Solving the riddle of round numbers on the 2018 election schedule

This article is a response to this article ( Analysis of the results of the presidential election in 2018. At the federal and regional level ).

In that article, the author’s phrase surprised me:

Instead of a normal or lognormal distribution, we see an interesting curve, with very strange peaks at round values ​​(70%, 75%, 80%, etc.), increasing by about -100% turnout and going far upwards by 100%.

The questions immediately arise:

• Why does the author believe that instead of “strange” peaks there should be a normal or lognormal distribution?
• Why are peaks considered “strange” in general?
• Where can “natural” peaks appear at round values?

That article is highly politicized and the comments in it are relevant. In this article we will discuss only mathematics, so I ask you to keep political views to yourself.

And as a bonus, at the end of the article, a key will be laid out to solve the riddle of round numbers on the 2018 election schedule.

Initial data

Database file (MongoDB) with voting results (parsing from the state website ), which was uploaded by the author of the original article:

File = 15-04-18.tar.xz
MD5 = 3a1c198cbc4ce102fbc074752fc0ca99

We will study the graph of the percentage of turnout and the number of PECs with this turnout . In the original article, it looks like this:

Introduction

Everyone can download the database and independently check for errors. My data from several PECs were randomly selected and checked by me from the obtained database, which allows us to state with some probability that the data were downloaded from the state. site correctly.


Official turnout calculation formula:

• The number of ballots issued at the polling station +
• The number of ballots issued off-site +
• The number of ballots issued ahead of schedule

All graphs in this article are constructed by the following parameters:
Turnout at each weekend = formula above

Rounding to the nearest integer:

10.2 => 10
10.5 => 11

Rounding to the first decimal place and subsequent ..:

0.22 => 0.2
0.25 => 0.3

Visualization # 1

Let's look at the graphs with visualization of the number of PECs by turnout percentage, additionally adding a line with the average PEC size to check for possible correlation:

Chart_1a:

X axis - Turnout percentage (interval = 1%)
Y axis (left) - PEC number
Y axis (right) - Average size of one PEC (number of registered voters) It can be seen that at point X = 100 the value is very high, while the average size of PECs decreases. In the discussion of fediq , a logical assumption was made:

High turnout is a normal occurrence for highly organized PECs such as traditional communities, security institutions, and military units.

Let's look at the first 10 regions by the number of PECs with 100% turnout:

K_ALL - the number of PECs in the region
K_100 - the number of PECs with 100% turnout
REGION - the name of the region Graph_1b: X axis - Percentage of turnout (interval = 0.1% ) Y axis (left) - Number of PECs Y axis (right) - Average size of one PEC (number of registered voters) + The point is not displayed 100%, because with her the rest of the values ​​are too small. The same peaks appeared on the round values. Chart_1c: X axis - Percentage of turnout (interval = 0.01%) Y axis (left) - Number of PECs + The point is not displayed 100%, because with her the rest of the values ​​are too small.

K_ALL K_100 REGION
393 346 foreign-countries
1580 213 primorsk
1911 165 dagestan
482 156 sakhalin
596 138 murmansk
2817 132 tatarstan
2052 128 st-petersburg
317 123 kamchatka_krai
948 67 arkhangelsk
854 60 khabarovsk




















As expected, with a decrease in the step, the number of peaks increases and they are located throughout the graph.

Graph_1d:

X axis - Percentage of turnout (interval = 0.001%)
Y axis (left) - Number of PECs
+ No 100% point is displayed, because with her the rest of the values ​​are too small. As we see the whole graph in peaks, i.e. on this scale, peaks are normal. Graph_1d: X axis - Percentage of turnout (interval = 0.001%) Y axis (left) - Number of PECs Increased area for a value of 80% We see that next to the round 80% there are many small values.

Visualization # 2

Now let's look at the same graphs, but from a different angle. Since the points (turnout percentage) located next to each other on the graph along the X axis are not connected in any way (completely different PECs with different geo-position, size and mood of voters, etc., fall into each such sample point) , then there’s no difference in what order they are, so we’ll sort them along the X axis not by increasing percentage of turnout, but by increasing the number of PECs.

Those. in the graphs above we saw how the number of PECs behaves with an increase in the percentage of turnout, and in this graph we will see how the percentage of turnout behaves with an increase in the number of PECs.

Graph_2a:

X axis - number of a point from the database in order
Y axis (left) - Number of PECs
Y axis (right) - Percentage of turnout (interval = 1%)



This is the same graph as above, but the X axis (white color) is now transferred to the Y axis (right) and displayed as a separate line, the Y axis (left), as before, displays the number of PECs, and on the X axis now simply displays the number of samples from the database (the number of the point along the X axis in order).

Explanation A

point along the X axis with number 60 corresponds to:
turnout percentage = 95
number of PECs = 1680

Graph_2b:

X axis - number of a point from the database in order
Y axis (left) - Number of PECs
Y axis (right) - Turnout percentage (interval = 0.1%)
+ The point of 100% is not displayed, because with her the rest of the values ​​are too small.



Let's find on this graph our 80% round dot (circled in red), which in the graph above looked like a peak with small values ​​around it.

Here she already looks less pretentious on a line with points close to her in terms of percentage turnout (yellow circles).

Graph_2c:

X axis - number of a point from the database in order
Y axis (left) - Number of PECs
Y axis (right) - Turnout percentage (interval = 0.01%)
+ Point 100% is not displayed, because with her the rest of the values ​​are too small. But this is already interesting, those who understand mathematics are probably starting to understand what the trick is. And the point of 80% can no longer be distinguished, because on this scale it is no longer visible. Graph_2d: X-axis - point number from the database in order Y-axis (left) - Number of PECs









Y axis (right) - Turnout percentage (interval = 0.001%)
+ The 100% point is not displayed, because with her the rest of the values ​​are too small. And this schedule is just a revelation. You can even count the number of steps in each period ...

