Benford's Law and Average Significands

Abstract: Sum invariance property and average significands are narrow connected, based on properties of logarithmic function. Theoretical values of average significands, together with theoretical values of minimal and maximal significands, are important for testing purposes. Theoreticaly, sum of significands with same leading digits is product of Benford's probability and average significand. By this it's possible to analyse goodness-of-fit for both values, specialy in big samples. For example, if sample sum of significands has a good fit with theoretical values sample average significand can significantly deviate from theoretical value, what can point to some kind of anomaly.