Category Archives: politics

Visualizing the Border Migration Crisis’ Surge in Detentions

Beginning as the Biden-Harris administration assumed power on 20 January 2021, the border migration crisis in the U.S. arose as the new administration significantly altered the nation's established border enforcement policies. Now, the number of detentions along the southwestern border of the United States with Mexico have hit a 21-year high.

Here's what that looks like in terms of the U.S. Customs and Border Protection (CPB) agency's monthly count of detentions at the U.S.-Mexico border:

CBP Detentions Along Southwestern U.S. Border with Mexico, October 2020 - April 2021

Here's where the problem originated:

Biden officials are trying to fulfill their campaign promises on immigration but have found that quickly reversing Trump’s policies can create an abundance of political headaches and contribute to a host of other problems, including trying to process and house a record number of unaccompanied children crossing the southern border.

That observation is backed up by the numbers, even as the Biden-Harris administration has started playing a shell game with how it manages the detained unaccompanied minors in its custody to try to conceal it.

COVID-19 and the 2021 Border Migration Crisis in Arizona

Three of the four high quality COVID-19 datasets we track for Arizona indicate a new adverse change in trends for the coronavirus pandemic in the state. Arizona's data for positive test results by date of sample collection, new hospital admissions, and ICU bed usage are confirming an increase in the incidence of COVID in the state. The fourth dataset for COVID-19 deaths by recorded date of death certificate does not as yet, but would not be expected to as yet because this data has the greatest lag from a change in the rate of incidence to the confirmation of a change in trend.

We've generated the following animated chart to cycle through each of the charts for these four datasets.

Animated Chart: Arizona COVID-19 Positive Test Results, New Hospital Admissions, ICU Bed Usage, and Deaths, 1 January 2020 - 24 April 2021

Considering the respective lags that apply for each dataset, the likely timing of a significant change in the rate of incidence for SARS-CoV-2 coronavirus infections in the state between 26 March 2021 and 30 March 2021. This period coincides with contemporary news accounts of the Biden administration moving migrants from overloaded Border Patrol migrant facilities in Arizona counties bordering Mexico to facilities and small towns in Maricopa and Pinal Counties.

We think this activity is showing up in Arizona's COVID-19 statistics because these migrants have been exposed to Mexico's higher incidence of COVID-19 infections. While those cases peaked in late January 2021, several weeks after they peaked and began falling in Arizona, the relative difference in infection rates between Arizona's population and the entering migrants is enough to affect the trends for Arizona's COVID-19 data.

On 26 March 2021, Senator Mark Kelly (D-AZ) stated the Biden administration did not have an effective plan for the border. It took another month for President Biden to acknowledge the migration crisis at the U.S. border with Mexico as a crisis.

On 21 April 2021, Arizona Governor Doug Ducey declared a state of emergency and sent Arizona National Guard troops to Arizona's border counties.

Coincidentally, that was the last date we updated our series on the COVID-19 pandemic in Arizona, where our analysis of the trends just before that point of time appeared very early in the morning....

Previously on Political Calculations

Here is our previous coverage of Arizona's experience with the coronavirus pandemic, presented in reverse chronological order.


We've continued following Arizona's experience during the coronavirus pandemic because the state's Department of Health Services makes detailed, high quality time series data available, which makes it easy to apply the back calculation method to identify the timing and events that caused changes in the state's COVID-19 trends. This section links that that resource and many of the others we've found useful throughout the coronavirus pandemic.

Arizona Department of Health Services. COVID-19 Data Dashboard: Vaccine Administration. [Online Database]. Accessed 25 April 2021.

Stephen A. Lauer, Kyra H. Grantz, Qifang Bi, Forrest K. Jones, Qulu Zheng, Hannah R. Meredith, Andrew S. Azman, Nicholas G. Reich, Justin Lessler. The Incubation Period of Coronavirus Disease 2019 (COVID-19) From Publicly Reported Confirmed Cases: Estimation and Application. Annals of Internal Medicine, 5 May 2020.

U.S. Centers for Disease Control and Prevention. COVID-19 Pandemic Planning Scenarios. [PDF Document]. Updated 10 September 2020.

More or Less: Behind the Stats. Ethnic minority deaths, climate change and lockdown. Interview with Kit Yates discussing back calculation. BBC Radio 4. [Podcast: 8:18 to 14:07]. 29 April 2020.

Reliving a Day of Panic in the Coronavirus Pandemic

Today marks the anniversary of the most pivotal moment in New York Governor Cuomo's COVID nursing home deaths scandals. Because one year ago today, Governor Cuomo and senior members of his administration reached the point of panic as they struggled to address the greatest challenge of his tenure in office.

