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Here are the precincts in Iowa that held Democratic caucuses last week. About 10 percent appear to have a mistake in their results. Some precincts bungled the complex rules for awarding delegates. And some awarded delegates to candidates who didn’t earn them. Note: Precinct locations are adjusted so that all precincts are shown without overlapping. Satellite precincts are not shown.

Since the troubled Iowa caucuses on Feb. 3, the state Democratic party has revised the results for about 100 of the state’s 1,765 precincts, and officials are still scrambling to verify dozens more precinct results after reports of widespread inconsistencies.

There has often been some fuzziness in the way the results of the Iowa caucuses were calculated and reported. But this is the first year that Iowa Democrats released raw vote counts. The transparency provided the public with its first opportunity to check the complex math that determines which candidates get the delegates they need to win the Democratic nomination.

And in many cases, the math did not check out. In such a close race — Pete Buttigieg, the former mayor of South Bend, Ind., is leading Senator Bernie Sanders of Vermont by a tenth of a percentage point — even small mistakes can add up.

Here are some of the inconsistencies and errors The New York Times uncovered in an analysis of the Iowa Democratic Party’s results.

Recorded vote totals do not add up: 79 precincts

Candidate First Alignment Final Alignment Delegates Candidate First Align. Final Align. Delegates 3 2 1 1 0

What’s wrong The Iowa caucuses happen over two rounds of voting. People may leave after the first round, but there should not be more people voting in the final round than in the first.

In more than 600 precincts, the vote totals did not match perfectly, and in 79 the final round votes were higher.

What it means When the totals don’t match up, it is difficult to determine which count is accurate.

The first count helps determine whether a candidate meets the viability threshold — a set share of total caucus attendance — and which candidates move on to the final round. The final count determines who wins delegates. If either of these counts is inaccurate, the results could be inaccurate, too.

An app designed to record caucus results would have flagged this discrepancy, a Times review of the app’s source code shows. But many people were unable to use the app or chose not to.

Candidates without enough supporters won delegates: 21 precincts

Candidate First Alignment Final Alignment Delegates Candidate First Align. Final Align. Delegates 2 1 1 0 0 0 Total 34 34 5

What’s wrong Each precinct sends a set number of delegates — real people chosen by caucus attendees — to a county convention, where each delegate casts a vote for one candidate based on the outcome of the caucus.

Only candidates who meet the viability threshold should win county delegates, but this wasn’t always the case this year.

What it means In 21 precincts, candidates who did not earn enough support to win any delegates appear to have won them anyway.

In the example above, from a precinct in Black Hawk County, the minimum votes required for viability was six. There were only five in former Vice President Joseph R. Biden Jr.’s group, but he still received one delegate.

Caucusgoers who initially chose Mr. Biden should have had to make another choice, or persuade a member of another group to join them. But since they did not, it’s impossible to know which candidate they might have chosen or how those choices would have changed the results.

Too many delegates awarded: 8 precincts

Candidate First Alignment Final Alignment Delegates Candidate First Align. Final Align. Delegates 1 1 1 1 0 0 0

What’s wrong In a handful of cases, precincts awarded more delegates than they were supposed to. This precinct in Webster County had three delegates to send to the county convention but awarded delegates to four candidates.

What it means Since Mr. Biden and Mr. Sanders were tied in the final alignment but both were viable, it should have come down to a coin toss over whose supporters should move to another viable candidate. Support for one of these candidates is overstated in the final results, but we can’t know which.

In a handful of other cases, precincts awarded fewer delegates than they were supposed to, understating support for at least some of the candidates there.

Delegates calculated improperly: At least 28 precincts

Candidate First Alignment Final Alignment Delegates Candidate First Align. Final Align. Delegates 2 1 0 0 0 0 Total 145 144 7

What’s wrong The rules for calculating delegates are complicated and not always intuitive. There are additional rules for ties, and contests can come down to a coin toss.

In this example from Dubuque County, because of errors in the delegate calculations, Mr. Buttigieg was awarded a delegate that should have gone to Mr. Biden.

Let’s walk through this case to see how challenging the process can be.

