Benford's Law by Kossovsky Alex Ely

Author:Kossovsky, Alex Ely.
Language: eng
Format: epub
Publisher: World Scientific Publishing Company

Figure 4.55 Distortion in Digits Absent Those Ubiquitous Anchors 0 and 1

First digits distribution for these 45 houses within the entire scheme is: {10/45, 0/45, 0/45, 0/45, 9/45, 8/45, 7/45, 6/45, 5/45} and calculated proportions are {22%, 0%, 0%, 0%, 20%, 18%, 16%, 13%, 11%}. Hence digit 1 still squeezed a narrow and unexpected victory, but a lot of focus is given to digit 5, standing at the second place in leadership, simply because ‘density’ starts with this digit! A clearer result would have been obtained had we assigned equal importance to all streets and performed a bit more complicated calculations as was done in the original Greek Parable. Now with the assumption of equal street importance, digit 5 earns its deserved status as having by far the most leadership, and digits distribution is {14%, 0%, 0%, 0%, 31%, 21%, 16%, 11%, 7%}. The similarity of either distribution above to the digital proportions of k/(x - 4) seen in Figure 4.54 is striking, and certainly expected. Without any benevolent intentions, the harsh and imposing king has provided us with a vivid flesh and blood narrative for this odd distribution. Clearly there is not even a resemblance to the logarithm for either proportion above, not even to Stigler’s Law. Absent here even that monotonically declining probabilities pattern. Such is the heavy price we must pay for blindly and mindlessly obeying the arbitrary decrees of the king and losing those crucial and natural anchors of 1 and 0.

On a more profound level, the failures of k/(x + 4) defined over (−3, 6) and of k/(x − 4) defined over (5, 14) to behave logarithmically emanate from the fact that their densities are not exactly inversely proportional to x. Such an exact relationship between x and its density height is unique to k/x distribution, and it constitutes an essential feature of its logarithmic behavior. For any k/x distribution, doubling x causes the density to be cut (exactly) in half. For example, for 0.4342945/x over (10, 100), density height or histogram count on x = 40 is exactly half the height or count on x = 20; while the two non-logarithmic distributions k/(x + 4) and k/(x − 4) lack this property. In this sense, the preferred view about the latter two distributions is not that they are positioned in an uncoordinated way along the x-axis in the narrower context of digits, but rather that they are not exactly inversely proportional to x in the more general quantitative sense.

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