Audits of the 2020 American election show an accurate vote count

Significance

This study quantifies the accuracy of vote counting in a collection of highly contested and consequential electoral contests. We compiled every available audit of votes cast in the 2020 United States elections at the county level or higher, and we used this election audit database to produce nation-scale numerical estimates of how accurately votes were counted. We find that the net error rate in counting presidential votes was on the order of thousandths of a percent, with similarly inconsequential errors for other state and federal contests. This nation-scale estimate of the accuracy of vote counting in a large federalist democracy suggests an approach for researchers to positively establish the legitimacy of fair electoral contests.

Abstract

After many elections, the accuracy of the vote count is assessed by retabulating a small percentage of ballots. These audits form one of the richest bodies of evidence regarding electoral legitimacy, which is particularly important in democracies where the accuracy of elections has been prominently questioned. In decentralized democracies such as the United States, however, there is tremendous variation in the conduct and reporting of audits, which are never compiled into one place to facilitate precise analysis of their results. Here, we introduce a nation-scale audit result dataset, which we use to estimate the error rate in vote counting during the 2020 U.S. election. The dataset includes all available postelection tabulation audits, spanning 856 regional governments across 27 states, with 71,702,471 individual votes and a further 1,210,528 ballots, including about 6.2% of all votes cast for Donald Trump and 6.9% of those cast for Joe Biden. We find that election audits shifted the net presidential vote count by only about 0.007%, with similarly minuscule errors across all major types of electoral contests. The construction of a nation-scale election audit dataset represents a novel approach to benchmarking electoral legitimacy and provides particularly direct and comprehensive evidence that Americans’ votes were counted correctly in 2020.

Democracy depends on the accurate counting of votes. In several established democracies that have enjoyed long periods of widespread confidence in the accuracy of vote counting, electoral institutions have recently been subjected to sustained questioning (1). One prominent example is the United States, where, for the first time in the country’s modern history, a sitting President focused substantial negative attention on the country’s election administration and vote counting institutions. This rhetoric has shaped contemporary American politics (2), not only fragmenting Americans’ trust in elections along partisan lines (3), but also appearing to affect electoral trust through spillover effects in other countries (4).

Questions about the legitimacy of the 2020 U.S. election prompted a broad response by scientists, journalists, and civil society. That response has tended to emphasize the lack of evidence for broadly inaccurate vote counting or electoral malfeasance in U.S. elections (5). While there is a clear absence of evidence of malfeasance, precise evidence of its absence has been somewhat more limited. This is the direct result of a gap in election science: Because votes are the primary measure of voter intentions, there often is no benchmark against which to gauge how accurately they are counted.

However, many states and localities across the United States conduct postelection audits that involve tabulating votes a second time, in order to assess the accuracy of the original vote count. We perform a nation-scale analysis of these election audit results. We first build a comprehensive audit dataset, which includes every state- or county-level election audit for which a government source announced sufficient information to estimate the rate of errors in the original 2020 vote count. 27 states^* provided such data, totaling 71,702,471 audited votes cast for any candidate, as well as the number of discrepancies in a further 1,216,179 ballots. Our dataset includes the audit results of about 6% of all votes cast for Donald Trump and about 6.9% of those cast for Joe Biden.^†

We find clear and consistent evidence that the vote count was exceptionally accurate. Across all regions and all types of contests, the median discrepancy between a candidate’s original vote total and their audited vote total is 0 votes. In jurisdictions that audited at least a thousand votes for a candidate, the typical error rate was only tens or hundredths of a percent of a candidate’s vote count, and almost never exceeded 1%. In jurisdictions where entire ballots were audited, discrepancies between the original and audited count were found in only about 0.04% of those ballots.

We pay particular attention to the candidate who has most prominently alleged systematic issues in vote counting, Donald Trump. Our dataset includes 10,379,419 votes cast in the presidential contest, with 5,584,297 cast for Joe Biden and 4,609,752 for Donald Trump. The margin of Trump’s loss in the closest state, Georgia, was about 0.23%, but in available audit results the mean shift in the presidential margin at the level of a county was about 0.007%. In the national popular vote, Trump lost by approximately 4.5%, but the total shift in the margin as a result of election audits was only about 0.00095%. Election audits found similarly small net shifts for gubernatorial elections and both federal and state legislative elections. The tight similarity of audited vote counts and the original vote count directly contradicts claims that the 2020 election results were affected by systematic errors or malfeasance.