We originally presented that story on 12 May 2020. Today, we're re-running that original article, in which we recreated critical information that influenced the most consequential decision Governor Cuomo made on that day. The deadly repercussions of what resulted from that day of panic are still rippling through New York and making national news a year later. Let's get started....

COVID-19 - Martin Sanchez via Unsplash:

We're fascinated with how politicians use data and models in setting the policies they pursue, where knowing both what they knew and when they knew it can explain a lot about why they made the choices they did at the time they made them.

To that end, we've been paying attention to how Governor Andrew Cuomo has been managing the difficult task of coping with the coronavirus epidemic in New York, and in New York City in particular, which has been the focal point for both the number of cases and the spread of the SARS-CoV-2 coronavirus across the United States. We've assembled a timeline of Governor Cuomo discussing the predictive models for how fast the coronavirus infection would spread within New York, which provides insight into how that information affected his decisions for how to allocate the limited health care resources over which he had influence during the worst part of the epidemic in his state.

We're going to pick up the action shortly after 7 March 2020, the date Governor Cuomo declared a state of emergency because of the coronavirus epidemic in New York, when the number of coronavirus cases within the state had 'soared' to 89. The following article is the earliest in which we find a reference to coronavirus modeling projections for New York City, which had been put together by New York City Mayor Bill de Blasio's staff:

9 March 2020: Coronavirus Cases in New York State Rise to 105:

Mayor Bill de Blasio said Sunday that the city had 13 confirmed cases, including a new case of a man in the Bronx. Based on modeling, his team estimated there could be 100 cases in the next two or three weeks, but for most people, the illness would result in very mild symptoms.

Three days later, New York City had nearly reached that total and was set to blast through it, prompting Governor Cuomo to ban all public events with more than 500 people in attendance and to require gatherings with fewer than 500 people to cut capacity by 50%. The faster than previously projected growth in the number of COVID-19 infections drove a change in public policy.

Four days after that, Governor Cuomo had clearly been presented with projections that showed the exponential growth in the number of cases that had gotten underway in New York.

16 March 2020 - Audio & Rush Transcript: Governor Cuomo is a Guest on CNN's Cuomo Prime Time:

"I see a wave and the wave is going to break on the health care system ... You take any numerical projections on any of the models and our health care system has no capacity to deal with it."...

"Yeah. I think you look at that trajectory, just go dot, dot, dot, dot, connect the dots with a pencil. You look at that arc, we're up to about 900 cases in New York. It's doubling on a weekly basis. You draw that arc, you understand we only have 53,000 hospital beds total, 3,000 ICU beds, we go over the top very soon."

At this point, Governor Cuomo was beginning to appreciate that the thousands of hospital beds across the state of New York were really a scarce resource. He expanded on that realization the next day after an overnight surge in the number of reported cases:

17 March 2020 - Video, Audio, Photos & Rush Transcript: Governor Cuomo Announces Three-Way Agreement with Legislature on Paid Sick Leave Bill to Provide Immediate Assistance for New Yorkers Impacted By COVID-19:

"There is a curve, everyone's talked about the curve, everyone's talked about the height and the speed of the curve and flattening the curve. I've said that curve is going to turn into a wave and the wave is going to crash on the hospital system.

I've said that from day one because that's what the numbers would dictate and this is about numbers and this is about facts. This is not about prophecies or science fiction movies. We have months and moths of data as to how this virus operates. You can go back to China. That's now five, six months of experience. So just project from what you know. You don't have to guess.

We have 53,000 hospital beds in the State of New York. We have 3,000 ICU beds. Right now the hospitalization rate is running between 15 and 19 percent from our sample of the tests we take. We have 19.5 million people in the State of New York. We have spent much time with many experts projecting what the virus could actually do, going back, getting the China numbers, the South Korea numbers, the Italy numbers, looking at our rate of spread because we're trying to determine what is the apex of that curve, what is the consequence so we can match it to the capacity of the health care system. Match it to the capacity of the health care system. That is the entire exercise.

The, quote on quote, experts, and by the way there are no phenomenal experts in this area. They're all using the same data that the virus has shown over the past few months in other countries, but there are extrapolating from that data.

The expected peak is around 45 days. That can be plus or minus depending on what we do. They are expecting as many as 55,000 to 110,000 hospital beds will be needed at that point. That my friends is the problem that we have been talking about since we began this exercise. You take the 55,000 to 110,000 hospital beds and compare it to a capacity of 53,000 beds and you understand the challenge."

