Let's take all possible values of the turnout and divide them into blocks: turnout from 0 to 1 %, from 1 to 2 % ... from 99 to 100 %. In each of the blocks, we will be putting the number of people who voted at polling stations with the corresponding turnout.

Let's imagine that the average turnout was 69 %. This means that the majority of people voted at polling stations with approximately the same turnout.

Slightly less amount of people voted at polling stations with a slightly different turnout from the average: for example, 66-72 %.

And much less people voted at polling stations with turnout significantly different from the average.

If we mark all the remaining polling stations in the same way, we will get an image of the dependence of the number of votes on the turnout.

Further, we will draw this statistics in the form of a line.

Analists believe that such a distribution should ideally resemble a bell. This is discussed, for example, by electoral analyst Sergei Shpilkin and HSE associate professor Alexei Kupriyanov. It's about the form, and not about the normal distribution in the strict mathematical sense.

Exceptions do happen and do not show the fact of falsification by themselves, but make us to have a closer look at the results. In some countries, charts never take the shape of a bell due to the global territorial differences: for example, north and south vote differently. This is not the case in Russia.

Here we can see a vivid confirmation - such a distribution of the turnout was in the 2000 presidential elections. Experts call them the most honest in the history of modern Russia. If we mark the votes for Vladimir Putin, then we get a shape that looks like a bell.

The same situation happens with votes for other candidates. Regardless of how many people came to the polling stations, the ratio of votes for Putin and other candidates remains the same. According to the theory of the analyst Sergei Shpilkin, this suggests that the voting took place without falsification.

Only the “tail” of the chart looks abnormal. At polling stations with a turnout of 78 % of votes for Putin, it becomes suspicious, but just a little bit more.

But we can see that a completely different picture emerges during the 2016 State Duma elections. This is called the “two-humped Russia” phenomenon. This is how the votes for United Russia look like, according to the Central Elections Commission data.

The distribution of votes for other parties is more like a bell we are used to.

Everything that is drawn on the chart immediately after the slightly dented bell (the so-called "second hump") is a reason to doubt the accuracy of the vote results.

The fact is that United Russia was gaining a suspiciously large number of votes at polling stations with a high turnout. This may indicate that the votes for the party in power were artificially "drew" there. This is how the turnout and the result grew.

Potential falsifications are also clearly visible at the 2021 elections.

The distribution of votes for United Russia in 2021 is very different from other parties.

To roughly estimate how many fair votes for United Russia were cast by voters, the rule of electoral analyst Sergei Shpilkin can be applied to these data, abbreviated to the phrase “in normal voting, the turnout should not affect the ratio of votes cast for different candidates.”

If the votes for United Russia distribute in the same way as for the other parties, then there will be less votes for the party in power.

Shpilkin suggested that the red-filled square on the chart indicate the amount of the stuffing-in of about 13.8 million votes. The abnormal peak of votes for United Russia with a turnout of 95% may be associated with the results of electronic voting in Moscow, the analyst said.

If we look at the dependence of the voting result on the turnout a little differently, we again see a suspicious abnormality. Let's put the turnout on one axis, and the results of the party on the other. If everything is in order, then the schedule will resemble a chaotic spot, like in the presidential elections in 2000, which analysts consider fair.

But the more falsifications, the more the chart resembles a diagonal line, like that of the results of United Russia in 2021 elections.

Analysts call this anomaly the "comet tail." A pile of cloud turns into an even tail, which indicates the dependence of the president's result on the turnout. This is suspicious.