Audits and the Accuracy of Vote Counting

Audits evaluate whether election procedures are correctly carried out, all voters are eligible to vote, and votes are counted accurately (6). In the United States, virtually every feature of election audits varies across election jurisdictions, including what types of audits are conducted. Before an election, “every election jurisdiction in the nation” conducts tests to ensure that their tabulators are working as intended (6). Most—but not all—states also require process audits to evaluate compliance with election procedures, as well as “postelection tabulation audits,” in which the accuracy of the vote count is evaluated by retabulating a share of real votes (7).

Different jurisdictions may adopt different rules regarding who performs the audit. Auditors may be organizations that are independently contracted (8), individuals nominated by local party officials (9), or local election officials (10), and there are often provisions for independent observers to witness the audit (6).

In a postelection tabulation audit, ballots might be tabulated through independent software (11), or entirely by hand (12). Usually such audits only tabulate a small share of ballots cast. The classic approach to selecting those ballots is a fixed-percentage random sample audit: Say, 1% of ballots are randomly selected and retabulated, and the result of this tabulation is compared to how those same ballots were adjudicated in the original vote count.

A classic question, however, is how to use the results of fixed-percentage random audits to judge the accuracy of the election. For example, if 1% of votes are audited, and 99% (or 95%, or 90%) of those ballots are counted the same way as in the original count, does that mean we should trust the results of the election? If the results fall below whichever threshold we choose, what should happen next? Questions like these motivated the development of the Risk-Limiting Audit (RLA), which uses a small initial sample of ballots to bound the probability that the wrong winner was declared, escalating to a 100% manual recount if necessary (13). 2020 was one of the first elections where entire states piloted or conducted RLAs.

Despite being one of the primary tools that governments use to ensure election security, “overall, the research literature into audits is still thin” (6). Most previous research on election audits has either focused on their conceptual properties (6), or piloted new ways to audit votes (14–16). Usually the accuracy of vote counting has been studied by other means (17, 18).

Why are audits not better-understood? One axis of variation that has hindered research into election audits is the tremendous range of data reporting, which affects both the quantities that election jurisdictions use to summarize the results of their audits and the format in which those data are released to the public. There is no national mandate that audit data be released to the public, and commonly they are not. States or counties may release these data as tabular data files; machine-generated Portable Document Format (PDF)s with typed vote totals; PDFs containing tables of handwritten vote totals; photographs of handwritten totals; or even press releases or quotes in news articles that summarize the results in prose.

Because of the limitations in data reporting, those studies which have used audit data to assess vote-counting accuracy have focused on an individual state or county (19, 20). Our research directly builds on two of these studies. Using recount data, Ansolabehere and Reeves showed that over the last century of New Hampshire elections, the percent difference between a contest’s original vote margin and the recounted result ranges from well under 1% to just under 2% (21). In 2022, Atkeson et al. found that a postelection tabulation audit of every vote cast in Leon County, Florida reported 99.9983% agreement with the original vote count (22).

What types of mistakes, malfeasance, or fraud can postelection tabulation audits provide evidence for or against? The essential point is that tabulating the same ballots a second time makes it possible to estimate the error rate of the original vote count. Audits are a replication exercise, usually conducted by an independent team; any issue that would make the original vote count nonreplicable would cause the results of a postelection tabulation audit to be dissimilar from the original count. In particular, if the vote count were tainted by systematic mistakes, then there should be substantial disagreement between the original count and the audit, and if malfeasance or fraud were at play, then the audit should produce a lower vote total for whichever candidate was supported by that fraud.

Among the most prominent allegations of fraud in the 2020 election, several fall into this category. For example, it was widely alleged that certain types of voting machines switched votes from one candidate to another (5). If that were the case, then audits that use independent software (such as Maryland’s) or manual tabulation (such as Georgia’s) should produce sharply different results, and these claims would imply that the results of the audit should be a higher vote total for Donald Trump.

