Benford's Law and the 2020 Election

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, Biden’s vote numbers in do not match Benford’s 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.

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

I’ve always encountered Benford’s law when there was a power law driving generation of the numbers. What’s the corresponding principle that would make the law applicable to election results?

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

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

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?

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.

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?

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).

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.

Quote: EvenBob

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

That is a reasonable argument he makes and the guy does know how to perform statistical tests.

His smoking gun at the end of the video is Biden votes in 755 voting jurisdictions in what he calls Swing States. Here is his table.

Number	Actual	Expected
1	205.00	227.28
2	138.00	132.95
3	94.00	94.33
4	102.00	73.17
5	44.00	59.78
6	54.00	50.54
7	43.00	43.78
8	40.00	38.62
9	35.00	34.55
Total	755.00	755.00

I did the math and he is right, the p value is 1.97%. There is a shortage of Biden vote totals that begin with 1 or 5 and a surplus that begin with a 4. However, the results of results this skewed or more is close to 2%. My response is 2% is not that low. You can slice and dice the data all kinds of ways and it won't be hard to find some test that has a value under 2%.

To make any accusations of voter fraud there should be evidence beyond a reasonable doubt and 2% is definitely more than enough to have reasonable doubt. At least to me.

Second, I don't necessarily buy that voting district voting populations make for a good Benford test. For example 21 out of Michigan's 83 counties have a voting total in the range of 12,080 to 18,135. That is 25.3% in that tight range. I suspect that given typical rural county sizes and population densities that there are a lot of counties that frankly look similar to each other, violating the assumption of independence in Benford's test.

To be more convinced, I'd like to see the same test applied to the same states but different years.

In conclusion, decent video and I don't dispute the math. However, I just don't agree that the results he presents look very fishy.

If total ballots and total trump look normal, that's a pretty strong indication that the Biden count is real but odd. Manipulating the Biden vote requires manipulating the total ballots and/or the trump vote.

Quote: rdw4potus

If total ballots and total trump look normal, that's a pretty strong indication that the Biden count is real but odd. Manipulating the Biden vote requires manipulating the total ballots and/or the trump vote.

Can you give some numbers to back this up, please.

Can statistical findings based on Benford's Law be used as court evidence ? I mean any court case not specifically to US Election 2020 case.

Quote: ssho88

Can statistical findings based on Benford's Law be used as court evidence ? I mean any court case not specifically to US Election 2020 case.

It is admissible at least in cases of showing fraud in a company’s financial statements.

It’s still not clear to me that Benford’s law should apply to county level election results. But it does pop up in unexpected places. Would love to see past elections analyzed to see how they lined up.

Census.gov has a spreadsheet of County Population Totals: 2010-2019. I previously speculated county populations may not adhere to Benford's Law well because there are many counties with populations between 10,000 and 20,000, at least in Michigan.

To explore that further, I downloaded the entire USA spreadsheet and applied Benford's Law to the first and second digits of the populations of all 3,142 counties. The following table shows the number of counties by the leading digit in the population for 2019. The expected column shows the expected total per Benford's Law.

First Digit	Total	Expected
1	954	945.84
2	590	553.28
3	371	392.56
4	307	304.49
5	230	248.79
6	205	210.35
7	163	182.21
8	175	160.72
9	147	143.77
Total	3142	3142.00

The result of a chi-squared test has a p value of 37.42%. So that looks very normal.

The next table does the same thing, but with the second digit in the population figures.

Second Digit	Total	Expected
0	377	376.03
1	328	357.84
2	355	341.92
3	322	327.80
4	307	315.17
5	302	303.76
6	324	293.38
7	284	283.89
8	274	275.15
9	269	267.06
Total	3142	3142.00

A chi-squared test of this table has a p value of 68.59%. So again, it conforms with Benford very nicely.

I found this paper on Benford’s law as applied to elections. I haven’t had a chance to read more than the abstract so far. But wanted to link it here in case others are interested. It’s from 2017 so shouldn’t have a politically motivated conclusion as applied to current election.



https://www.cambridge.org/core/journals/political-analysis/article/benfords-law-and-the-detection-of-election-fraud/3B1D64E822371C461AF3C61CE91AAF6D

Quote: unJon

I found this paper on Benford’s law as applied to elections. I haven’t had a chance to read more than the abstract so far. But wanted to link it here in case others are interested. It’s from 2017 so shouldn’t have a politically motivated conclusion as applied to current election.

Here is a more direct link. I skimmed the paper. It basically uses Benford to analyze some elections in the Ukraine that were suspected of being fixed. I suspect similar academic papers will be written about this election.