Faced with the potential shortage of needing 110,000 beds and only having 53,000 to provide care to coronavirus patients in New York, Governor Cuomo lobbied President Trump for support, which resulted in President Trump ordering the U.S. Navy's hospital ship USNS Comfort to sail to New York City the next day, and also lobbied for the U.S. Army's Corps of Engineers to begin identifying public facilities in New York City to be converted for use as temporary hospitals to handle the projected overflow of coronavirus patients from regular hospitals.

USNS Comfort would arrive in New York City on 30 March 2020, and the Army Corps of Engineers would have 1,000 beds ready at New York City's Javits Center ready on 27 March 2020, and were working to expand it to a 2,500 bed temporary hospital facility by 1 April 2020. But during the time in between, the updated projections of the coronavirus models led Governor Cuomo to panic.

24 March 2020: Andrew Cuomo: Apex of coronavirus outbreak in NY two or three weeks away:

Cuomo, speaking at his daily COVD-19 briefing in Manhattan, said the state's projection models now suggest the apex of the coronavirus crisis could hit New York within 14 to 21 days, rather than the 45 days the state projected late last week.

He likened it to a "bullet train" headed for New York, urging the federal government to deploy as many ventilators and as much protective medical gear it can to the state as quickly as possible.

"Where are they?" Cuomo said. "Where are the ventilators? Where are the masks? Where are the gowns? Where are they?”

At this point, we should show what one of the more influential coronavirus models that Governor Cuomo was using looked like. The following chart is taken from the Institute for Health Metrics and Evaluation (IHME)'s 25 March 2020 projections showing its estimates of the minimum, likely, and maximum number of additional hospital beds that would be needed in the state of New York to care for the model's expected surge of coronavirus patients.

IHME Forecast of All Hospital Beds Required for COVID-19 Care Beyond Available Capacity in New York State, Projection from 25 March 2020

This is just one of several coronavirus models whose projections were being combined and presented to Governor Cuomo by consultants from McKinsey & Co., where the IHME's coronavirus model's projections for New York are consistent with the figures and timing of a peak cited by Governor Cuomo in the days preceding his panic.

Faced with what appeared to be an imminent shortage of hospital beds and other medical resources, the Cuomo administration appears to have adopted an emergency triage strategy, one that would have devastatingly deadly consequences. Here, to free up as many beds as possible in New York's near-capacity hospitals, the Cuomo administration would try to move as many patients infected with the SARS-CoV-2 coronavirus as they could out of these facilities into others, even though they could still be contagious and present the risk of spreading infections within the facilities to which they would be transferred.

25 March 2020: The facilities in which they chose to place them were predominantly privately run nursing homes, where a directive issued by the state's Department of Health on 25 March 2020 mandated they must admit them into their facilities, where refusals could mean the loss of their New York state-issued licenses to operate.

New York Department Of Health Directive to Nursing Homes Mandating Admission of Coronavirus-Infected Patients, 25 March 2020

Flashing forward to the end of March 2020, the coronavirus epidemic forecast models Governor Cuomo was using in making his decisions were pointing to the peak still being ahead:

Cuomo said various predictive models being used by New York indicate the apex of the surge for hospitals will come anywhere from 7 to 21 days from now.

“The virus is more powerful, more dangerous than we expected,” Cuomo said. “We’re still going up the mountain. The main battle is on top of the mountain.”

Four days later, the coronavirus models were predicting the peak was almost upon New York:

While giving an update Saturday on the frantic work to ready New York hospitals for the most intense period of the coronavirus (COVID-19) crisis, Gov. Andrew Cuomo said that the state’s models put the so-called apex about four-to-eight days out.

“By the numbers, we’re not yet at the apex. We’re getting closer,” he said at his daily press briefing. “Depending on whose model you look at, they’ll say four, five, six, seven, days, some people go out 14 days. But our reading of the projections is that we’re somewhere in the seven-day range. Four, five, six, seven, eight-day range.”

“Part of me would like to be at the apex, and just, let’s do it,” Cuomo continued. “But there’s part of me that says it’s good that we’re not at the apex because we’re not yet ready for the apex, either. We’re not yet ready for the high point...the more time we have to improve the capacity, the better.”

But on 6 April 2020, the IHME model revised its estimates for New York and the U.S. downward, indicating the peak Governor Cuomo feared would overwhelm New York's hospitals was not going to come anywhere close to what it had previously projected. On 8 April 2020, it indicated New York had already passed its peak in number of daily new cases.