Another prominent allegation focused on absentee ballots, asserting that “irregularities in the absentee vote counting procedure” led to Biden’s victory in key counties (5). In many states, postelection tabulation audits are required to include a certain share of ballots cast by each mode, including both votes cast absentee and in-person on election day. If absentee ballots were counted in a systematically incorrect manner, then audits of those ballots should systematically diverge from the original totals.

Finally, many claims of systematic fraud in this election focused on differences between different kinds of counties, and especially that county or state governments where elected offices were predominantly held by Democrats fraudulently added votes to Joe Biden’s total or removed them from Trump (5). Then, at a minimum, we should expect to see large discrepancies in counties where Democrats hold the balance of power but the postelection tabulation audit is conducted by bipartisan or nonpartisan auditors.

Postelection tabulation audits, of course, would not turn up every possible type of issue with the original vote count. Often counties report the total number of votes that candidates received at some level of geographic aggregation in the original vote count and in the audit, which means that mistakes could cancel each other out to produce a similar vote count. Crucially, however, this would not prevent us from detecting systematic issues or systematic fraud, which would appear as a large net difference between each candidate’s vote total in the original count compared to the audit.^‡

We must finally acknowledge that, of course, the auditors and the original vote counters could conceivably conspire to repeat the same malfeasance. To evaluate this possibility, it is useful to consider the scale of audits in United States, where thousands of regional governments across dozens of states oversee largely independent audits, conducted by teams who are independent from each other and from the original vote counters, witnessed by independent observers, with independent random draws to select the audited ballots, and often using different methods from the original vote count to tabulate and report the results. If postelection tabulation audits across the country closely match the original vote counts, this is a strong signal that the vote count was fair.

Data

We introduce a nation-scale dataset containing as many postelection tabulation audits as possible from the 2020 U.S. election. Our corpus includes the results of every postelection tabulation audit at the county or state level for which data were publicly available, or were made available to us on request. Ultimately we obtained data from 856 regional governments across 27 states. In SI Appendix file, we specify the types of audits we include in our dataset (SI Appendix, Data Eligibility), outline our procedure for obtaining their results (SI Appendix, Obtaining Data), and report the availability of those data state-by-state (SI Appendix, Data Availability).

Each county falls into one of four categories: It either conducted an audit and made the data available; conducted an audit and did not make the data available; did not conduct an audit at all; or conducted an audit that produces a type of data which cannot straightforwardly be used to compute an error rate (for example, if the state conducted a type of RLA that does not involve comparing how a ballot was originally counted to how that same ballot is tabulated in the audit). Which counties belong to which category is shown in Fig. 1. SI Appendix, Fig. S1 shows the distribution of the number of votes and number of ballots at the county level as a kernel density plot.

Fig. 1.

Whether a postelection tabulation audit of 2020 general election votes took place in each county in the United States, and whether the results of that audit were eligible and available for inclusion in our dataset. “Data Available” means that at least one vote from the county appears in our dataset. “Not Applicable” includes states that conducted and reported audits which cannot be used to calculate an error rate, or the counties that were not selected for an audit in a state that audited random counties.

The most common data format reports the number of votes that each candidate for public office received in the original vote count, and how many they received after the election audit (e.g., “candidate A received 100 votes originally and 101 votes in the audit”). This format, which was used for audits in 23 states, makes it possible to compute the net difference between the original vote count for a candidate and their total after the audit. We will call this candidate-level data. The other format, which we encountered in 8 states (states are double-counted here if different jurisdictions used different formats), reports the number of ballots included in the audit, and the number of discrepancies found, without identifying which office or candidate the discrepancy affected (e.g. “100 ballots were audited, and 1 discrepancy was found”).^§ We call this ballot-level data.

Because audits are typically performed by counties or equivalent forms of regional government, and important features may vary across counties, we will take counties as our main geographic level of analysis. In counties that produce candidate-level data, our unit of analysis is each candidate’s county-level vote total. We will focus our analysis on the change between each candidate’s original vote total among the votes that were selected for a postelection tabulation audit, and the vote total obtained when those same ballots were audited. For ballot-level data, our unit of analysis is the total number of ballots audited in a county, and our analysis will focus on the number of discrepancies encountered. We use “vote” to refer to an individual candidate selection on a ballot; so, a ballot may have many votes.

