Wizard
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Wizard
Joined: Oct 14, 2009
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November 9th, 2020 at 11:03:51 AM permalink
Before getting into the topic at hand, let me remind the forum about it's policy on discussing the 2020 election. As a remind, rule 19 states the following:

Quote: Rule 19

In an effort to keep the focus of the forum on gambling, Vegas, and math, comments of a political, racial, religious, sexual, or otherwise controversial nature are not allowed. We recommend taking such discussion elsewhere (Added 8/13/19).



However, since there is such significant betting on the election, that we do allow discussion of that. As a general rule, we allow mathematical and scientific discussion of anything, to include the election and the coronavirus. Such discussion must be of an academic nature. A good rule of thumb if a post about a sensitive topic is allowable if it does not betray the personal opinion of the poster.

With that out of the way, there is an accusation out there that the Michigan election results were faked as evidenced by allegedly not being in compliance with Benford's Law. Briefly, that suggests that the digits of numbers of a random nature tend to follow certain distributions. A good way to catch falsified numbers, for example on an income tax return, is if the digits do not follow expected distributions. Human beings are notoriously bad at randomizing without the aid of computers.

What got me interested in the topic was this post by Evenbob at DT, alleging the vote counts by county in Michigan were he says in part, "Applied to the election in MI, Bidenís vote numbers in do not match Benfordís law at a 99.999% level." I have surfed around a bit and the best argument I have found about this is There is Undeniable Mathematical Evidence the Election is Being Stolen at The Red Elephants. This articles makes a number of points, just one of them about Benford's Law. I will not endeavor to argue the other points here.

First, here is a list of the county by county election results in Michigan, which allegedly has the fake vote totals. At this time (10:47 AM Nov 9, 2020) 99% of the vote is in, so these totals can be expected to change slightly.

County Biden Trump Total
Alcona County 2,142 4,848 6,990
Alger County 2,053 3,014 5,067
Allegan County 24,447 41,381 65,828
Alpena County 6,000 10,686 16,686
Antrim County 7,289 9,783 17,072
Arenac County 2,773 5,928 8,701
Baraga County 1,475 2,512 3,987
Barry County 11,804 23,473 35,277
Bay County 21,718 30,919 52,637
Benzie County 5,480 6,600 12,080
Berrien County 37,438 43,518 80,956
Branch County 6,161 14,066 20,227
Calhoun County 28,417 35,900 64,317
Cass County 9,122 16,686 25,808
Charlevoix County 6,939 9,841 16,780
Cheboygan County 5,435 10,171 15,606
Chippewa County 6,651 10,682 17,333
Clare County 5,199 10,861 16,060
Clinton County 21,963 25,095 47,058
Crawford County 2,612 4,955 7,567
Delta County 7,605 13,206 20,811
Dickinson County 4,569 8,469 13,038
Eaton County 31,297 31,797 63,094
Emmet County 9,662 12,135 21,797
Genesee County 120,082 99,199 219,281
Gladwin County 4,524 9,893 14,417
Gogebic County 3,573 4,600 8,173
Grand Traverse County 28,682 30,502 59,184
Gratiot County 6,693 12,104 18,797
Hillsdale County 5,883 17,037 22,920
Houghton County 7,755 10,380 18,135
Huron County 5,349 11,949 17,298
Ingham County 94,221 47,640 141,861
Ionia County 10,899 20,655 31,554
Iosco County 5,371 9,760 15,131
Iron County 2,493 4,216 6,709
Isabella County 14,072 14,815 28,887
Jackson County 32,004 47,381 79,385
Kalamazoo County 83,674 56,823 140,497
Kalkaska County 3,003 7,436 10,439
Kent County 186,753 165,318 352,071
Keweenaw County 672 862 1,534
Lake County 2,288 3,946 6,234
Lapeer County 16,368 35,480 51,848
Leelanau County 8,793 7,915 16,708
Lenawee County 20,916 31,539 52,455
Livingston County 48,218 76,980 125,198
Luce County 842 2,109 2,951
Mackinac County 2,589 4,258 6,847
Macomb County 225,561 264,535 490,096
Manistee County 6,107 8,321 14,428
Marquette County 20,465 16,287 36,752
Mason County 6,802 10,207 17,009
Mecosta County 7,373 13,265 20,638
Menominee County 4,315 8,117 12,432
Midland County 20,493 27,675 48,168
Missaukee County 1,967 6,648 8,615
Monroe County 32,975 52,710 85,685
Montcalm County 9,703 21,815 31,518
Montmorency County 1,628 4,171 5,799
Muskegon County 45,508 44,544 90,052
Newaygo County 7,874 18,864 26,738
Oakland County 433,982 325,916 759,898
Oceana County 4,944 8,892 13,836
Ogemaw County 3,475 8,253 11,728
Ontonagon County 1,391 2,358 3,749
Osceola County 3,214 8,928 12,142
Oscoda County 1,342 3,466 4,808
Otsego County 4,743 9,779 14,522
Ottawa County 64,566 100,511 165,077
Presque Isle County 2,912 5,343 8,255
Roscommon County 5,166 9,670 14,836
Saginaw County 51,068 50,784 101,852
St. Clair County 31,363 59,184 90,547
St. Joseph County 9,262 18,128 27,390
Sanilac County 5,966 16,194 22,160
Schoolcraft County 1,589 3,090 4,679
Shiawassee County 15,371 23,154 38,525
Tuscola County 8,713 20,310 29,023
Van Buren County 16,800 21,591 38,391
Washtenaw County 157,130 56,241 213,371
Wayne County 587,074 264,149 851,223
Wexford County 5,838 12,102 17,940


