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I have taken a closer look at the info on my 23andMe match with my known 1st cousin. It says we share 360 Centimorgans, and the Centimorgan charts I've looked at suggest this is potentially outside the "normal" range of a 1st cousin.

I know this falls outside of what seems to be accepted as the "normal" range for a 1st cousin match, and that it would seem genetically more likely she's my half 1st cousin.

Could this simply be a weird case of how DNA is passed on, and we just happen to have inherited less DNA than typical cousins? Could it have anything to do with the fact we didn't inherit the same X DNA (except for a tiny little piece)?

I've attached an image of our shared DNA. enter image description here

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This website presents information from a simulation study of first cousin DNA matches done by an expert statistician, Philip Gammon.

https://dna-explained.com/2017/09/27/first-cousin-match-simulations/

The first graphic shows the distribution of centimorgans among 200,000 (simulated) first cousin matches. The height of each bar gives the chances of each category of centimorgans among (true) first cousins. The range of centimorgans is from the category 250-275 centimorgans to 1600-1625 centimorgans for (true in the simulation) first cousin. As seen in the figure, these categories are rare—not many first cousins share this few (250-275) or this many (1600-1625) centimorgans, but the simulations shows that even these extreme values can occur for true first cousins.

As described at this website:

Academically calculated expectations suggest first cousins should share 850 cM. The data collected by Blaine showed an actual average of 874 cM, but varied within a 99th percentile range of 553 to 1225 cM utilizing 1512 respondents.

The use of the word “should share 850 cM” is misleading. It would be more correct to state that “on average” first cousins share 850 cM. Further, using the 99th percentiles for the academically calculated expectations, 99% of first cousins DNA matches will be from 553 to 1225 Cm. But that means that, even using the academically calculated data on 1512 respondents, 1% of first cousins will have less than 553 shared centimorgans or more than 1225 shared centimorgans.

In the description of the data from the simulation of 200,000 first cousin matches, Philip Gammon, the statistician, further states that:

At the lower end I have 787 events (about 0.4%) with fewer than 510 shared cM and a minimum of 272 cM.

The finding that you and your first cousin share only 360 centimorgans is very unusual but it is not impossible. It is at the extreme lower end of the probability distribution of shared centimorgans for first cousins.

The second graphic shows the (simulated) distribution of shared centimorgans considering the sex of the siblings who are the parents of the first cousins.

First cousins whose fathers were brothers are about two and a half times as likely to either share less than 8% or more than 17% than first cousins whose mothers were sisters.

It is not completely clear from the post whether you and your first cousin had fathers who were brothers. If your father and your first cousin’s father were brothers, then being in a “tail” of the distribution of shared centimorgans is even more understandable.

You ask ask:

Could this simply be a weird case of how DNA is passed on, and we just happen to have inherited less DNA than typical cousins?

Both the empiric data and the simulation results suggest that the answer is “yes” although weird may not be entirely

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The Shared cM Project 4.0 tool v4 advice for 360cM suggests that 360cM is too low for a full first cousin because 1C does not appear amongst the relationships with greater than 0% probability.

For full first cousins a match in the range of 396 to 1397 (average 866) cM is to be expected.

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