The available candidate-level data contained 71,702,471 audited votes for 2,317 distinct candidates (or other ballot options, such as “yes” or “no” in a referendum). The ballot-level data identified the number of discrepancies across an additional 1,188,377 ballots.^¶ The number of audited votes (for candidate-level reporting) or ballots (for ballot-level reporting) is shown per state on the Left side of Fig. 2. The Right side of Fig. 2 shows the number of distinct electoral offices that were audited in each state.

Fig. 2.

The number of votes (candidate-level) or ballots (ballot-level) included in election audits in each state (Left), and the number of distinct elected offices included in the reported candidate-level audits (Right).

Details about the extensive process required to clean these data, and some related caveats for their interpretation, are available in SI Appendix, Data Cleaning. There are two essential caveats. First, we will present summary statistics over data generated by numerous different processes and in different places. Different types of audits may systematically produce different outcomes, so they cannot be directly compared to one another to assess relative accuracy (no two states have exactly the same audit processes, so if state A uncovers a lower error rate than state B, did state A conduct a more accurate election, or a less rigorous audit?). Our results should be interpreted as summary statistics across all types of audits, net the specific attributes of different auditing methods. The second crucial caveat is that audit data are missing not at random, so the conclusions we draw about the 27 states in our dataset may not apply to the other 24.

Results

What was the error rate of vote counting in the 2020 election? Did audits of the 2020 election uncover reasons to suspect that Donald Trump was credited with fewer votes than he should have been in the original vote count? Fig. 3 displays on its y-axis the change Δ in vote margin between the two major presidential candidates (defined in Eq. 1 in Materials and Methods) for each available county, with a rectangle representing the total number of votes that were included in a county’s audit on the x-axis. Each rectangle represents the number of votes in one county that found a certain shift, but because multiple counties may have the same shift (in particular, many counties found exactly the same result in the audit and original count), we horizontally stack those rectangles to display the total number of votes audited in counties that found a certain shift.

Fig. 3.

The net change in presidential vote share at the level of each county as a result of the election audit. The exact quantity displayed on the y-axis is defined in Eq. 1 of Materials and Methods. Because multiple counties can have the same net shift—and in particular, many counties found a shift of exactly zero—we display a cumulative sum of votes audited in counties that found that margin in a stacked bar, with one rectangle for every county that found that margin.

So, for example, many counties’ audits found a net shift of 0 votes in the presidential contest, and the total number of votes audited in those counties was nearly a million; in contrast, only one county found a net shift larger than 0.6 percentage points toward Trump, and that county audited fewer than 2,000 votes. To illustrate how many counties saw a net shift of exactly zero or very close to it, SI Appendix, Fig. S2 shows the distribution of county-level shifts in candidate margin as a kernel density plot.

The median change in the Biden-Trump vote margin within a county as a result of election audits was exactly zero. The mean change in Trump’s net vote share as a result of election audits was about $\mu (\mathrm{\Delta })=7.3·{10}^{-5}$. So while the average effect on the vote share across counties was to shift votes toward Donald Trump, the net shift was two orders of magnitude smaller than the smallest presidential vote margin in any state (≈0.23%). The largest shift toward Trump as the result of any county’s audits was about 0.65%, and the largest shift toward Biden was about 0.74%. Only five counties had a shift of 0.5% or more. Of these 5, 4 were counties in Georgia, which participated in the largest single audit of ballots in U.S. history, and the state emphasized that these margins were in line with the expected level of variation when millions of votes are subjected to a full hand count (23). The other was in Butte County, Idaho, where 1,139 presidential votes were audited and 9 extra votes for Trump were discovered in the course of the audit; in this unusual case, “officials attributed the ballots not getting counted to ballot sorting and storage policies” (24).

Above all else, this is evidence that vote counting was not stacked against either candidate. Fig. 3 does not just show that errors were centered around zero without systematic bias toward one candidate or another; it is also noteworthy that the proportional errors are all substantively quite small. Every net change in presidential vote share at the county level was smaller than 1%. When tens or hundreds of thousands of votes were recounted, the net shifts were nearly all hundredths of a percent or smaller.

