Benford’s Law can recognize the probabilities of highly likely or highly unlikely frequencies of numbers in a data set. The probabilities are based on mathematical logarithms of the occurrence of digits in randomly generated numbers in large data sets.
What are the Right Circumstances for Using Benford’s Law?
Almost from the beginning, proponents of Benford’s Law have suggested that it would be a beneficial tool for fraud detection.
Benford’s Law can recognize the probabilities of highly likely or highly unlikely frequencies of numbers in a data set. The probabilities are based on mathematical logarithms of the occurrence of digits in randomly generated numbers in large data sets. Those who are not aware of this theory and intentionally manipulate numbers (e.g., in a fraud) are susceptible to getting caught by the application of Benford’s Law.
Modeling observations: Our simulation model, that produced the data used in this post, is a simplification of what is inherently a complex process that varies state to state. We simulated counties casting a number of votes based on what was reported by NYT for Wisconsin in 2020.
At each time step, we assumed that a poisson-distributed number of votes were counted with an expected rate of 1000 votes/step. In reality, the rate is likely somewhat higher for larger counties, but we note that this assumption nonetheless captures the phenomenon of larger counties taking longer to count.
The rate limits, in combination with a large number of smaller counties, leads to early counties being approximately the same order of magnitude. At this stage, the assumptions that lead to Benford’s law are violated leading to the patterns generated in the Figure 1 above.
Only once all counties have been counted does the distribution approach something consistent with Benford’s law, seen in Figure 2.
Even at this stage, the distribution of county sizes still makes it unlikely to exactly match expectations. A more complete model might include non-random voting patterns whereby rural counties lean a different direction than urban ones.
This, compared with the relationship between the rate of vote counting and county or precinct size would probably cause more drastic violations of assumptions. As this is a rapid response, incorporating this complexity was impractical. For a more rigorous discussion of Benford’s law and its limited utility for detecting electoral fraud see Deckert, Myagkov, Ordeshook 2011.
Benfords law has been used on many occasions to prove election fraud (2000&2004 elections, 2003 California gubernatorial election, etc.).
Let me know if you would elaboration but the numbers can be computed in a spreadsheet using data taken from Michigan’s election data tracker from individual precincts. For context, Biden scored a 0.0000017 and trump scored a 0.58.
While Benford’s Law is not 100% proof that fraudulent data has been manufactured, “In the United States, evidence-based on Benford’s law has been admitted in criminal cases at the federal, state, and local levels.”
It was also used to determine fraud in the 2000 and 2004 U.S elections. I conducted a Chi-test comparing Michigan’s precinct vote counts to Benford’s law and found that Biden/Haris votes returned a, 0.000017% (Statistically significant, especially with a very large sample) whereas Trump/Pence votes returned a score of,
53.059791% and whilst looking at my data set I noticed there were 0 write-in votes in Michigan.
Very odd stuff.
This is the source I used for my data, https://detroitmi.gov/webapp/election-results.
Here are some pictures of the spreadsheets with a graph comparison of the expected results vs what actually happened.
Trump’s data matches the law
(not manufactured vote counts):
Biden’s data differs greatly (fake data)
The highlighted percentage value tells you how sure you can be that fraud was committed, typically a score below 0.05 is statistically significant, and Biden scored a 0.00000017
NEDL Applying Benford’s law to US 2020 presidential race: Analysing elections with maths!
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