Revisiting the Benford law: When the Benford-like distribution of leading digits in sets of numerical data is expectable?

The Benford law states that the frequencies of decimal digits at the first place of numbers corresponding to various kinds of statistical or experimental data are not equal changing from 0.3 for 1 to 0.04 for 9. The corresponding frequencies’ distribution is described by the logarithmic function. As is shown in the present article, the Benford distribution is a particular case of a more general mathematical statement. Namely, if a function describing the dependence between two measurable quantities has a positive second derivative, then the mentioned above frequencies decrease for digits from 1 to 9. The exact Benford distribution is valid for the exponential function only. Explicit expressions for frequencies of leading digits are obtained and specified for the power, logarithmic, and tangent functions as examples. The kinematic experiment was performed to illustrate the above results. Also the tabulated data on thermal conductivities of liquids confirm the proposed formula for frequencies’ distribution. •The Benford distribution is a particular case of a more general mathematical statement.•The exact Benford distribution is valid for the exponential function only.•Explicit expressions for frequencies are obtained for the power, logarithmic, and tangent functions.•The kinematic experiment illustrating proposed distributions is reported.•Analysis of leading digits frequencies in thermal conductivities of liquids data is presented.