Translated material
Saratov phenomenon (entertaining statistics)
Probably, most of the readers of the Trinity Option - Science heard about the Saratov anomaly. About 140 sections out of 1885, the share of votes set for United Russia lies in the range of 62.1–62.3%. This phenomenon was discovered by Alexander Kireev [1]. People involved in the calculation of the results in the Saratov region called the anomaly "mathematical coincidence." Other people appreciated the probability of this “mathematical coincidence” -estimates range from 10 -55 to 10-100, depending on the approach. We proceed from the fact that an interesting cognitive material can be made from any shameful phenomenon of our life. Therefore, we talk about how to assess the probability of the Saratov “mathematical coincidence” and how to interpret it.
The probability of a statistical emission depends on the so -called zero hypothesis - how the true distribution without emission should look. Suppose we do not have a reasonable zero hypothesis, and we want to put a restriction on top: what is the maximum probability of the Saratov ejection with arbitrary zero hypotheses that do not contradict the laws of nature. Such an assessment was made by Boris Ovchinnikov [2]. Two assumptions are required for her:
- 62.2% - the true share of voters of the Saratov region, voting for United Russia.
- These voters are evenly mixed in all areas and vote independently of each other and from external factors.
Then the distribution of votes behind the “EP” in each section will be with good accuracy of the Gaussian (more accurately binomal) with an average of 62.2 and a width equal to a square root from the number of the “EP” in the area (in fact are different areas, so the distribution will be the sum of humps of different widths). The same cannot be a distribution, this is the basics of mathematical statistics. But such a hump, close to Gaussian, is still much wider than the Saratov peak, and, according to Boris Ovchinnikov, the probability of such an emission is 10-55. It would seem that this can be stopped: the upper assessment is vanishingly small, you can initiate a criminal case of malicious falsification. But we, as is customary with physicists, will go further and try to give a realistic assessment, albeit not so strictly justified.
In the real elections of the Gaussian distributions with the dispersion, the “root from N” does not exist-they are wider, due to the fact that the voting are not independent of each other (families, neighboring companies) and objective external factors (standard of living on the site, the history of the district, etc.). We can try to borrow zero hypothesis from life. The simplest and most natural is to take the results of the elections in the Saratov region, cutting out the notorious peak from the distribution. It will be our zero hypothesis, and the peak is a statistical emission, the probability of which we will evaluate. “But the rest of the curve can be falsified,” the penetrating reader will say. Of course, but the zero hypothesis is precisely in the assumption of the honesty of elections.
Take the real distribution from Fig. 1, built by Sergey Romanchuk. In two bina 62.1–62.3 fell 140 sites. According to Romanchuk, in nearby bins there are an average of three sites on average, that is, there should be approximately 6 sites under the ejection (we do not need better accuracy due to the absurdity of the task). And we observe 140. The probability of emission is well described by the distribution of Poisson (when the total number of areas is many more than their number in the emission). This is the distribution:
Here A is the expected average, n is the dropped number. At A = 6 and N = 140, we have a probability of ~ 10-135 (Romanchuk, without using the distribution of Poisson, received “on the fingers” 10-100, which can be considered a good coincidence).
What is 10-135? How to imagine the meaning of such a “mathematical coincidence”? Let's resort to the mental experiment. In order to make a similar coincidence with a high probability, you need to make approximately 10135 equal tests. That is, to hold exactly so many elections in the millionth regions, divided into a thousand sites. A lot of them were carried out on Earth, only 102 regions participated in these Duma elections (rounded to orders). And all over the world, a dozen elections, referenda and votes of a similar scale are held annually (we throw more order). And this is how most of the 20th century happens - we will accept in 100 years. Taking numbers with a large margin, we will receive105 similar votes in the entire earthly history. There is not enough 130 orders of value.
About very large numbers, such as, for example, 1013, they say the "astronomical number." Then 10135 is already something hyperastronomic. Let's move on to the next cosmological scale. In the observed part of the universe, about 1012 galaxies. Each of them has 1011 stars (take excess). Total 1023 stars in the observed part of the universe. By the way, this is more than grains of sand on all beaches of the Earth - about 10 thousand km3 of sand. Of these, about 1022 stars are close in their luminosity and mass to the sun. Of these, according to the Kepler telescope, one tenth, or approximately 1021, have global planets in the zone of abode. Suppose that life appears on all such planets and evolves to rational creatures that ripen to democracy requiring regular votes. The history of earthly democracy barely has 100 years, but maybe civilizations live and hold elections for a very long time - an average of billion years. Then, 1021 (planets) x 103 (per year) x 109 (years) = 1033 votes similar to the scale of what took place in the Saratov region passed through the entire history of the universe in its observed part. But we need much more - there are not enough 102 orders!
Is this probability not physically realized?
In fact, not everything is so bad. The theory of cosmological inflation states that the observed area inside the horizon is only a microscopic part from the entire giant universe, which at its first moments grew in exponent, and where this exhibit was broken - one god is known. It can be 100 times, and 50 orders are larger than the size of our horizon. If we lack 102 orders, then it is enough to assume that the size of the universe is about 34 orders of magnitude larger than the distance to the horizon (how much the distance to the horizon is larger than the small bacterium). And in such a universe, densely inhabited by billion-year-old civilizations, diligently adhering to democratic procedures, with the probability of unit order during its existence, a similar result will fall somewhere in the elections. And this honor went to the Saratov region! And Ella Pamfilova says that this fact is not worth a damned egg. Wow egg!
Boris Stern