DNA Cousins are more distant than they appear

When interpreting autosomal DNA statistics, one must be careful to distinguish between the distribution of shared DNA for a given relationship and; the distribution of relationships for given amounts of shared DNA.

Distribution of shared DNA for a known relationship


Data has been collected for the distribution of shared DNA by relationship by the Shared cM Project 3.0. These were crowdsourced by Blaine Bettinger. Most were likely from tests conducted on known relatives, others from a test that found a match which was subsequently identified. The figure is an example of the full distribution for first cousin once removed.

The table below summarizes the centimorgan (cM) results for a wider range of relationships. The more distant the relationship the less shared DNA, and also the fewer the data reported so there is greater sampling error. For relationships more distant than third cousin the Shared cM Project 3.0 obtained insufficient data for a full analysis. Perhaps there was a relationship but the amount shared was too small for it to be recognized. That also shows in the ratio of the average cM to the amount expected which increases for more distant relationships.

Relatives	1C	1C1R	1C2R	2C	2C1R	2C2R	3C	3C1R	4C	4C1R
1st percentile	486	131	47	47	0	0	0	0	0	0
Average (Expected)	884 (850)	440 (425)	232 (213)	232 (213)	123 (106)	75 (52)	75 (53)	49 (27)	36 (13)	29 (7)
99th percentile	1761	851	517	517	317	229	229	175	122	118

Distribution of relationships for given amounts of shared DNA
Extended families vary in size. In informal polls I've found anything from zero to over 80 first cousins. According to a study cited by ISOGG https://isogg.org/wiki/Cousin_statistics the average British person has an estimated 5 first cousins, 28 second cousins and 175 third cousins.

Even if the expected amount of shared DNA is relatively small more people with a more distant relationship will increase the chance of finding a match — you catch more fish if there are more fish to catch. The weighting depends on the number of people with the relationship, and that depends on whether families had few or many children who themselves went on to have few or many children, and how many survived.

Simple Scenarios
Take the situation of 200 cM of shared DNA where the only relatives with tests are of the same, one or two generations younger. This would be the case for a senior where those in the older generations are deceased while there are no people in the great-grandchildren’s and subsequent generations yet born. Assume every family in the tree consists of the same number of children and there are no half relationships, endogamy or other complicating factors. Everyone survives and has that same number of children.  The table shows the percent probabilities from the Shared cM Project and for families which uniformily have two and three children.

Percent Probability	1C	1C1R	1C2R	2C	2C1R	2C2R	3C	3C1R
Shared cM Project	0	2	34	37	19	4	4	>0
Two Family	0	1	22	24	31	9	10	3
Three Family	0	1	19	18	34	12	12	5

The peak probability shifts from second cousin to second cousin once removed. Probabilities of closer relationships decrease and of more distant relationships increase. The trend is larger for larger family size — unsurprising as, for instance, there are 64 third cousins in the Two Family, 432 in the Three Family.

Conclusions
Simple population scenarios show that the distribution of relationships for given amounts of shared DNA differs from the distribution of shared DNA for a known relationship as found in the Shared cM Project. Taking into account the population distribution moves probabilities to more distant relationships, more so for larger family size.
It’s important to appreciate the limitations of the analysis. The statistics in the Shared cM Project are less reliable where there are fewer data reported, for more distant relationships and in the tails of the distribution. Further, given the variability of individual extended families the scenarios in the study will only be broadly representative for any individual's result.
While it would be interesting to have a similar analysis for lesser amounts of shared DNA the statistics at present are too noisy and do not extend to sufficiently distant relationships to yield reliable results.