During his first full day in office, Mr. Trump grossly overestimated the size of his inauguration crowd, saying that up to 1.5 million people attended, when, in fact, aerial pictures showed it was much less.

His press secretary at the time, Sean Spicer, doubled down on Mr. Trump’s assertion and said the president drew “the largest audience to ever witness an inauguration.” Photographs of Mr. Obama’s inauguration in 2009 disproved that claim, as well.

In May, Mr. Trump said that when he appeared on the CBS program “The Late Show,” the broadcast had its highest ratings ever. But he was off by a million or so viewers on that, too. The episode that aired when Mr. Trump was a guest in September 2015 was watched by 4.6 million people, whereas the first “Late Show” that Stephen Colbert hosted drew 6.6 million viewers, according to a CNN report.

In late December, Mr. Trump said his “so-called low approval rating” was approximately the same as Mr. Obama’s at the same point in that presidency. Real Clear Politics research, however, shows that Mr. Obama’s average approval rating across major polls was about 10 points higher than Mr. Trump’s in the December of his first year in office.

Just last month, Mr. Trump falsely claimed that his approval rating among black Americans had doubled (it has actually declined). In July, he also tweeted that his 40 percent approval rating, as recorded by an ABC/Washington Post poll, was “not bad” at this point in his presidency. The same poll shows that Mr. Trump’s approval rating at six months was the lowest recorded since World War II.