So far, I have yet to see anything in 2020 that fails a Benford test beyond a reasonable doubt.

Simply put the numbers thru this analyzer!

https://www.dcode.fr/benford-law#0

I'm thinking about this more today. How does Benford work with numbers that are derived from each other? Say I start with county totals. They're random. If I split those county totals into two subsets, is it appropriate to expect each of those sub-sets to also conform to the original county-level expected digit distribution? I don't think it is. If 30% of the original set starts with a 1, I think it's much harder for 30% of each subset to also start with a 1.

Here's a guy who does Benford's Law for
a living and he found 'massive fraud' in
GA. You can skip the first half where he
mostly explains how this works.

Quote: EvenBob

Here's a guy who does Benford's Law for
a living and he found 'massive fraud' in
GA. You can skip the first half where he
mostly explains how this works.

This guy just does an eyeball test of the results with statements like, "You can just tell there are a lot of anomalies in here." So I had to do a chi-squared test myself on his table of the first digit for Georgia. Here are the same tables he has at the 9:30 point.

This first table is the first digit of the Biden vote total in Georgia's 159 counties.

Digit	Count	Expectations
1	47	47.86
2	40	28.00
3	16	19.87
4	21	15.41
5	7	12.59
6	9	10.64
7	9	9.22
8	5	8.13
9	5	7.28
Total	159	159.00

Here are the results of a chi-squared test:

chi-squred statitistic = 12.600898
p value = 0.126339

That p value is 1.14 standard deviations south of an 0.5 average.

Next, here is the same table for Trump votes.

Digit	Count	Expectations
1	43	47.86
2	36	28.00
3	12	19.87
4	15	15.41
5	9	12.59
6	17	10.64
7	9	9.22
8	9	8.13
9	9	7.28
Total	159	159.00

chi-squred statitistic = 11.230501
p value = 0.188978

That p value is 0.88 standard deviations south of an 0.5 average.

With p values of 12.6% and 18.9% for the two tables, this is a little more skewed than expectations, but nothing that screams fraud. Show me something that is more than three standard deviations from expectations and I'll start to take fraud accusations seriously.

What things might affect distribution besides fraud? For instance, parties gerrymandering their district certainly creates artificial population distribution of the electorate.

Quote: Wizard

and I'll start to take fraud accusations seriously.

This is why statistical math is such a
joke. I have a friend in his 80's who
was a math teacher for 55 years and
still makes money tutoring on the
net. He wants nothing to do with
stat or probability math because
he says you can manipulate it to
say anything you want it to say.
Four people will come up with
four different conclusions using
the same input data. To him this
isn't real math, it's voodoo. I
understand now what he's talking
about. It's joke math.. Like the math
people who tell me you can't beat
roulette. I just smile and answer
'whatever you say' and go out and
beat it every time.

Quote: EvenBob

This is why statistical math is such a
joke.

I, too, throw tantrums when the stats don’t reaffirm my prior beliefs.

The sheer scale of the purported voter fraud is mind-boggling. It would take thousands of people in hundreds of county Boards of Election across several states. Scores of republicans, including at least one republican secretary of state, all involved in a coordinated effort to defeat Donald Trump! And some people really think that's more likely than the less popular candidate losing a free and fair election.

People interpret statistics however they want, not manipulate. The nation would be a lot less dumb if our educational system put students on a statistics and probability path rather than calculus path by default.

Quote: mcallister3200

People interpret statistics however they want, not manipulate. The nation would be a lot less dumb if our educational system put students on a statistics and probability path rather than calculus path by default.

Mark Twain said 'There are three kinds of lies:
lies, damned lies, and statistics.' He was
mainly talking about political stats. I can
guarantee that if I post 10 different sources
for voter fraud using Benford, all the math
people here will say nope, their math is
wrong, nothing to see here. So there is
no point in even discussing it.

Wayne Allen Root, famous Vegas oddsmaker,
says the fix was in and a few insiders made
millions by betting Biden at the right minute.
Just before Fox called AZ for Biden, Trump's
odds of winning with bookmakers went all
the way to 8 to 1, and right then millions
worldwide was bet on Biden. After Fox
called AZ the odds went back to even
money immediately. Root says a few very
rich insiders knew the election was fixed
and took full advantage of it.

Quote: EvenBob

Mark Twain said 'There are three kinds of lies:
lies, damned lies, and statistics.' He was
mainly talking about political stats. I can
guarantee that if I post 10 different sources
for voter fraud using Benford, all the math
people here will say nope, their math is
wrong, nothing to see here. So there is
no point in even discussing it.

