certain reliability (resolution) is inversely proportional to the square root of the number of measurements employed. However, if measurements for these two values are mixed together, they need to be clustered before being averaged. Distinguishing mixed clusters with small separations can. be thought of as a problem of deciding the number of components in a finite mixture model. Using the likelihood ratio, the second-moment estimator, and the k-means clustering methods, we will show that a similarly defined resolution for the mixed scenario is, approximately, inversely proportional to the fourth-root of the number of measurements. The observed fourth-root law is explained in terms of some more intuitive properties of the problem. We also conclude that, assuming that the fourth-root law is universal, the methods reported here are near-optimal.