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It would seem that heated debates about the statistical analysis of election results are a thing of the past. The assessments (see, for example, [1], October 11, 2009), made «on the hot tracks» according to which there were falsifications, were met with criticism [2, 3]. While the first of these works can be described as a conflict of interest (one of the authors served as the chairman of the CEC at that time), the second one was written by a well-known and very qualified mathematician. In some cases, artifacts of data presentation (peaking at 50%) were taken as indications of falsification. In others, the «non-normality» of the distribution was taken as evidence of falsification. Additional confusion arose also because «normality» in the mathematical sense (proportionality of the Gauss function) is easily mixed with «normality» in the everyday sense («the election was normal»).

Since then, the situation has been clarified: presentation artifacts were identified, and various hypotheses explaining possible histogram shapes were tested in subsequent elections. Emotional blog posts were supplemented by detailed papers published in scientific journals in statistics [4, 5, 6], in which statistical evidence of falsification did not rely in any way on the assumption of normality (in the sense of Gauss) of distributions. The question of statistical analysis of elections (as a scientific, not public question) seemed more or less closed.

The peer-reviewed paper again raises the question about the incorrectness of the conducted statistical analysis. But it is based on a misunderstanding: the authors rightly point out and confirm with numerous examples that histograms of elections may well be very different from «Gaussian» and in the absence of falsification. They have probably not seen the papers cited above and suggest that the conclusions about fraud so far are based on a deviation from «Gaussianity».

The authors make political accusations as well: «to speak about “bona fide delusion” of the authors of “mathematical theories” of electoral behavior estimations, who disseminate unreliable information, in this case it is not necessary» (p. 4). In this connection, one can recall the speeches of Lysenko's students, who stated that «the terms, scales, and, most importantly, the results envisaged by the theory of Mendelism are unsuitable for our Soviet reality» [7], «as Lenin wrote, … statistics, which leads to depersonalization, turns into the emptiest and most harmful “game of numbers”» [8]. To this Kolmogorov, having made a correct statistical analysis of the works of the «Lysenkovians» and having established that they are quite consistent with the «pea laws» allegedly refuted by them [9], briefly remarked: «Kolman's work … is entirely based on a misunderstanding of the circumstances stated in our note».

It would be possible to follow his example and limit ourselves to what has been said (the authors argue against the thesis of «a priori Gaussianity» on which the analysis does not rely), but it is also possible to note the extreme carelessness of the authors in preparing the text. Reproducing one of the criticized graphs, they write (p. 17): «The results of voting in Chernogolovskaya TEC of the Moscow Region» raise “suspicions” of the authors-compilers of the charts. Meanwhile, on the graph they reproduced it is written «0 thousand anomalous votes» — as it is obvious, the authors not only did not understand the data, but even did not read the inscriptions on the figure they included in their work. In their defense, it can be noted that this inscription is in small (though quite legible) font.

The authors found it necessary to include in their paper «mathematical proof» of the well-known fact that convergence to the Gaussian distribution occurs when the system size tends to infinity (Section 5, «Peculiarities of mathematical modeling of the process»). However, it is difficult to make sense of this section even with the most sympathetic attitude and with a lot of experience in checking the work of twos. The authors write on p. 45, that «for all possible contributions of voting voters who cast ‘votes’ in each precinct» to be equally likely to be accounted for, the number of precincts must be equal

(where n — precinct size) and that «this unambiguously imposes a condition on the total number of voters equal to *N = nY*» (where*Y = 2 ^{n}* , the authors instead write the sum of the binomial coefficients, probably unaware that it is equal to

Note that statistical methods by their very nature say nothing about «legitimacy of elections». They answer a more technical question: how plausible is the «null hypothesis» that the published election results are obtained by correct counting, and the available evidence indicates that for most of the votes in recent years (including the «constitutional amendments» and the «Crimean referendum») the null hypothesis does not look plausible. But if we talk about methods of «delegitimization of elections» it would be worth mentioning two possible (and apparently effective) methods: (1) falsification of their results, as well as (2) publication of illiterate texts as a refutation of quite correct, even if not read, works.

In the last paragraph of the peer-reviewed paper, the authors recommend «to continue the work on improving the legal culture of voters, paying attention to the issues of criteria for the authenticity of election results and assessment of the reliability of information. » Joining this wish, I express the hope that this review is a step in the specified direction.

**Literature list**

[ 1 ] Graph of vote distribution by districts, blog entry, uborshizzza.livejournal.com/674242.html .

[ 2 ] Churov V.E., Arlazarov V.L., Solovyov A.V. Election results. Analysis of electoral preferences. In: Proceedings of the Institute of System Analysis of the Russian Academy of Sciences. Collection: mathematics and management. Edited by Corresponding Member of the Russian Academy of Sciences, Professor V.L. Arlazarov and Doctor of Technical Sciences, Professor N.E. Emelyanov, LKI, 2008.

[ 3 ] Yury Neretin, On statistical researches of parliamentary elections in Russian Federation, 04.12. 2011, preprint (January 2012), https://arxiv.org/abs/1205.1461

[ 4 ] Peter Klimek, Yuri Yegorov, Rudolf Hanel, Stefan Thurner, Statistical detection of systematic electoral irregularities, Proceedings of the National Academy of Sciences, 109(41), 16469–16473 (2012)

[ 5 ] Dmitry Kobak, Sergey Shpilkin, Maxim S. Pshenichnikov, Integer. Pshenichnikov, Integer percentages as electoral falsification fingerprints, The Annals of Applied Statistics, 10(1), 54–73 (2016)

[ 6 ] Dmitry Kobak, Sergey Shpilkin, Maxim S. Pshenichnikov, Statistical fingerprints of electoral fraud?, Significance, 13(4), 20–23 (2016)

[ 7 ] Ermolaeva N.M., Once more about «pea laws», Yarovizatsiya, 1939, vol. 2(23), pp. 79–86.

[ 8 ] Colman E., Perversions of mathematics in the service of Mendelism. Yarovization, 1939, vol. 3(24), pp. 70–73.

[ 9 ] Kolmogorov A.N., On one new confirmation of Mendel's laws, Reports of the Academy of Sciences of the USSR, 1940, vol. 27, 38–42.

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