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When getting a DNA test, how many generations back will a specific ethnicity show up?

Our family tree shows that our 5th great grandfather was African.

My cousin just received her DNA results and it shows only 2% African.

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It depends somewhat on the ethnicity and how distinguishable it is from the predominant ethnicity. For example, it seems that trace amounts of DNA from Native American ancestors is more difficult to identify in the DNA of a person of mostly European ancestry than trace amounts of DNA from African ancestors.

In the case you present of DNA from African ancestors in (I'm assuming) a person of primarily European descent, it can a surprising number of generations. This is because back to about 5 generations back, we generally have some DNA from all of our ancestors in that generation. For each of those ancestors though, we will have only a few blocks of DNA from that person scattered around our total DNA. When we have a child, because they only get half our DNA, they may get one of say, three blocks we have from a particular ancestor, or they may even not get any. When we get down to having only one block from a distant ancestor, each child may or may not get that block. These blocks are large enough that we can discern them when matching distant relatives, for example, but they are small enough that they are unlikely to get split in half by the crossover mechanism. So, past about 5 generations, it's not that we get tinier and tinier chunks of DNA from ancestors at each subsequent generation, it's that we either do or don't get the chunk(s) our parent has for a particular ancestor. At about the 6th generation back, we get to a situation where some ancestors are no longer in our DNA at all in detectable amounts. Simulations show that our DNA has room to represent about 128-256 ancestors at any particular generation. So, you could have a detectable block of DNA from an ancestor 10 generations back, but you have detectable blocks of DNA from only about 10% of those 10th generation ancestors.

At distance you mention, an African ancestor's DNA will likely be detectable, though there's a small chance it might not. If detectable, the amount will vary, because of the random nature of inheritance. If you have multiple blocks of DNA that correlate to African ancestry, you should also consider the possibility you have more than one African ancestor, with more distant ancestors you haven't discovered besides the ancestor you have identified. If you have Colonial southern US ancestry, this is especially true. In early colonial times, mixed race unions were not all that uncommon.

You can read more here:

How much of your genome do you inherit from a particular ancestor?

Everyone Has Two Family Trees – A Genealogical Tree and a Genetic Tree

How many genetic ancestors do I have?.

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Each of us has 128 x 5th great grandparents (assuming no inbreeding) which, if only one of them was African, suggests that your cousin might be about 0.78% African.

Consequently, a result of 2% African, is a little more than might be expected, but certainly a reasonable value.

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