Source: Michigan presidential results at politico.com.

Next, here are frequency figures of the first and second digits of, shall we say, random large numbers.

Digit First digit expectations Second digit expectations
0 0.00% 11.97%
1 30.10% 11.39%
2 17.61% 10.88%
3 12.49% 10.43%
4 9.69% 10.03%
5 7.92% 9.67%
6 6.69% 9.34%
7 5.80% 9.04%
8 5.12% 8.76%
9 4.58% 8.50%
Total 100.00% 100.00%


The next three tables shall show the first digit frequency for total votes, Biden votes, and Trump votes, as well as the expected total by digit according to Benford's Law. I shall also state the chi square statistic and p value. The p value is the probability that random results would be less skewed that those observed.

Before I continue, an argument could be made that a total of 83 counties is not enough to apply the chi-squared test. This argument has some merit. The rule of thumb is there should be an expected total of each digit of at least 5. In this case, the expected total of nines is only 3.80. For the limited data, I can't think of a better test, but am all ears to ideas.

That said, here is the table for the first digit of total votes.

Digit Count Expectations
1 30 24.99
2 14 14.62
3 9 10.37
4 5 8.04
5 6 6.57
6 7 5.56
7 3 4.81
8 7 4.25
9 2 3.80
Total 83 83.00


chi-squred statitistic = 6.110748
p value = 0.634828

Here is the table for the first digit of Biden votes.

Digit Count Expectations
1 15 24.99
2 17 14.62
3 9 10.37
4 8 8.04
5 11 6.57
6 9 5.56
7 5 4.81
8 4 4.25
9 5 3.80
Total 83 83.00

chi-squred statitistic = 10.080136
p value = 0.259446

Here is the table for the first digit of Trump votes.

Digit Count Expectations
1 22 24.99
2 13 14.62
3 11 10.37
4 11 8.04
5 7 6.57
6 2 5.56
7 3 4.81
8 7 4.25
9 7 3.80
Total 83 83.00

chi-squred statitistic = 9.134425
p value = 0.331083

All three p values look very normal. So, how about the second digits.

Here is the table for the second digit of total votes.

Digit Count Expectations
0 9 9.93
1 8 9.45
2 10 9.03
3 3 8.66
4 7 8.33
5 11 8.02
6 9 7.75
7 11 7.50
8 8 7.27
9 7 7.05
Total 83 83.00


chi-squred statitistic = 7.33793
p value = 0.60198

Here is the table for the second digit of Biden votes.

Digit Count Expectations
0 7 9.93
1 12 9.45
2 8 9.03
3 8 8.66
4 10 8.33
5 8 8.02
6 8 7.75
7 8 7.50
8 9 7.27
9 5 7.05
Total 83 83.00


chi-squred statitistic = 3.11012
p value = 0.95977

Here is the table for the second digit of Trump votes.

chi-squred statitistic = 3.11012
p value = 0.95977
Digit Count Expectations
0 14 9.93
1 9 9.45
2 8 9.03
3 8 8.66
4 6 8.33
5 4 8.02
6 14 7.75
7 7 7.50
8 6 7.27
9 7 7.05
Total 83 83.00


chi-squred statitistic = 9.81754
p value = 0.36546

Bottom line is these election results look perfectly compliant with Benford's Law to me.

Of course, I welcome all comments and arguments to the contrary, so long as they are mathematical in nature and don't betray personal political opinions.
It's not whether you win or lose; it's whether or not you had a good bet.
EvenBob
EvenBob
Joined: Jul 18, 2010
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November 9th, 2020 at 11:46:34 AM permalink
This guy seems legit and he found
significant deviations from Benfords
Law in the election results for Biden.

"It's not enough to succeed, your friends must fail." Gore Vidal
unJon
unJon
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November 9th, 2020 at 11:56:13 AM permalink
Iíve always encountered Benfordís law when there was a power law driving generation of the numbers. Whatís the corresponding principle that would make the law applicable to election results?
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
darkoz
darkoz
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November 9th, 2020 at 12:15:13 PM permalink
To apply Benfords law to this election would it not make sense to first examine results in other elections?