Is this absence of any net change in the result a peculiarity of the presidential contest, or is it true of other offices as well? To perform this comparison, we assign party labels to every candidate in our dataset contesting any of the six state-level or federal offices that are always partisan: U.S. President, U.S. Senate, U.S. House, or any state governorship, state senate, or state house (Nebraska holds the only nonpartisan state legislative elections, but had no audits in 2020). Then, we sum the votes received by a candidate of each major nationwide party or any minor party (Democrat, Republican, Libertarian, any other party, or no party affiliation), both in the original vote count, and in the audited vote count. Fig. 4 visualizes the total number of votes in each category, as well as the percent change from the original vote to the audit as defined in Eq. 2 in Materials and Methods. Fig. 4 shows both the total number of votes received by candidates of these party affiliations within each office category among the votes that were audited, as well as the net percent change in votes rounded to one decimal place.

Fig. 4.

Net change in candidates with each party label in each partisan office category: Democrat, Republican (GOP), Libertarian, all others, and no party. Within each combination, the bar on the Left is the total votes those candidates received among audited ballots in the original vote count, the bar on the Right is the number of votes they received in the audit, and the percentage above is the change from the original vote count to the audit.

Fig. 4 shows remarkably little change in candidates’ net vote totals by party as a result of election audits. For Democrats and Republicans, that change, rounded to one decimal place, was 0.0% for U.S. President, U.S. Senate, U.S. House, state senate, and state house. Among all of these office categories, the smallest number of votes for any major party-office combination was for Democratic candidates for governor, who received just over 6,000 votes (the small number of votes in this category is simply a quirk of what elections occurred in 2020 among states that released election audit data), and this is the only category where a major party saw any change in vote percentage as a result of election audits: The audits revised their vote total down by about 0.1%.

Evidently, there were no meaningful net shifts in vote by party. However, we have only examined the rounded values of net changes in vote share, aggregated to the level of candidates or entire parties. The actual changes in vote totals are not always exactly zero, so how large do they tend to be?

Fig. 5 shows the precise shift in individual candidates’ vote totals as a result of election audits. The x-axis shows how many votes were audited, total, for a given candidate. The points represent the error rate that the audit found in a candidate’s vote total, and the diagonal line represents the proportional error that a single vote change would make; for example, if a candidate’s vote total shifted from 99 to 100, that 1 vote change represents a 1% shift in the official total of 100 votes cast for the candidate, so the result would be represented by a circle at the coordinate (x = 100, $y={10}^{-1}$).

Fig. 5.

This figure displays the number of changes in the candidate-level vote totals in two different ways. The bars are the number of times that a candidate’s vote total did not change as a result of an election audit, on a log–linear scale. So, a bar at $x={10}^3$ with a height of y = 125 would mean that 125 candidates who had about 1,000 votes audited saw no change in their vote total as a result of the election audit. The width of the bars represent an equal range of votes within each order of magnitude. The circles represent the absolute value of any change in candidate vote totals greater than zero. Because the smallest nonzero change is one vote, the smallest nonzero error rate is a function of the number of votes audited. The diagonal black line denotes a change of 1 vote.

An important feature of this variable is that the number of votes audited places a minimum bound on the error rate. There can always be an error rate of zero, but the smallest net discrepancy other than zero votes changing is one vote changing, so when N ballots are audited the smallest positive error rate is $\frac{1}{N}$. So, in Fig. 5, we plot the number of votes audited against the proportional change in vote total for each candidate as a result of the election audit. We also color the dots by Democratic, Republican, or any other candidates. A histogram in the Bottom-Left of the figure shows the number of candidates who had a certain number of votes audited and whose total number of votes received did not change as a result of the audit.

There are two particularly noteworthy patterns in Fig. 5. First, regarding the overall accuracy of vote counting, if at least 100 of a candidate’s votes are audited, that candidate’s vote total almost never shifts by more than 1%. With a single exception, whenever at least 1,000 of a candidate’s votes were audited, the error rate was always well below 1%. Between 1,000 and 10,000 votes audited, error rates tended not to exceed 0.1%, and likewise with 10,000 votes, audited error rates tended to be below 0.01%.

