On the uncertainty of the mean of digitized measurements

摘要

Two equations relating to the effect of finite resolution on the uncertainty of a sample mean give the impression of being contradictory. One suggests that knowing the data to be digitized increases uncertainty while the other might be thought to imply the opposite. This issue is clarified and relevant concepts are discussed. This discussion leads to the development of a new approximation relating to the uncertainty of an estimate obtained by taking the mean of a set of digital measurement results. This result is consistent with others published recently in implying that resolution error can usually be ignored in such a situation. The solution to this problem obtained in a Bayesian analysis with the usual non-informative prior distribution is shown to give an inappropriate result when all the observations are equal and to exhibit anomalous behaviour in certain other circumstances. One of the appendices describes an estimator that has smaller mean-square error than the sample mean when averaging over the set of measurement problems of this type.