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Since the advent of cheap DNA tests, several companies have sprung up which will tell you ancestral origins according to your DNA.

23andMe claims,

Where in the world are you from?

Find out where your DNA comes from around the world. Your DNA can tell you where your ancestors lived more than 500 years ago. See a percentage breakdown by region, including eastern Asia, Sub-Saharan Africa, Europe and more. We look at 31 populations worldwide, and we will continue to refine your results as our database grows.

AncestryDNA makes similar claims

Nigerian? Scandinavian? What are you?

AncestryDNA can estimate your origins to 26 ethnic regions, 9 in Africa alone—more than anyone else. And we've got Europe covered 9 ways.

Reading through their advertising claims, I am having a hard time determining what exactly it means to be 72.3% Northern European, or 24% Ghana/Ivory Coast. At what point in history was there an archetypal group of people who were 100% Ivory Coast? My historical knowledge of human migration and population growth is pretty spotty, but I am under the impression that it was a very messy process. Their site has a veneer of science, but is clearly targeted at a lay audience. This question could be improved by finding a scientific discussion of what the company's scientists think their ancestry reports mean.

I am skeptical of the claims partly because they seem difficult to falsify. If a DNA test says that my ancestors are 19% asian, who can argue with that? The ancestors in question are dead.

What evidence shows that the DNA tests from 23andMe and AncestryDNA provide accurate and meaningful results? Is this evidence sufficient to justify their claims?

Note: I have to tip my hat to @Statsanalyst who originally asked a similar question that lacked a specific answerable claim.

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    I am not sure this question has a specific claim either. It seems to be more about asking "What does 23andMe mean when they say 72.3% Northern European?", rather than doubting the tests are accurate in their way. – Oddthinking Jul 11 '17 at 5:04
  • (I have wondered about this question before when this video was doing the viral rounds. My cynical supposition was that the producers were manipulating the subjects, knowing that, for example, the man who thought his ancestors were Icelandic for many generations could never get a "100% Icelandic" result from the tests that were run, no matter what his genes. I don't know enough about the statistical methods used to know if this is true or not.) – Oddthinking Jul 11 '17 at 5:07
  • @Oddthinking: Iceland has a small population and an excessive amount of public genealogical information so virtually everyone is a cousin and can easily discover it. But ultimately they are of predominately Nordic and Celtic descent plus some later migrants and this should show up in these tests, which are essentially clever correlation calculations involving other people who have taken the test in other countries – Henry Jul 11 '17 at 8:09
  • Let's take this to chat. – Oddthinking Jul 11 '17 at 10:57
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    Sorry, I'm with Oddthinking. "accurate and meaningful" only can be tested if there's a fixed benchmark for determining accuracy or specific fixed meaning to compare to. This could (imho) be a great question on Biology.SE with only minor wording tweaking; but i don't think it fits nearly as well with Skeptics – user5341 Jul 11 '17 at 13:02

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