This points to the second noteworthy pattern: When more votes are audited, the share of votes that change grows smaller. That is, audits of more votes tend to find a more accurately counted election, with a steady decline in error rate as the number of votes counted grows. Of the 2,317 candidates (or overvotes, undervotes, and other ballot options) in our dataset, 1,435 (about 62%) saw no change in their votes received as a result of election audits. In some cases, there are large error rates for some candidates who had very few votes audited. These may sometimes be due to random error, but they are mostly the result of write-in options being resolved in the course of the audit, where there is opportunity to reevaluate whether they are actually intended votes for a more commonly chosen candidate. When a substantial number of votes was included in the election audit, the shift in a candidate’s vote total was nearly always quite small, with typical shifts in the candidate’s vote total on the order of tens of votes per million votes cast. Among candidates with more than 1,000 votes audited, in no case was the net shift larger than 1 vote for every 100 counted toward that candidate. For these candidates the median net shift in their vote total was about $5.91·{10}^{-5}$, or an adjustment of about 59 votes for every 1,000,000 cast. Compared to realistic election margins, this is a negligible change.

What of the ballot-level data? SI Appendix, Fig. S3 reports results from the audits in 15 states that reported the total number of ballots counted in the original vote count, and the total number counted in the audit. We compute the net change in the total number of ballots at the level of each state-office combination. We find that when fewer than 10,000 ballots were audited, the total very rarely changed. In state-office combinations with over 10,000 ballots audited, it was common for the total number of ballots to increase or decrease by up to 100 ballots either way. One outlier is Georgia’s full manual audit of the presidential election, where millions of ballots were audited and the total number changed by about 5,000. SI Appendix, Fig. S4 shows the number of discrepancies encountered in ballot-level data. Across the country (adding together the number of discrepancies found overall and dividing it by the number of ballots), only about 0.04% of ballots audited contained any issue. In about 81% of counties that conducted ballot-level audits, no discrepancies were found.

Discussion

No analysis has demonstrated that there was systematic fraud in the 2020 U.S. election (5). And yet, many Republicans’ adherence to this idea, and the resulting counterreactions by Democrats, have become enduring features of American public opinion and party behavior. To lend empirical specificity to this debate requires addressing the pressing scientific problem of exactly how accurately votes are counted. Our ability to produce such estimates has been severely hindered by the scattershot way that election audit records are kept and reported to the public. Inconsistent rules about whether and how to report audits means that estimating the accuracy of vote-counting at the scale of the entire country involves combing through thousands of pages of structured, semistructured, and even handwritten records and compiling them into one dataset for analysis.

We report such an effort to produce a nation-scale dataset of postelection tabulation audits in the United States. The dataset of over 70 million vote choices and more than 1 million additional ballots shows that the 2020 U.S. election, the locus of so much controversy, was meticulously well-counted around the country. The regional governments that recounted votes cast in the 2020 presidential contest overwhelmingly found no net shift toward one candidate or another, with no county’s shift as large as even just 1%, and a median change of zero. This pattern of minuscule shifts, typically on the order of tens or hundredths of a percent, was repeated for both parties throughout every major type of electoral contest in every state that released election audit results. To the absence of evidence of electoral malfeasance in the 2020 American election, these findings add clear evidence that systematic errors were absent from the vote count.

Materials and Methods

In this section, we describe the methods used to calculate the descriptive statistics in the text. More information is available in SI Appendix.

We mention that our dataset includes information from “71,702,471 votes.” This is the sum of the number of candidate selections in available audits. Suppose that a county audited votes in two precincts, and reports that “in the original count, Trump received 100 votes in precinct A and 200 votes in precinct B, and a candidate for U.S. Senate received 50 votes in precinct A and 40 votes in precinct B; in the audit, Trump received 101 votes in precinct A and 200 votes in precinct B, and that U.S. Senate candidate received 50 votes in precinct A and 40 votes in precinct B.” The total number of audited “votes” in this example would be 101 + 200 + 50 + 40 = 391. So, 71,702,471 is the number obtained by computing this sum across all candidates, offices, and geographies in the audit dataset. It may be thought of roughly as the number of “bubbles filled in” for any candidate. In SI Appendix, section 8, we show that these votes were cast by at least 11,591,334 voters.

