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.

Both Mr. Ahrens and Mr. Freeman cited a 2014 New York Times article as evidence for their claims. I wrote that article. So let me just say this: They’re wrong.

They both mistakenly treat DNA as a matter of simple fractions. We each have two parents, the thinking goes, so therefore we inherit half of each parent’s DNA. From each grandparent we inherit precisely a quarter of our DNA, and so on by the powers of two back into the mists of time. This is how they came up with the 1/1024th figure — two parents, each with two parents, going back 10 generations.

This misguided way of thinking has a history that extends far beyond the discovery of DNA. For centuries, people thought of ancestry in terms of blood, and fractions of it. People were pure-blooded or half-blooded. When the United States government set up rules for deciding who could be members of Native American tribes, it called the system “blood quantum.”

Slavery, too, led to an obsession with increasingly tiny fractions of ancestral blood, reaching the absurd extreme of the “one drop” rule. A single black ancestor — no matter how far back in the family tree, no matter how tiny the mythical drop of blood he or she contributed — was enough to make a person black.

Blood still soaks our language about heredity. Reporting Monday on Senator Warren’s test, The Atlanta Journal Constitution wrote that “she may have Cherokee blood.”

But DNA is not a liquid that can be divided down into microscopic drops. It’s a string-like molecule, arranged into 23 pairs of chromosomes, that gets passed down through the generations in a counterintuitive way.

Eggs and sperm randomly end up with one copy of each chromosome, coming either from a person’s mother or father. In the process, some DNA can shuffle from one chromosome to its partner. That means we inherit about a quarter of our DNA from each grandparent — but only on average. Any one person may inherit more DNA from one grandparent and less from another.

Over generations, this randomness can lead to something remarkable. Look back far enough in your family tree, and you’ll encounter ancestors from whom you inherit no DNA at all.

The geneticist Graham Coop of the University of California, Davis, and his colleagues have studied how DNA disappears. If you pick one of your ancestors from 10 generations back, the odds are around 50 percent that you carry any DNA from him or her. The odds get even worse beyond that.

Even if you get no DNA from many of your ancestors, they are still your ancestors. “Genetics,” Dr. Coop has noted, “is not genealogy.”

To understand Senator Warren’s ancestry, the geneticist Carlos Bustamante of Stanford University examined the sampling of DNA she inherited from some of her ancestors. He used software he and his colleagues have developed to compare stretches of DNA in one person to those in different populations.

Dr. Bustamante’s study had some clear limits. For example, Native Americans in the United States have not participated much yet in genetic studies, so he couldn’t include them in his comparison.

But all indigenous people in the Americas share a common genetic heritage because they descend from people who came from Asia at the end of the Ice Age. And Dr. Bustamante found that five small segments of Senator Warren’s DNA were all similar to segments from indigenous people in Mexico, Peru and Canada.

Dr. Bustamante’s software is not designed to offer precise percentages of genetic ancestry. But it’s safe to say that some fraction of 1 percent of Senator Warren’s DNA comes from a Native American ancestor.

How many European-Americans are like Senator Warren, with a small amount of Native American ancestry? Scientists can’t say for sure. The best clues to date come from a 2014 study carried out by researchers at 23andMe. They looked at the DNA of 160,000 customers who described themselves as being of European, African or Latino ancestry. Across all the European-Americans in the study, the average amount of Native American ancestry was 0.18 percent.

But once again, averages can be deceptive. The researchers found that 2.7 percent of their European customers had 1 percent or more Native American ancestry. The vast majority had no detectable Native American ancestry at all. It’s impossible to directly compare Senator Warren’s results to the 23andMe customers because they were produced with different software.

What we can say is that Senator Elizabeth Warren is not unique. There are millions of European-Americans who have a small amount of Native American ancestry. But it would be wrong to claim that all European-Americans do.

What these results will mean to Senator Warren’s sense of herself is for her to work out. For her part, she has said that she would not claim membership in any tribe based on these results. Everyone else getting ancestry tests also will have to decide what their results mean for themselves. But if we want to come to a full reckoning with our DNA, we have to get a lot more familiar with the science behind it.

Carl Zimmer writes the Matter column for The New York Times and is the author of “She Has Her Mother’s Laugh: The Powers, Perversions, and Potential of Heredity.”

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