A new quantity for statistical analysis: "Scaling invariable Benford distance"

For the first time, we introduce “Scaling invariable Benford distance” and “Benford cyclic graph”, which can be used to analyze any data set. Using the quantity and the graph, we analyze some date sets with common distributions, such as normal, exponent, etc., find that different data set has a much different value of “Scaling invariable Benford distance” and different figure feature of “Benford cyclic graph”. We also explore the influence of data size on “Scaling invariable Benford distance”, and find that it firstly reduces with data size increasing, then approximate to a fixed value when the size is large enough.