How Mathematics Helps to Identify Fraud in Favour of Poroshenko in the Election

Like any process in which many objects (for example, people) take part, elections – including the Ukrainian presidential election – is subject, among other things, to statistical analysis.

And this analysis shows that during the 2019 presidential election, to put it mildly, it’s not everywhere that the vote took place correctly. Explicit distortions towards the current president have been observed.

Figures outside of politics

Let’s take, for example, such an indicator as voter turnout within this or that constituency. The turnout is, in fact, the percentage of the registered number of voters who visited polling precincts in this district.

However, it is obvious that active voters are distributed unevenly within the district only in view of the fact that their distribution is a random process. In other words, in one district the turnout will be slightly more than average, and in another one it will be slightly less. And this is not because these districts differ in something among themselves (there are, of course, such districts, but this is a separate conversation), but just because the people are different, and nobody especially “settles” active voters on the territory of one district, and passive voters on another.

If we analyse how the turnout between polling precincts was distributed, then the following will be a natural state of affairs. At the greater number of stations we will see a turnout that will be close to average. For example, if the average turnout is 60%, then most stations will show a turnout near the mean value – let’s say, from 58% to 62%.

Of course, there will be stations with either a higher or lower turnout. But as we move away from the district averages, we will see how the number of stations that showed a turnout in this range will decrease.

Let’s emphasise: this happens just because active and passive voters are distributed throughout the district randomly. All random processes in nature are subject to exactly the same rules.

It is precisely for this reason that the distribution of the number of stations versus the turnout that is recorded at them is subject to the standard rule known as Gaussian distribution, which (because of its universal prevalence in nature and life) is also called normal distribution.

Gaussian curve – the reliable controller 

The diagram of this distribution looks like an inverted bell: a pronounced peak in the center – and at first smoothly, and then sharply falling “limbs” on both sides of the peak.

In the most general case it looks like this:

Something similar should be observed also during the election (generally speaking, at any election). And practice confirms this fact. Here is, for example, what the relationship between the number of stations and the turnout registered at them in the second round of the 2005 Polish presidential election looked like.

X axis = turnout; Y axis = number of stations

On the diagram a certain “sawtooth” roughnesses, of course, is visible – this is due to the fact that it concerns, after all, rather (by the standards of statistics) a small number of studied objects, and a role is played by fluctuations (random deviations from the statistically expected values). However, the general principle remains, and the characteristic “Gaussian” form of the curve is perfectly visible.

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If we see such a diagram, then we can note: what happened during the election was completely subjected to general statistical rules, i.e., the vote was free, without any organised influence over its process and, respectively, results. Any essential deviations from this form will mean that the process of voting (or its results) suffered from some external interference.

Let’s add that analysis of the purity of elections by studying of distribution of turnout at stations is the conventional method that is widely used in many countries of the world.

Let’s look at what happened in this regard at the last Ukrainian election. Data is taken from the official website of Central Election Commission: anyone, if they have the desire (and the existence of a large amount of free time) can reproduce our constructions independently.

Also, we will make a reservation that while processing data we immediately excluded special stations from consideration (courts, military units, prisons, hospitals, and so on). It is obvious that the turnout at such stations is subjected to slightly different rules, and they will only confuse us.

Here, for example, is district No. 137 (Baltsky, Kodymsky, and Podolsky districts of the Odessa region).

Seemingly, the diagram looks almost normal. However, it was not without anomalies here: for example, at station No. 510083 in the village of Borsuky in the Baltsky district 100% of voters somehow voted, and in the village of Murovana in the Podolsky district – 85% of voters somehow voted. However – let’s just assume that they are statistical deviations: well, could it so happen that surprisingly disciplined voters live in Borsuky?

The diagram for district No. 133 (Kievsky district of Odessa) looks even better – it is almost ideal!

Perhaps, even the most captious analyst will not see any strangenesses here.

Alas, at this election it was far from being like this everywhere.

Reason for doubts

Here, for example, a similar distribution for district No. 143 (Izmail and Reniysky district) looks far worse.

Pay attention: in the diagram there is not one, but three peaks at once. More precisely, where the number of stations should grow monotonically as the turnout increases, it grows, then sharply “falls” down, and then again creeps up. This is more than suspicious.

At the same time, having studied the results of the vote, it is possible to be convinced that at stations with a smaller turnout (to the left of the main peak) the number of people who voted for Poroshenko was noticeably smaller than it was at stations with a bigger turnout.