If I remember correctly Mondale versus Reagan 84 resulted in defeat 49:1 statewide. How does Benfords law apply there.

I'm not of the mathematicians club so forgive me if that question is a stupid one
For Whom the bus tolls; The bus tolls for thee
terapined
terapined
Joined: Dec 1, 2012
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November 9th, 2020 at 12:43:25 PM permalink
Quote: darkoz

To apply Benfords law to this election would it not make sense to first examine results in other elections?

If I remember correctly Mondale versus Reagan 84 resulted in defeat 49:1 statewide. How does Benfords law apply there.

I'm not of the mathematicians club so forgive me if that question is a stupid one


I think it's a mathematical formula to determine if numbers were randomly generated as they would in counting vote totals in a county
Or
A human input a vote total for all the counties trying to use random numbers
"Everybody's bragging and drinking that wine, I can tell the Queen of Diamonds by the way she shines, Come to Daddy on an inside straight, I got no chance of losing this time" -Grateful Dead- "Loser"
rdw4potus
rdw4potus
Joined: Mar 11, 2010
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November 9th, 2020 at 1:03:01 PM permalink
Quote: EvenBob

This guy seems legit and he found
significant deviations from Benfords
Law in the election results for Biden.



Go to the 20 minute mark. Look at the rows for 4 and 5. The results are skewed because Biden has too many 4s and too few 5s. Now recall that this is a first digit test. Ask yourself: is it GOOD to replace 5s with 4s at the start of numbers when you're cheating and trying to win?
"So as the clock ticked and the day passed, opportunity met preparation, and luck happened." - Maurice Clarett
EvenBob
EvenBob
Joined: Jul 18, 2010
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November 9th, 2020 at 2:52:51 PM permalink
Quote: rdw4potus



Go to the 20 minute mark. Look at the rows for 4 and 5. The results are skewed because Biden has too many 4s and too few 5s. Now recall that this is a first digit test. Ask yourself: is it GOOD to replace 5s with 4s at the start of numbers when you're cheating and trying to win?



Is this in a foreign language? I have
zero math training.
"It's not enough to succeed, your friends must fail." Gore Vidal
sabre
sabre
Joined: Aug 16, 2010
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Thanks for this post from:
rdw4potusams288
November 9th, 2020 at 2:59:09 PM permalink
Quote: EvenBob

Is this in a foreign language? I have
zero math training.



If you don't
realize that 4<5
then how can
you say that
someone
seems legit?
EvenBob
EvenBob
Joined: Jul 18, 2010
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November 9th, 2020 at 4:15:34 PM permalink
Quote: sabre

If you don't
realize that 4<5
then how can
you say that
someone
seems legit?



From the comments section of the video:

Mathematician here. I'm afraid you've made a big mistake in this analysis by not first plotting a histogram of the magnitudes of the dataset. Looking at it, it seems this is very restricted. Almost all numbers come in at 4 or 5 digits. As a result, Benford's law would not be expected to fit well.
This problem then worsens as you look at specific areas, and renders your chi squared tests invalid.
In order to properly test against an expectation of Benford you need data which ranges nicely over several orders of magnitude.

Answer by the video author:

NEDL
12 hours ago
Hi, and many thanks for a thought-provoking comment. I am aware of these applicability limitations of the Benford's law. However, there are two redeeming counterarguments one can make here. First, all of the tests have at least 30 (and generally much more) observations for all digits, basically a textbook case for Chi-squared applicability. Some alternative tests like Kolmogorov-Smirnov or Kuiper would be most relevant in case of smaller datasets (Chi-squared has higher power in larger samples). Second, as for orders of magnitude, the data does span seven, with most coming in 3 to 5 digits (3 is actually more frequent than 5). The more important question is whether you would have expected such data to abide by Benford's law under the null. This can be evidenced by Monte Carlo simulations and past elections. I plan to do a second video on the topic later in the week, addressing most notable suggestions and comments, including yours (will certainly do histograms, Monte Carlo, past elections, second-digit tests, and, if time allows, alternative goodness-of-fit tests).
"It's not enough to succeed, your friends must fail." Gore Vidal
terapined
terapined
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November 9th, 2020 at 4:20:40 PM permalink
Just playing devils advocate for math reasons
Aren't there ways of cheating to avoid being caught by Benfords law
Such as raising each county of Bidens total by the same exact amount.
It still generates a random number because you are adding the exact same number of votes to an already random numbers keeping them looking random.
"Everybody's bragging and drinking that wine, I can tell the Queen of Diamonds by the way she shines, Come to Daddy on an inside straight, I got no chance of losing this time" -Grateful Dead- "Loser"

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