Wayne Allen Root, famous Vegas oddsmaker,
says the fix was in and a few insiders made
millions by betting Biden at the right minute.
Just before Fox called AZ for Biden, Trump's
odds of winning with bookmakers went all
the way to 8 to 1, and right then millions
worldwide was bet on Biden. After Fox
called AZ the odds went back to even
money immediately. Root says a few very
rich insiders knew the election was fixed
and took full advantage of it.

So now you're saying Fox is in on the fix? They were the only news organization to call AZ early, so I guess that's really the only way your story could possibly work. Wow, that's a very odd co-conspirator.

Quote: EvenBob

Mark Twain said 'There are three kinds of lies:
lies, damned lies, and statistics.' He was
mainly talking about political stats. I can
guarantee that if I post 10 different sources
for voter fraud using Benford, all the math
people here will say nope, their math is
wrong, nothing to see here. So there is
no point in even discussing it.

Wayne Allen Root, famous Vegas oddsmaker,
says the fix was in and a few insiders made
millions by betting Biden at the right minute.
Just before Fox called AZ for Biden, Trump's
odds of winning with bookmakers went all
the way to 8 to 1, and right then millions
worldwide was bet on Biden. After Fox
called AZ the odds went back to even
money immediately. Root says a few very
rich insiders knew the election was fixed
and took full advantage of it.

I had Biden at -130 bet the day before election. Election night when Florida was going Trump I tried to double down on Biden at +400. Couldn’t get it down though. There was a crazy swing in the market that I thought was an overreaction. It was.

Quote: EvenBob

This is why statistical math is such a
joke. I have a friend in his 80's who
was a math teacher for 55 years and
still makes money tutoring on the
net. He wants nothing to do with
stat or probability math because
he says you can manipulate it to
say anything you want it to say.
.

One of my favorite books is How To Lie With Statistics. It is on one of my bookshelves.

https://en.wikipedia.org/wiki/How_to_Lie_with_Statistics#:~:text=How%20to%20Lie%20with%20Statistics%20is%20a%20book%20written%20by,to%22%20articles%20as%20a%20freelancer.

I just ran across a new video on Benford's Law and the 2020 Election by one of my favorite YouTubers, Matt Parker. He mainly looks at applying Benford's Law to the Chicago election results.


Direct: https://www.youtube.com/watch?v=etx0k1nLn78.

I've read about 5 articles from
people who are experts in the
field and they all say Benford's
Law used in elections is useless
because the "spread of orders of
magnitude" of the numbers
involved are not large enough.
It has to do with precinct size.
Precincts don’t have that much
size variation in them.

So that's that. Never mind.

Quote: Wizard

I just ran across a new video on Benford's Law and the 2020 Election by one of my favorite YouTubers, Matt Parker. He mainly looks at applying Benford's Law to the Chicago election results.

I just watched this as was going to post a link. Matt makes math fun and entertaining.

Quote: EvenBob

I've read about 5 articles from
people who are experts in the
field and they all say Benford's
Law used in elections is useless
because the "spread of orders of
magnitude" of the numbers
involved are not large enough.
It has to do with precinct size.
Precincts don’t have that much
size variation in them.

So that's that. Never mind.

That is Matt's point in the video I just posted.

In my opinion a good place to start is to do a Benford test on the total population in each jurisdiction or the total voting population. Doing that in the various counties in Michigan or the combined United States seems to pass a Benford test. These are the only two regions I've done the test on. If that seems to indicate a natural spread in population numbers, it's fair to consider a Benford test on vote totals.

This is historic, I think it's the first time you've changed your position on something.

Quote: Wizard

This is historic, I think it's the first time you've changed your position on something.

I never had a position to change.
I had never heard of this stupid
'law' before I posted about it. I
just assumed it was true because
it had the word 'law' attached to
it. I should have realized that in
math 'law' is just a meaningless
word.

Quote: EvenBob

I never had a position to change.
I had never heard of this stupid
'law' before I posted about it. I
just assumed it was true because
it had the word 'law' attached to
it. I should have realized that in
math 'law' is just a meaningless
word.

The word "law" in mathematics seems to mean something that is always true and self-evident. For example, the Commutative Law says that a+b=b+a. It's just obvious and fundamental. Not something that I think can be formally proven as it is part of the foundation on which math is built.