Ordinarily, that would be a good thing. Except, Governor Cuomo had taken an action by which he intended to avoid the spectacle of having pictures of sick New Yorkers not able to get medical treatment in the media, but instead ensured the state's death toll from its coronavirus epidemic would no longer be small. That part of the story has its own special timeline, which we've moved here from the bottom of the article where we had previously been piecing together this part of the story of COVID-19 in New York....

Image credit: unsplash-logoMartin Sanchez

The explosion of Cuomo scandal news has prompted us to launch a new blog to host the timeline we had been updating regularly in this space! We officially launched the new site a week ago. If you haven't yet seen it, may we introduce A Timeline of New York Governor Andrew Cuomo's Nursing Home Scandals.

The Governor Who Kills Grandmas?

Now serving all your Cuomo nursing home scandal news needs!

A Tale of Two States and the Coronavirus

Although California and Arizona are geographically next to each other, both states have had very different experiences with the coronavirus.

That's because unique circumstances within both states have affected the progression of SARS-CoV-2 coronavirus infections within each during the pandemic.

That observation is driven home when we compare the reporting of newly confirmed COVID-19 cases in both states. Here, because California doesn't make the same kind of high quality data available to the public that Arizona's Department of Health Services does, we're turning to Johns Hopkins CSSE COVID-19 Data as our data source for both states. Using this data will allow for a more apples-to-apples comparison for applying the back calculation technique we've developed in analyzing Arizona's experience with the coronavirus pandemic.

That technique involves identifying turning points that changed the trajectory of newly reported cases following events that changed the rate of incidence of new infections. These turning points begin appearing in the rolling seven-day moving averages for newly reported cases some 9 to 11 days after the events that changed the viral infections rate of incidence, which corresponds with when 95% of all infections following an initial exposure event have developed. This process identifies a specific window of time in which contemporary news reports involving large gatherings may be reviewed to identify any "superspreader" events that prompted the change in the rate at which the coronavirus spreads.

That's the background - let's look at what happened in California during the last four months of 2020.

Newly Reported COVID-19 Cases in California, 1 September - 31 December 2020

In this chart, the vertical dashed lines indicate when the trajectory of newly reported cases definitively broke from the seven day moving average trend immediately preceding it, as indicated by the heavy blue line. The orange-shaded vertical bands indicate the period some 9-11 days earlier in which the events triggering the change in trend would have occurred. We've labeled five events in the chart with letters from A through E, the following section describes the corresponding events and their subsequent effect on California's reported incidence of COVID-19 cases.

Major Milestones for COVID-19 in California

Event A: L.A. Lakers win NBA Championship

After several weeks of following a flat-line trend averaging 3,200 new cases per day, the incidence of COVID-19 cases in California notched up to average roughly 4,400 new cases per day beginning on 22 October 2020, with the increase largely contained within Los Angeles County. This change occured 10 days following the L.A. Lakers NBA championship on 12 October 2020, which saw large crowds in L.A. celebrating the victory. Compared to what happened later however, the Lakers victory had a comparatively low impact.

Event B: L.A. Dodgers win World Series

The Dodgers victory over the Tampa Bay Rays in the 2020 World Series was the real spark that ignited California's late surge of new coronavirus cases. Here, we see the change in trend with the number of cases rising definitively above their post-Lakers championship high on 6 November 2020. Ticking the calendar backward by 9 days puts us at 28 October 2020, which is when the Dodgers became baseball's world champions. Much larger crowds in L.A. celebrated the event, pointing to baseball's much larger fan base in the region. Following the Dodgers victory, Los Angeles County led California for growth in the number of new cases, with the surrounding counties that make up L.A.'s greater metropolitan area following suit.

Event C: "Emergency Brake" Restrictions Go Into Effect

The next significant change in trajectory took hold beginning on 25 November 2020 (Thanksgiving Day), 9 days after California Governor Newsom imposed new restrictions on the operations of California businesses. The restrictions appear to have been somewhat successful in sharply slowing the rate of increase that began with Dodgers' world series win, but proved to be short lived.

Event D: A Clumsy Curfew and Governor Newsom's Loss of Credibility

On 2 December 2020, the number of new COVID-19 cases in California began increasing faster than they had before. Tracking significant events backwards, we find Governor Newsom's announced month-long curfew was the trigger for the increase, although it occurred 12 days earlier, on 23 November 2020.

That's because the curfew announcement sparked a popular backlash against Governor Newsom, with large political protests beginning the following day, which falls within the expected 9-11 day window. The protests came as public outrage spiked following the publication of photos of Governor Newsom disregarding his COVID-19 rules at a dinner held at the exclusive French Laundry restaurant less than two weeks earlier.

