Category Archives: election

When Benford’s Law Breaks Down

Can you use Benford's Law to reliably detect election fraud?

Benford's Law Leading Digit Distribution

This question has been raised often since the U.S. elections on 3 November 2020. The earliest mention of it we can find where it was used to question results in this election dates to 6 November 2020, at this Twitter thread, which featured this infographic looking at the city of Milwaukee's election results in the U.S. presidential race. That was followed by tweets looking at vote tallies in Michigan, Chicago, Allegheny County (Pittsburgh), and other cities and states with long reputations for election irregularities.

Much of the analysis behind these posts appears to have originated with Rajat Gupta's DIY Election Fraud Analysis Using Benford's Law tutorial from September 2020, which provides a guide for how to apply Benford's Law to evaluate election results using spreadsheet applications like Microsoft Excel.

As such tutorials go, it is relatively well done, providing step by step instructions that anyone with basic spreadsheet programming experience can follow to conduct their own analysis to see if their dataset of interest follows Benford's Law and generate a chart showing the outcome. But the tutorial has a huge problem, in that it skips over a huge factor that determines whether Benford's Law can even be successfully applied to evaluate whether the data represents the results of a natural outcome or has potentially been artificially manipulated.

To be successfully applied to detect potential fraud, the dataset must contain values that span several orders of magnitude. In the case of election results from voting precincts, the number of voters in individual precincts would have to number from the 10s, to the 100s, to the 1,000s, to the 10,000s, and so on.

But when you look at the precinct voting data covering the geographies in question, it most often covers data that predominantly falls within a single order of magnitude. This limitation makes Benford's Law an unreliable indicator for detecting potential fraud in this application.

Matt Parker, avid student of math gone wrong and currently the #1 best selling author in Amazon's Mathematics History subcategory, has put together the following video to explain why Benford's Law doesn't work well under these limiting circumstances. It is well worth the 18 minute investment of time to understand where Benford's Law can be successfully applied to detect potential fraud and where it cannot.

Benford's Law has recently been more successfully applied to challenge the validity of the number of coronavirus infections being reported by several nations. The difference in determining its success comes down to the virus' exponential rate of spread, which quickly generates an escalating number of cases that satisfies the requirement that the range of values subjected to analysis using Benford's Law span several orders of magnitude.

With that condition unmet for the raised election examples, we find that using Benford's Law to detect potential voter fraud is unreliable. If the authors of the analyses identifying voter fraud were aware of the deficiency of data used to support their findings, we would categorize these results as the product of junk science. We don't however think that harsh assessment applies for these cases, as most seem to be following an otherwise useful guide that omits presenting the conditions that must be satisfied to properly apply Benford's Law. That makes this situation different from other examples of junk science where the offenders clearly knew their data's deficiencies and failed to disclose them, sometimes with catastrophic effect.

We'll close by providing a short guide to our work covering related aspects of the topics we've discussed and pointing to academic work that provides more background.

Previously on Political Calculations


Deckert, Joseph; Myagkov, Mikhail; and Ordeshook, Peter C. Benford's Law and the Detection of Election Fraud. Political Analysis, Volume 19, Issue 3, Summer 2011, pp. 245-268. DOI: 10.1093/pan/mpr014.

Mebane, Walter R. New Research on Election Fraud and Benford's Law. Election Updates. [Online Article]. 23 August 2011.

Mebane, Walter R. Inappropriate Applications of Benford's Law Regularities to Some Data from the Presidential Election in the United States. [Online Article (PDF Document)]. 10 November 2020.

Brown, Michelle. Does the Application of Benford's Law Reliably Identify Fraud on Election Day? Georgetown University Graduate Theses and Dissertations - Public Policy. [PDF Document]. 2012.

Is The Typical American Household Better Off Today Than Four Years Ago?

"Are you better off today than you were four years ago?"

That question first became famous when asked in 1980 by then-presidential candidate Ronald Reagan. Every four years since, polling firm Gallup has asked that question whenever a presidential election is held in the U.S.

In 2020, the year of the coronavirus pandemic and a deep recession, they received a surprising response when asking that question of registered voters:

Gallup's most recent survey found a clear majority of registered voters (56%) saying they are better off now than they were four years ago, while 32% said they are worse off.

Gallup provides a chart showing the graphical results of their polling in the fourth year of the first terms of Presidents Ronald Reagan (1984), George H.W. Bush (1992), George H.W. Bush (2004), Barack Obama (2012), and Donald Trump (2020).