In addition to these votes, we report the inclusion of 1,210,528 ballots. These are nearly all ballot-level data, reporting the number of ballots audited and the number of discrepancies found. We discuss two edge cases, Texas and Iowa, in SI Appendix, section 8.

Of the candidate-level votes audited for U.S. President, 4,609,752 were found in the audit to be votes cast for Donald Trump, which was about 6.2% of his total number of votes. Many jurisdictions audit a very small share of votes (only 20 states reported the results of auditing Trump votes, and half of these audited fewer than 10,000 of Trump’s votes). The total rises into the millions because a few states audited a very large number of votes. Georgia’s full manual audit included 2,462,857 votes for Trump, while Maryland reported 976,668 and Florida audited 587,217 votes for Trump. Similarly, we find 5,584,297 audited votes for Joe Biden, about 6.9% of his total votes received in the election. Again, a small number of states contribute disproportionately to this total, with 20 states reporting audits of Biden’s votes, half of those reporting fewer than 10,000 votes audited, and Georgia and Maryland together auditing nearly 4.5 million votes cast for Biden. SI Appendix, Table S2 shows the total number of audited votes for president in each state, the number of Biden votes audited, and the number of Trump votes audited.

In the Abstract and Introduction we cite, and in Results, we heavily use, a measure Δ of the difference between the margin in the audited votes and the margin when those same votes were counted in the original vote count, as a share of the number of audited votes. In the case of the Trump–Biden contest, this measure is defined as

$$\mathrm{\Delta }\equiv \frac{(T_a-B_a)-(T_g-B_g)}{T_a+B_a},$$ [1]

where T_a is Trump’s audited vote count, T_g is Trump’s original vote count, B_a is Biden’s audited vote count, and B_g is Biden’s original vote count. So for example, if the original vote count reported that Biden won the county by 2%, but an audit determined that Biden had actually only won the county by only 1%, Δ would be +1%.

Δ, calculated within each county, is the value displayed on the y-axis of Fig. 3. The mean Δ over the 447 counties for which Trump and Biden votes were audited, which we represent by $\mu (\mathrm{\Delta })$ in the text, is $\mu (\mathrm{\Delta })\approx 7.3·{10}^{-5}$. This should be interpreted as the average increase in the margin for Trump against Biden as a result of the election audit in a county. The median of Δ, or the median shift toward Trump as a result of a given county’s election audit, was 0 votes.

To calculate the net change in vote margin at the national level, denote it ${\mathrm{\Delta }}_n$, we compute Δ using the sum of all votes for each candidate (so, T_a is the total number of votes for Trump found in any audit across the country, B_a is the total number counted for Biden in any audit across the country, and so on). The resulting value is ${\mathrm{\Delta }}_n\approx 9.5·{10}^{-6}$.

In Fig. 4, we show the percent change in the number of votes that any candidate of a given party received for each office. We compute this by summing the number of votes cast for any member of a given party across the country for a certain office as tabulated in the election audit, call it $T_r^a$, as well as the total received by that party in that office in the original count, $T_r^g$, and then computing the percent change h as

$$\begin{array}{c}\hfill h\equiv \frac{T_r^a-T_r^g}{T_r^a}·100\%\end{array}$$ [2]

Finally, Fig. 5 shows the error rate (the difference between the original and audited vote counts, as a proportion of the number of votes counted) at the level of individual candidates. To obtain this measure, we first combine the state, the office being contested, and the candidate’s name into a unique ID (we verified manually that this is always a unique combination in our dataset). Then, we subtract the audited vote total from the original vote total, at the level of unique candidates, and take the absolute difference, which represents the net discrepancy between the two measures of the candidate’s vote total. Finally, we divide that discrepancy by the number of votes that were audited for that candidate, representing the net error as a proportion of that candidate’s votes.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Datasets have been deposited in Harvard Dataverse (Harvard Dataverse: https://dataverse.harvard.edu/dataset.xhtml?persistentId=doi%3A10.7910/DVN/ZJAVZD) (25) under the name “Replication Data for: Audits of the 2020 American Election Show an Accurate Vote Count”).