Governor Newsom was far from the only Calfornian official to shred their own credibility through outright hypocrisy during this time. On 4 December 2020, Governor Newsom announced a new stay-at-home order for California residents, while local governments announced they would ban outdoor dining, but instead of reducing the state's upward COVID-19 trajectory as might be expected in the period from 13 through 15 December 2020, the number of cases rose sharply as many small restaurant owners and their customers began protesting the officials' apparent determination to destroy their businesses. These actions were further undermined when California's top public health official confirmed the state and local governments' new restrictions lacked any support from scientific data indicating they would be effective in slowing the rate of COVID-19 infections.

Update 22 January 2021: Infectious disease experts have identified the ban on outdoor dining as a primary cause of California's winter surge of excess coronavirus cases, hospitalizations, and deaths.

Event E: California's ICUs Fill Up with COVID Cases and Christmas Travel

It's not until 22 December 2020 that we see any reversal in California's COVID trends, which corresponds to events that transpired from 11 through 13 December 2020. Here, we think the key event that altered California's COVID-19 case trajectory were reports of ICUs in the state nearing their capacity. We think these stories combined with the caution that Californians planning to travel to celebrate Christmas with their families began adopting in advance of the holiday, where increased social distancing contributed to reversing the surge provoked by Governor Newsom's hypocrisy.

Meanwhile, During the Same Time in Arizona...

Arizona's coronavirus experience is very different from California's. The following chart tracks four significant events in the state's coronavirus experience in the period from 1 September 2020 through 31 December 2020, which we've labeled as Events F through I.

Newly Reported COVID-19 Cases in California, 1 September - 31 December 2020

Note the very different vertical scale in the chart as compared to California's! The following section describes the four events identified on the chart:

Major Milestones for COVID-19 in Arizona

Event F: High risk businesses reopen

Following its early summer surge in cases, Arizona's number of new COVID-19 cases had bottomed out with roughly 550 new cases reported each day through September. The number of new cases however began rising slowing after 3 October 2020, which points to events that occurred from 22 through 24 September 2020 as leading to the increase. There was no large single event in this case, but this period marks when most businesses believed to have high exposure risk for spreading COVID-19 infections were allowed to reopen in the state, though with some restrictions on their operations.

Events G and H: 2020 Political Campaign Events

2020 was unusual in that Arizona was a swing state for the U.S. presidential election. Unlike in California, both Republicans and Democrats held significant events across the state on two weekends preceding the 3 November 2020 election. The changes in trend some 9-11 days after these two weekends suggests these were superspreader events that boosted the rate of growth of new COVID-19 infections within Arizona, regardless of political party affiliation.

Event I: Arizona's ICUs Fill Up with COVID Cases and Christmas Travel

That upward trend in new cases continued until 19 December 2020, after which the number of new cases has begun decreasing. As with California, this change in trend follows widespread news reports from 9-11 days earlier indicating that ICUs in Arizona were nearing capacity.

Although they are adjacent to each other, the events that have contributed to the fall and winter surge in coronavirus cases in Arizona and California are very different from one another. At the same time, the two states have adopted very different strategies for coping with the coronavirus pandemic. Generally, Arizonans have been more free to engage in commerce and other activities than Californians have, with little apparent downside. Californians however have faced an increasing level of heavy-handed restrictions on their activities through these four months, with little apparent benefit. There are many lessons to be taken from this tale of two states and the coronavirus.

Bonus Update: 8 January 2021: Since we mentioned Arizona has higher quality COVID-19 data available, we'll add charts showing its data for newly confirmed cases by sample collection date, daily ICU bed usage, deaths by death certificate date, and new hospital admissions, covering the period from 3 March 2020 through 5 January 2021. Although the lag from exposure to change in trend for each chart is different, the back calculation method for each confirms the identification of significant events we discussed for Arizona above.

Arizona COVID-19 Confirmed Cases by Sample Collection Date, 3 March 2020 - 5 January 2021
Arizona COVID-19 ICU Bed Usage, 3 March 2020 - 5 January 2021
Arizona COVID-19 Deaths by Death Certificate Date, 3 March 2020 - 5 January 2021
Arizona COVID-19 New Hospital Admissions, 3 March 2020 - 5 January 2021

Of these charts, only the chart showing Arizona's ICU Bed Usage is fully current. Data for the other three charts are incomplete, where the most recent three weeks shown will be subject to revision during the next few weeks, especially for the most recent dates indicated on the charts.

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.