The key to solving the riddle of round numbers

Well, and as a bonus, it is dedicated to fans of “conspiracy theories” and mathematical puzzles: This is a beautiful pattern that I found in “round numbers”. The bottom line is that starting from point 40% and then in increments of 5%, the number of registered voters in PECs is always a multiple of 2, 5 or 10. Actually, the table above displays this. PROCENTX - turnout percentage KOLVO - total number of PECs with exactly this percentage without any rounding X2 - number of PECs in which the number of registrations. voters in multiples of 2 X5 - the number of PECs in which the number of registered. voters in multiples of 5 X10 - the number of PECs in which the number of registered. voters of 10 Next, I decided to check the multiplicity in steps of 1 ...

PROCENTX KOLVO X2 X5 X10
25 2 2 0 0
40 5 2 5 2
45 3 3 3 3
50 70 70 15 15
55 10 10 10 10
60 108 57 108 57
65 34 34 34 34
70 57 57 57 57
75 140 140 29 29
80 122 62 122 62
85 36 36 36 36
90 78 78 78 78
95 64 64 64 64
100 2613 1370 582 324












PROCENTX KOLVO X2 X5 X10
25 2 2 0 0
34 1 1 1 1
36 1 1 1 1
40 5 2 5 2
42 1 1 1 1
44 2 0 2 0
45 3 3 3 3
46 1 1 1 1
47 1 1 1 1
48 2 0 2 0
50 70 70 15 15
51 3 3 3 3
52 7 4 7 4
53 4 4 4 4
54 4 4 4 4
55 10 10 10 10
56 10 6 10 6
57 5 5 5 5
58 9 9 9 9
59 4 4 4 4
60 108 57 108 57
61 3 3 3 3
62 18 18 18 18
63 1 1 1 1
64 23 10 23 10
65 34 34 34 34
66 14 14 14 14
67 8 8 8 8
68 22 10 22 10
69 2 2 2 2
70 57 57 57 57
71 4 4 4 4
72 17 5 17 5
73 6 6 6 6
74 8 8 8 8
75 140 140 29 29
76 23 11 23 11
77 4 4 4 4
78 10 10 10 10
79 2 2 2 2
80 122 62 122 62
81 10 10 10 10
82 14 14 14 14
83 6 6 6 6
84 24 11 24 11
85 36 36 36 36
86 10 10 10 10
87 3 3 3 3
88 23 8 23 8
89 4 4 4 4
90 78 78 78 78
91 4 4 4 4
92 31 17 31 17
93 6 6 6 6
94 13 13 13 13
95 64 64 64 64
96 25 11 25 11
97 6 6 6 6
98 17 17 17 17
99 4 4 4 4
100 2613 1370 582 324

It turns out that such a pattern is observed except for 100 on all integers (in which there is exactly an integer value, if there is no integer value, it is omitted from the list).

And finally, the total breakdown by the number of registered voters x2-10 for all PECs: KOLVO - the total number of PECs in which elections were held X2-10 - the number of PECs in which the number of registered. of voters is a multiple of 2-10 And the total breakdown by the number of voters who came to the polls is x2-10 for all PECs: KOLVO - the total number of PECs in which elections were held X2-10 - the number of PECs in which the number of voters who came in was a multiple of 2-10 Well , and then usually in such cases mathematicians write ... the solution is trivial :) Update:

KOLVO X2 X3 X4 X5 X6 X7 X8 X9 X10
97699 49413 32753 24724 20283 16649 13923 12464 10917 10411






KOLVO X2 X3 X4 X5 X6 X7 X8 X9 X10
97699 49268 32712 24634 20608 16492 14085 12192 10938 10752

Conclusion

Oddly enough, but my article turned out to be rather ambiguous for those who joined the discussion without reading the main article to which the answer was made and the discussion that was conducted there. Therefore, I will add explanations here, in the form of answers to the questions posed by me at the beginning of the article.

Why does the author (of the original article) consider that instead of “strange” peaks there should be a normal or lognormal distribution?
“I never received an answer.”
But, my opinion is - to expect that on the voting schedule (where a reasonable choice is involved) there must be a normal or lognormal distribution - this is not true.

Why are peaks considered “strange” in general?
- In the original article, this was the version “But this can be explained by the fact that, when falsified, the chairmen preferred to take“ round ”percent numbers.”, As well as “Because I don’t see the natural reasons for their appearance on round numbers.”

Where can “natural” peaks appear at round values?
- This was already my question to myself, for the solution of which I asked the author of the original article to upload a database file for analysis.

Actually, this article answers it - according to the graphs of Graph_1c, Graph_1d we see that the presence of peaks on the graph under discussion is quite natural, and the reason for their appearance lies in the fraction of the
number of voters_ who came to the elections / number of voters of the registered

The value of which is more likely to be "round" than others, this is confirmed by the data from the section "The key to solving the riddle of round numbers". But we must keep in mind that here, “round” values ​​are understood to mean “exactly round” values ​​without any rounding, i.e. Illustrated peaks on Graph_1d and Graph_1c. On Graph_1b - the values ​​are already very rounded and the values ​​from the nearest neighborhood are added to the “exactly round” values, therefore the “naturalness” of the peak on the “exactly round” values ​​has much less influence than the “reasonable choice”. Therefore, the appearance of high peaks at round values ​​in real voting, the “naturalness” shown by us does not explain. But, the fact that they exist and can be enhanced by a “reasonable choice” is quite likely.



Update2:


Update3:


PS
And yet, in view of the negative karmic situation, I can’t immediately answer everyone in the commentary in the article, so sorry, I haven’t answered anyone yet :)