People have always told stories about their ancestral origins. But now millions of people are looking at their DNA to see if those stories hold up. While genetic tests can indeed reveal some secrets about our family past, we can also jump to the wrong conclusions from their results.

The reception of Senator Elizabeth Warren’s DNA results is a textbook case in this confusion.

On Monday morning, Senator Warren released an analysis on her DNA showing that six to 10 generations back she had a Native American ancestor. Within hours, Michael Ahrens, an official at the Republican National Committee, dismissed the results in a tweet:

“So Elizabeth Warren is possibly 1/1024 (0.09 percent) Native American. Scientists say the average European-American is 0.18 percent Native American. That’d make Warren even less Native American than the average European-American.”

By Monday afternoon, James Freeman, an assistant editor of The Wall Street Journal’s editorial page, had fleshed out Mr. Ahrens’s arithmetic. The DNA analysis, he wrote, “suggests that the senator is somewhere between 1/64th and 1/1024th Native American.” He added: “Her genetic makeup is perhaps similar to that of the average white person in the U.S.”

These numbers then began to ricochet around social media. They carried a clear implication: that Elizabeth Warren was no different in her Native American ancestry than a great many other white Americans.