Gallup (7 October 2020): Americans' Views on Whether They're Better or Worse Off Than Four Years Ago

How could that possibly be? Thanks to state and local government lockdown orders that shuttered businesses and required Americans to stay-at-home in late March 2020, the U.S. economy has been experiencing one of the sharpest, deepest recessions in its history, from which it has only begun recovering in recent months as those lockdowns have been lifted. And yet, when asked during the two week period from 14 September through 28 September 2020, a clear majority of Americans stated they and their families were better off than four years ago.

We have unique data that explains that outcome, at least as it applies to the typical American household. Median Household Income is the measure of total money income earned by the American household at the exact middle of the nation's income earning spectrum. 50% of American households have higher incomes, 50% of American households have lower incomes.

Tracking how median household income changes over time can tell us a lot about the state of the typical American household. Not only can it tell us whether nominal incomes are rising or falling with economic conditions, if we adjust it for consumer price inflation, it can tell us a lot about the buying power of the incomes Americans earn.

We've visualized that information in a single chart that shows median household income measured in both these ways.

Median Household Income in the 21st Century: Nominal and Real Modeled Estimates, January 2009 - August 2020

The chart covers the period from January 2009 through August 2020, which captures the eight years of President Obama's tenure in office and most of President Trump's. With the 2020 presidential election a race between Joe Biden, who served as Vice President during President Obama's terms in office, and Donald Trump, who is running for reelection, it seems most appropriate to focus on this period to evaluate the effect of their respective policies on the welfare of the typical American household.

We see that nominal median household income, shown as the red data series in the chart, declined from $50,608 in January 2009 to $48,559 in early 2010, which then rose at a steady rate through 2016, before accelerating after January 2017. It ultimately peaked at $66,639 in February 2020 as the U.S. economy peaked before the onset of the coronavirus recession in March 2020. Through August 2020, median household income has fallen to $65,602.

The inflation adjusted data, shown as the blue data series, tells a similar story but with meaningful differences. In terms of constant August 2020 U.S. dollars, the Obama-Biden era began with median household income at $62,299, which then fell to $57,209 in early 2011. The buying power of the median income-earning U.S. household then stayed flat until mid-2014, when it finally began recovering.

Inflation-adjusted median household income would go on to slightly surpass its January 2009 level in early 2016, and then largely stagnated for the rest of the year before peaking in December 2016 at $62,440. A few months after President Trump assumed office in January 2017, the stagnation ended and the buying power of the typical American household rose above the levels recorded throughout the Obama-Biden era. The inflation adjusted median household income ultimately peaked at $67,131 in early 2020, but has since fallen with the coronavirus recession to its current level of $65,602.

These outcomes help explain the difference in Gallup's polling results for both 2012 and 2020. For 2020, a clear majority of Americans are answering that they are better off than four years ago because they are better off, even with the negative impact of the coronavirus recession.

Speaking of which, we see indications the recessionary trend for median household income in the U.S. reached a bottom in August 2020. We anticipate September 2020's data will show the first increase in this measure since the coronavirus recession began, as the economic recovery gains traction. The data for September 2020 will become available on 30 October 2020.

Analyst's Notes

Sentier Research suspended reporting its monthly Current Population Survey-based estimates of median household income, concluding their series with data for December 2019. In its absence, we are providing monthly estimates of median household income based upon our alternate methodology. Our references and data sources are presented in the following section.


Sentier Research. Household Income Trends: January 2000 through December 2019. [Excel Spreadsheet with Nominal Median Household Incomes for January 2000 through January 2013 courtesy of Doug Short]. [PDF Document]. Accessed 6 February 2020. [Note: We've converted all data to be in terms of current (nominal) U.S. dollars.]

U.S. Department of Labor Bureau of Labor Statistics. Consumer Price Index, All Urban Consumers - (CPI-U), U.S. City Average, All Items, 1982-84=100. [Online Database (via Federal Reserve Economic Data)]. Last Updated: 10 September 2020. Accessed: 10 September 2020.

U.S. Bureau of Economic Analysis. Table 2.6. Personal Income and Its Disposition, Monthly, Personal Income and Outlays, Not Seasonally Adjusted, Monthly, Middle of Month. Population. [Online Database (via Federal Reserve Economic Data)]. Last Updated: 1 October 2020. Accessed: 1 October 2020.

U.S. Bureau of Economic Analysis. Table 2.6. Personal Income and Its Disposition, Monthly, Personal Income and Outlays, Not Seasonally Adjusted, Monthly, Middle of Month. Compensation of Employees, Received: Wage and Salary Disbursements. [Online Database (via Federal Reserve Economic Data)]. Last Updated: 1 October 2020. Accessed: 1 October 2020.