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For example, in the range of turnout from 20% to 55% the proportion of voters who voted for Petro Poroshenko is 5.7%, and at stations with a turnout from 60% to 65%, 8.1% of voters voted for Poroshenko.

A candidate from the current government relying on turnout, by the way, is suspicious in itself. Of course, in theory it can be explained by the fact that Poroshenko’s supporters, apparently, were more active than those who supported other candidates. Although this is doubtful: pre-election polls showed that in terms of the level of expected activity, Poroshenko’s supporters lagged behind, for example, the supporters of Tymoshenko, Zelensky, and Boyko. But both factors (an interrupted diagram showing turnout and the result being dependant on turnout) give more than strong grounds to claim that this took place not without external interference.

It is not a fact that the direct falsification of the election took place (stuffing, the wrong calculation, or forged protocols). It is quite probable that the bias towards a higher turnout and the increased number of supporters of Poroshenko at stations with a high turnout is explained, for example, by the result of voter bribery: it is obvious that a person who is promised remuneration for arriving at the station is, generally speaking, a more motivated voter than those who vote free of charge.

One more very doubtful Odessa district – No. 139 (Southern, Razdelnaya, Limansky, and Zakharyevsky districts).

The diagram of turnout here looks very bad: there is a little “saw” around the mean turnout values, and also a completely inexplicable “tooth” in the area of 70% turnout.

Let’s switch from the Odessa region to Donbass. Here, for example, is district No. 47 with its center in Slavyansk.

The central peak looks almost not bad, but what is that suspicious “saw” from districts in regions with a big turnout? And of course, at the stations in the “sawtooth zone” (a turnout of 70-95%) Poroshenko’s average result is 25.3% of votes, whereas in the peak zone of 55-65% he collects on average only 8% of votes.

Even less beautiful is the chart for district No. 59 (Marinka). We will repeat: we excluded the special stations, including the ones created for military personnel. But the rest show a very strange picture.

Here we see as many as two “saws” – both in the area of big and small turnout. And in the “small saw” results for Poroshenko are below the district average (12% vs 16%), and in the “big saw” – on the contrary, it is higher than 26%.

Summary and conclusion

But the most intriguing and “promising” is district No. 52 (Bakhmut). Let’s start at least with the fact that this district is “very fast” at counting votes: at the time when this text was written (02:00 on April 2nd) only 58.92% of votes had been counted, although in general across Ukraine at this time 97.38% of votes had already been counted.

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Because there is no data for nearly a half of stations in the district, we will not even try to form a Gaussian curve for it. However, even without it the station looks extremely interestingly. Let’s begin with the fact that even 42.7% of voters whose votes have already been counted here, allegedly, voted for Poroshenko – while on average in the Donetsk region only 12.59% of people support the acting head of state.

The turnout here is also striking: there are two stations with a turnout of 99%, at the same time 6 stations with a turnout of 80% and above, and so on.

And the proportion of the voters who voted for Poroshenko at many stations exceeds 65%! It is only possible to imagine what “Gaussian curve” this district will produce when the calculation there is complete!

Unfortunately, it is possible to note: the data on the results of voting at a number of polling stations that we have considered above allow us, to put it mildly, to doubt the fairness of the elections held here. And these are not some idle statements: these are not our words, but those of a mathematician who does not care either about Poroshenko nor about the fate of the Ukrainian election in general.

Of course, this is not observed at all stations, and where there are well-founded grounds for suspicion, the size of falsifications is unlikely to be large: it is impossible to say for sure, but they are unlikely to exceed 3-5%. However, in conditions when the current president Petro Poroshenko is only 2.5% away from dropping out of the race (according to the results of counting 97.39% of votes), even such distortions can become, without understatement, decisive.

What to do with all of this? In principle, such a question should be immediately asked to the Central Election Commission. Plotting Gauss curves, as we have just done, is a completely standard control method of honesty of elections. And according to all canons, the CEC already had to do this procedure for those districts where vote counting has already been completed. And having seen what was already seen by us, members of the CEC should have a think and do something to confirm or disprove the arisen doubts – for example, to carry out a vote recount at the most suspicious stations, and even in all districts in general.

Whether this will be done by the Central Electoral Commission – most of its members are said to be controlled by the Presidential Administration – is a separate and very interesting question.

Yury Tkachev

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