Candidate First Round Final Round Vote Share Unrounded Delegates Rounded Delegates Delegates Candidate 1st Final Vote Share Raw Round 0.32 0.3 2.22 2.2 2 0.31 0.3 2.17 2.2 2 0.21 0.2 1.45 1.4 1 0.16 0.2 1.11 1.1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total 145 144 — — 6 Candidate First Round Final Round Vote Share Unrounded Delegates Rounded Delegates Delegates Candidate 1st Final Vote Share Raw Round 0.32 0.3 2.22 2.2 2 0.31 0.3 2.17 2.2 2 0.21 0.2 1.45 1.4 1 0.16 0.2 1.11 1.1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total 145 144 — — 6 Candidate First Round Final Round Vote Share Unrounded Delegates Rounded Delegates Delegates Candidate 1st Final Vote Share Raw Round 0.32 0.3 2.22 2.2 2 0.31 0.3 2.17 2.2 2 0.21 0.2 1.45 1.4 1 0.16 0.2 1.11 1.1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total 145 144 — — 6 Candidate First Round Final Round Vote Share Unrounded Delegates Rounded Delegates Delegates Candidate 1st Final Vote Share Raw Round 0.32 0.3 2.22 2.2 2 0.31 0.3 2.17 2.2 2 0.21 0.2 1.45 1.4 1 0.16 0.2 1.11 1.1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total 145 144 — — 6 Candidate First Round Final Round Vote Share Unrounded Delegates Rounded Delegates Delegates Candidate 1st Final Vote Share Raw Round 0.32 0.3 2.22 2.2 2 0.31 0.3 2.17 2.2 2 0.21 0.2 1.45 1.4 1 + 1 = 2 +1=2 0.16 0.2 1.11 1.1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total 145 144 — — 7 Candidate First Round Final Round Vote Share Unrounded Delegates Rounded Delegates Delegates Candidate 1st Final Vote Share Raw Round 0.31 0.3 2.17 2.2 2 0.16 0.2 1.11 1.1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total 145 144 — — 7









Caucus volunteers tally the final round votes. Then they divide the county delegates among the viable candidates based on their vote shares. They must round down or up to the nearest whole number to award the delegates. Sometimes rounding leaves the precinct with too many or too few total delegates. This precinct ended up with six total delegates after rounding, but had seven delegates to award. Who should win the final delegate? The caucus guide says that an additional delegate should be awarded to the candidate whose unrounded delegate count is closest to but less than 0.5 in the decimal place. Got that? In this case, that would be Mr. Biden, who had .45 in the decimal place. That would bring Mr. Biden’s delegate count up to two and the total number of delegates to seven. But it appears this precinct awarded the delegate to Mr. Buttigieg instead. Note: Displayed vote shares were rounded to two decimal places but calculations were done with figures rounded to four decimal places.

What it means These kinds of miscalculations are easy to make. The caucus math worksheet, which caucus volunteers use to record votes, does not offer instructions for every scenario, and volunteers must consult a handbook for advice on the less common ones. Moreover, the handbook does not appear to cover every potential situation, and delegates could mistakenly be awarded to the wrong candidates.

The Times checked the delegate math in more than 1,200 precincts. The vast majority of these appeared to follow the rules. But at least 28 precincts appear to have gotten it wrong, with consequences for the candidates. (About 500 precincts had some inconsistency or complication that precluded a check for delegate calculation errors.)

Steve Drahozal, the Democratic party chairman for Dubuque County, confirmed that the 36th precinct appeared to have made a mistake in its delegate math. Here’s a copy of the precinct’s worksheet:

Courtesy of the Dubuque County Democratic Party.

Mr. Drahozal said he was alerted on Sunday to errors in one other precinct in Dubuque County where delegates were awarded improperly. “I know with 100 percent certainty that was a good-faith mistake,” he said.

He had not reached out to the state party about the error, assuming officials there would contact him instead. As of Thursday afternoon, he’d heard nothing.

Data entry mistakes: At least 100 precincts

Candidate First Alignment Final Alignment Delegates Candidate First Align. Final Align. Delegates 3 2 0 0 0 Total 199 190 9 Note: These results have since been corrected by the Iowa Democratic Party.

What’s wrong When the reporting app failed on caucus night, the state party was flooded with calls from volunteers trying to submit their results. The people taking the calls had to copy down the results and enter the data by hand, leaving plenty of room for error.

In this example, in Indianola’s Second Precinct in Warren County, the party released results showing that Senator Elizabeth Warren of Massachusetts and Mr. Sanders had zero votes in the first round but picked up several dozen each in the final round, a highly unlikely scenario. Meanwhile, the billionaire Tom Steyer and former Gov. Deval Patrick of Massachusetts met the viability threshold in the first round but lost all of their votes in the final round.

Courtesy of the Warren County Democrats.

The precinct’s worksheet shows vote counts that make more sense. It appears that whoever entered the data into the party’s database switched the votes for Ms. Warren and Mr. Sanders with those for Mr. Steyer and Mr. Patrick.

What it means There is no way to know how many of these errors occurred without a full recanvass of every precinct worksheet. The party had to revise results for dozens of precincts in an early release of results because of a transposition error.

The party agreed last week to review 92 precincts at the request of the campaigns and county chairs. On Sunday, it issued corrections to 54 of them — including Indianola’s Second — to bring its reported results in line with what was written on precinct worksheets.

But if the worksheets showed any mistakes in counting votes or calculating delegates, the party did not correct them. The party said the worksheets were legal documents and could not be revised.

“We do not believe that we should be altering the official record of what happened in the room,” Troy Price, the party chairman, said Monday at a news conference. On Wednesday, Mr. Price resigned.

The Sanders and Buttigieg campaigns have requested a further review of some precincts. If the campaigns decide the problems are enough to shift the race, they could request a recount.