To be honest, I am now sure why Bedford's Law is a "law." It is a true statement, but it can be proven with logarithms. It seems more like a theorem to me. Perhaps one of the other math-heads on the forum can take this further.

Are a “property” and “law” interchangeable terms in mathematics?

Also what’s the deal with some people calling it math, and some people calling it maths, is that just like the thing where some people insist on inserting unnecessary “u’s” in inappropriate places after an o like color/colour, behavior/behaviour etc?

Quote: Wizard

It seems more like a theorem to me.

It's a theory that says if the
numbers are just the right
kind of groupings, this
thing might work. Or not.
It's just more pretend
probability math that real
math people don't go near.

Quote: DRich

One of my favorite books is How To Lie With Statistics. It is on one of my bookshelves.

Fascinating book. Was required reading in a stats class I took in 1989. I really should re-read. Thanks!

Quote: Wizard

I just ran across a new video on Benford's Law and the 2020 Election by one of my favorite YouTubers, Matt Parker.

Thank you for sharing. I had not heard of Matt, but that was a great video. I'll certainly start diving through more.

Two issues might cause this analysis to be suspect ... First, 83 data points is a fairly small universe to observe a genuine Benford distribution. But second, and much more importantly, it's Michigan. and at the risk of turning this into a political discussion, you can count on the data from most counties being honest, but when it comes to Wayne County (i.e. Detroit and the surrounding suburbs) the levels of graft and corruption there are stunning. (I know, I lived there for the first 23 years of my life) So as a test, I think it's more realistic to look at the 1165 individual precinct reports from Wayne County rather than the overall county totals. Of course the data for each precinct can't be found anywhere on the internet (though I had no problem finding it for my home state of NH) ...

So, at least as a check for the honesty of data coming from Wayne County, I thought I might take a look at the 2016 election ... and the results were "interesting" to say the least. First, they don't make it easy. Transparency laws require the data be published (eventually) but the data for Detroit was scanned as image data into a PDF which required OCR software to turn back into "real" data. And then a check to be sure the OCR didn't mess with too many data points. Finally I got this:

For President Trump the table looks like this (Wish I could show the graph, it's more informative)

Digit	Count	Expectations
0	22	0
1	252	350.7
2	137	205.2
3	164	166.5
4	153	112.9
5	145	92.3
6	89	77.9
7	94	67.6
8	59	59.6
9	50	53.4

As we can see the data looks a little wonky ... the first thing to notice is 22 precincts reported zero votes for President Trump ... virtually a statistical impossibility, even in Detroit ... but it appears the numbers my have been manipulated .... So we begin to wonder, since President Trump won Michigan in 2016, might his campaign have been manipulating the data?

Then we have a look at the data for Secretary Clinton ... and we find the data is totally wacked .... (again, the Excel chart shows it better)

Digit	Count	Expectations
0	0	0
1	92	350.7
2	179	205.2
3	279	166.5
4	232	112.9
5	163	92.3
6	107	77.9
7	64	67.6
8	33	59.6
9	16	53.4

Benford analysis shows that the data was clearly manipulated for Secretary Clinton, and probably for President Trump ... but it really doesn't indicate how the data was changed or who was supposed to be the beneficiary .... Thoughts?

Quote: mcallister3200

Are a “property” and “law” interchangeable terms in mathematics?

Also what’s the deal with some people calling it math, and some people calling it maths, is that just like the thing where some people insist on inserting unnecessary “u’s” in inappropriate places after an o like color/colour, behavior/behaviour etc?

You mean English English, as opposed to 'simplified English' for American's $:o)

Quote: pjt36

For President Trump the table looks like this (Wish I could show the graph, it's more informative)

Digit	Count	Expectations
0	22	0
1	252	350.7
2	137	205.2
3	164	166.5
4	153	112.9
5	145	92.3
6	89	77.9
7	94	67.6
8	59	59.6
9	50	53.4

As we can see the data looks a little wonky ... the first thing to notice is 22 precincts reported zero votes for President Trump ... virtually a statistical impossibility, even in Detroit ... but it appears the numbers my have been manipulated .... So we begin to wonder, since President Trump won Michigan in 2016, might his campaign have been manipulating the data?

Then we have a look at the data for Secretary Clinton ... and we find the data is totally wacked .... (again, the Excel chart shows it better)

Digit	Count	Expectations
0	0	0
1	92	350.7
2	179	205.2
3	279	166.5
4	232	112.9
5	163	92.3
6	107	77.9
7	64	67.6
8	33	59.6
9	16	53.4

May I see the raw data? I'd like to to a Bedford test on the total combined votes. I suspect this is a situation like Matt Parker described in Chicago where the precincts are so small and consistent in population that you expect first digit clumping.

