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The uniform distribution produces a uniform distribution of first- and second place digits, as was to be expected.
Third, and most importantly, the examples demonstrate the link between the distribution of a random variable on the one hand and the first-and second place digits' distributions on the other hand.
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When the cards are added together, the digits in the tens places are dropped, making the ones place digit the score.
The exponential distribution produces first- and second-place digits' distributions coming close to the Newcomb-Benford distribution.
Analyzed were the distributions of the following five variables plus their first- and second-place digits' distributions.
As for the sample sizes, results will be given in a reduced manner, namely for the first- and the second-place digits only.
The first- and second-place digits' distributions are shown in Figure 3 on the right, whereby observed values are represented by bars, values expected according to the NBL by a line.
The closer the shape of the distribution of a random variable comes to that of a survival distribution or a distribution behaving like a survival distribution, the better follows the first- and second-place digits' distributions the NBL.
Nevertheless, based on the Pearson chi-squared goodness-of-fit test (5% significance level), all of the first- and second-place digits' distributions are compatible with the NBL, with one exception: the first-place digit's distribution of the bibliography data clearly fails to fit the NBL; see Table 9.
Thereby, the fit to the NBL was tested formally by chi-square goodness-of-fit tests of the first- and second-place digits' distributions, whereas the fit of the observed variable's distribution to a survival distribution of unspecified form was informally assessed by visual inspection.
Further, it generalizes to the second-place digit.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com