President Trump has chartered a Commission on Election Integrity to investigate his claim that millions of voters, including undocumented immigrants, voted illegally in 2016. Although no evidence has been offered to support this allegation, it does evoke some popular histories of election fraud in the United States.

One of the most famous examples — cited by proponents of Trump’s commission — is the 1960 presidential election, when Chicago Mayor Richard J. Daley allegedly “stole” Illinois for the Democratic presidential campaign of John F. Kennedy.

But the story of the stolen 1960 election rests on several myths. When myths are replaced with evidence, it’s not clear that the election was stolen at all.

Myth 1: Illinois put Kennedy over the top

Popular accounts sometimes claim that winning Illinois was what put Kennedy in the White House. For example, in a recent op-ed, Virginia State Sen. Richard Black (R) claims that “Mayor Daley’s corrupt machine delivered huge tallies for Democrats, changing the election outcome.”

But Illinois didn’t change the national outcome. Although the 1960 election was the closest since 1884 according to the popular vote, Kennedy’s margin in the electoral college was more than large enough to survive the loss of Illinois.

Kennedy won 303 electoral votes; his Republican opponent Richard Nixon won 219. (The remaining 15 electors voted for segregationist Democrat Harry Byrd.) If Illinois’s 27 electoral votes shifted from Kennedy to Nixon, Kennedy would still have won, with 276 votes to Nixon’s 246. Only 269 were needed to win in 1960 (vs. 270 today).

A more accurate claim is that Nixon would have won, with 270 electoral votes, if he had flipped both Illinois and Texas. But Texas wasn’t nearly as close as Illinois. While Kennedy won Illinois by just 8,858 votes (0.2 percent), he won Texas, the home state of his running mate Lyndon Johnson, by 46,627 votes (2 percent). Out of the 22 states that Kennedy won, there were seven that he won more narrowly than Texas.

Myth 2: Kennedy’s margin of victory in Cook County was extraordinary

Another myth, alluded to in Black’s op-ed, is that Kennedy’s margin of victory in Cook County, where Daley chaired the Democratic Central Committee, was “huge” or even impossible without fraud. In fact, Kennedy’s margin was pretty typical for the time and about what he should have expected.

Kennedy’s margin in Cook County is sometimes overstated as 450,000 votes (or more precisely 456,312), but that figure is limited to the city of Chicago. Cook County also includes a large suburban area, which Nixon won. Across all of Cook County, Kennedy’s margin was 318,736 votes.

For comparison, Dwight Eisenhower had won Cook County by a similar margin in 1956 (315,402 votes). In 1964, Lyndon Johnson would win Cook County by more than twice as many votes (641,463).

Of course, 1956 and 1964 were landslides. How did Kennedy rack up such a margin in a squeaker like 1960? The key is Cook County’s demographics, which were very favorable to Kennedy. Thirty-nine percent of Cook County residents were Catholic, and 20 percent were black. Black and Catholic voters heavily favored Kennedy, who was Catholic and had recently helped to secure Martin Luther King Jr.’s release from a Georgia prison.

Given Cook County’s demographics, Kennedy’s margin was about what he should have expected. I built a simple statistical model that predicted the 1960 election results in every U.S. county outside the South using four factors: the results of the 1956 election, the county’s 1960 population size, and the fraction of residents who were black or Catholic. In Cook County, the model predicted Kennedy’s results within half a percent. You don’t need fraud to explain how Kennedy got the numbers that he did.

Myth 3: Daley held back votes

Part of the legend of 1960 is that Daley held back results from Cook County until he knew exactly how many votes were needed to offset Nixon’s margins “downstate” (outside Cook County).

The historian Edmund Kallina has questioned whether this really happened. He reviewed written accounts of the 1960 election and found little agreement on when results from Cook County or downstate were released.

In fact, if you watch NBC’s 1960 election night coverage, Chicago and Cook County actually come in ahead of the rest of the state. By 12:30 a.m., two-thirds of Chicago’s precincts had reported, compared to just half statewide. When the Republican Chicago Tribune went to press, 79 percent of Cook County’s 5,199 precincts were in vs. just 62 percent of Illinois’s 10,015 precincts overall. A little after 7 a.m. Eastern, NBC reported that 850 (8.5 percent) of Illinois precincts were still out, vs. just 400 (7.7 percent) of Cook County precincts, and 200 (5.3 percent) of precincts in Chicago.

If Chicago and Cook County reported early, they can’t have been used to erase Nixon’s lead. In fact, Nixon never led, and it was Kennedy’s lead that shrank as the night went on. At 12:30 a.m., Kennedy led by 290,000 votes. When the Chicago Tribune went to press, Kennedy’s led by 85,000 votes. His final margin was 8,858. It was Nixon, not Kennedy, who benefited from votes that came in late.

Myth 4: There was no recount

None of this means there was no fraud in Illinois. The fact that simple demographics can predict the vote within half a percent offers small comfort when the winning margin was 0.2 percent. It wouldn’t take a massive fraud to swing an election that close. Just a little would do.

One way to bring fraud to light is with a recount, and there is a persistent myth that there was no recount in Cook County. In fact, there were two recounts — one in November 1960, before the state vote was certified, and one after Kennedy’s inauguration in 1961.

According to Kallina’s history, both recounts whittled Kennedy’s margin in Cook County, but not by enough to erase his lead statewide.

Dissatisfied Republicans pointed out that some forms of fraud — such as buying or coercing votes — were not the kind that could be corrected by simply recounting the votes. Democrats countered that the downstate counties, which hadn’t been recounted, might have had fraud favoring Nixon, but Democrats did not offer evidence to support their accusation.

We may never know who received the most votes the 1960 election in Illinois. But the available evidence suggests that there wasn’t enough fraud, at least in the counting, to give Illinois to Kennedy. And in any case Kennedy didn’t need Illinois to win the White House.

Paul von Hippel is an associate professor of public policy and data science at the University of Texas. He has developed fraud detection models for major U.S. banks.