Quote: EvenBob

It's a theory that says if the
numbers are just the right
kind of groupings, this
thing might work. Or not.
It's just more pretend
probability math that real
math people don't go near.

Well not really. It works anytime there is some exponential growth or power law that generates the numbers. It’s just a question of asking whether that assumption is applicable to a given situation. The neat thing is it pops up in unexpected places like a normal distribution does.

Here are the 2016 Wayne county results. https://www.waynecounty.com/elected/clerk/november-8-2016-general.aspx

I did two very simple simulations based on the county sizes and how many people voted in 2016 (I know more people did in 2020 but it probably doesn't matter whether 39% or 50+% voted) and ran ten elections and listed each of the county results. I had four candidates and assumed the voting was (i) close 49.5% 48.5% 1.8% 0.2% (ii) less close 51%/47%/1.8%/0.2% with a slight random variation added. Since I don't exactly know what one is supposed to notice, other than the ratios being associated with the logarithms. I might run other simulations with a higher turnout later.

	1st digit	2nd digit
0	0	7 530	0.000%	11.983%
1	18 095	6 702	28.795%	10.665%
2	11 124	6 596	17.702%	10.496%
3	8 475	6 485	13.487%	10.320%
4	6 119	6 379	9.737%	10.151%
5	5 358	6 246	8.526%	9.940%
6	4 227	5 978	6.727%	9.513%
7	3 381	5 764	5.380%	9.173%
8	3 137	5 553	4.992%	8.837%
9	2 924	5 607	4.653%	8.923%

	1st digit	2nd digit
0	0	7 555	0.000%	12.023%
1	18 099	6 884	28.802%	10.955%
2	11 177	6 608	17.786%	10.516%
3	8 408	6 543	13.380%	10.412%
4	6 157	6 426	9.798%	10.226%
5	5 343	6 046	8.503%	9.621%
6	4 268	5 859	6.792%	9.324%
7	3 387	5 751	5.390%	9.152%
8	3 115	5 718	4.957%	9.099%
9	2 886	5 450	4.593%	8.673%

Quote: miplet

Here are the 2016 Wayne county results. https://www.waynecounty.com/elected/clerk/november-8-2016-general.aspx

Just eyeballing this table of Wayne County election results in 2016, each precinct has total votes in the three digits. It is not going to be appropriate to apply Bedford's Law to such data.

Exactly the same issue as Matt Parker discussed about Chicago.

Charlie, if you have the vote totals in a spreadsheet, can you share them please.

Is there a way to attach a spreadsheet here? ..
miplet linked to the Wayne County data, but those are the scanned images (pita to work with)
You're almost certainly right that this data suffers the same shortcomings as Parker's analysis of Chicago
The Benford analysis for the Total vote in Wayne County shows a more reasonable Benford distribution, but you can see the spike at 2-3 from Secretary Clinton's returns easily overlayed, even on that curve.

Digit	Count	Expectations
0	0	0
1	296	350.7
2	151	205.2
3	192	166.5
4	153	112.9
5	101	92.3
6	92	77.9
7	64	67.6
8	58	59.6
9	58	53.4

I also had a look at 2016 precinct data from Los Angeles County (4500+ datapoints) and saw similar results .... though Secretary Clinton's graphed peaked at 5 there rather than 2.... But I suspect, again, that this data is similar to the Chicago and Detroit data. This will probably be true for any large metropolitan area as their precincts are more uniform in size. Statewide analysis at the precinct level may work, but this data's more likely to be "honest" as it includes large areas not controlled by any one political faction.

Finally, took a look at the "last two" digits for Secretary Clinton's returns which showed a similar "random" distribution as Parker's Chicago data.
---
edit: Let me just add here that the expected "random" distribution for the two digit trailing test should be 11.65 for any single instance but this does show spikes above 20 for 93 and 98 .... probably nothing :)
---
.....
So since we've apparently put the "Human Generated" manipulation issue to bed (or at least failed to demonstrate it via Benford), anyone want to have a look at the Edison Research data to see if the machines themselves have been compromised ;-P
---
2nd Edit: ... an interesting presentation by Dr. Shiva Ayyadurai addresses the "Machine" issue here;

MIT PhD Analysis of Michigan Votes Reveals Unfortunate Truth of US Voting Systems

.... One significant conclusion is that whatever the math, the systemic issues raised are real ... and if those aren't fixed, bookmakers are going to stop taking bets on elections.