Cousin Chart and Shared DNA Centimorgan (cM) Math Explained

Few phrases at a family reunion cause more confusion than “second cousin once removed”, and few surprises in genetic genealogy cause more alarm than discovering that a documented fourth cousin shares 0 cM (zero centimorgans) of autosomal DNA with you on an ancestry test. Both puzzles are governed by exact, straightforward mathematics.

Once you count the generational steps (g_A and g_B) from two relatives up to their Most Recent Common Ancestor (MRCA), four simple equations give you their exact kinship label, Sewall Wright’s Coefficient of Relationship (r), their expected shared autosomal DNA in centimorgans (cM), and the Donnelly–Coop Poisson probability that random chromosome recombination left them with zero overlapping DNA segments. You can run all four calculations interactively in Tab C of our Tombstone, Census & Kinship Calculator.

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How Cousin Degree (C) and Generations Removed (R) Work in Plain English

Every collateral kinship label in English genealogy is defined by two integers:

• g_A: The number of generational steps (parent-child links, or meioses) from Person A up to the Most Recent Common Ancestor (g = 1 for parent, g = 2 for grandparent, g = 3 for great-grandparent, g = 4 for 2×-great-grandparent).
• g_B: The number of generational steps from Person B up to that same Most Recent Common Ancestor.

When both relatives are at least two generations below their shared ancestor (min(g_A, g_B) ≥ 2), their cousin relationship is defined by two equations:

Cousin Degree (C)       = min(g_A, g_B) - 1
Generations Removed (R) = |g_A - g_B|

1. Cousin Degree (C) Is Set by the Shorter Path (Count the “G’s”)

The cousin number (1st, 2nd, 3rd) is set by whichever person is closer in generations to the shared ancestor (min(g_A, g_B) − 1). Because Grandparent (g = 2) has 1 “G”, Great-Grandparent (g = 3) has 2 “G’s”, and Great-Great-Grandparent (g = 4) has 3 “G’s”, simply count the capital G’s in the shorter path’s ancestor title:

• Shared Grandparents (g_A = 2, g_B = 2): C = 2 − 1 = 1 → First Cousins (1C).
• Shared Great-Grandparents (g_A = 3, g_B = 3): C = 3 − 1 = 2 → Second Cousins (2C).
• Shared Great-Great-Grandparents (g_A = 4, g_B = 4): C = 4 − 1 = 3 → Third Cousins (3C).

2. “Removed” (R) Means Different Rows on the Family Tree

In genealogy, removed (R = |g_A − g_B|) counts how many generations apart the two cousins sit from the common ancestor:

• First cousin’s child (g_A = 2, g_B = 3): Your grandparents (g_A = 2) are that child’s great-grandparents (g_B = 3). Here C = min(2, 3) − 1 = 1 and R = |2 − 3| = 1 → First Cousin Once Removed (1C1R).
• Parent’s first cousin (g_A = 3, g_B = 2): Your great-grandparents (g_A = 3) are your relative’s grandparents (g_B = 2). Again C = 1 and R = 1 → First Cousin Once Removed (1C1R).
• First cousin’s grandchild (g_A = 2, g_B = 4): C = 1 and R = 2 → First Cousin Twice Removed (1C2R), whereas a Second Cousin (2C, g_A = 3, g_B = 3) sits on your own generation, even though both have d = 6 meioses and share 217.8 cM on average.

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Sewall Wright’s Coefficient of Relationship (r) and Inbreeding (F)

In 1922, geneticist Sewall Wright formulated the path coefficient of relationship (r), measuring the expected fraction of autosomal alleles two relatives share Identical by Descent (IBD). Each meiosis transmits an allele with probability 1/2, so a path of d = g_A + g_B steps from Common Ancestor i has weight (1/2)^(g_A + g_B):

Wright's Coefficient of Relationship:
r = sum_i [ (1/2)^(g_A,i + g_B,i) × (1 + F_i) ]

1. Half Cousins (m = 1 shared ancestor): Share one grandparent (r = (1/2)^(g_A + g_B)). For Half First Cousins (g_A = 2, g_B = 2), r = (1/2)^4 = 1/16 = 0.0625 (6.25%).
2. Full Cousins (m = 2 shared ancestors): Share both grandparents (r = 2 × (1/2)^(g_A + g_B)). For Full First Cousins (g_A = 2, g_B = 2), r = 2 × (1/2)^4 = 1/8 = 0.125 (12.5%).
3. Double First Cousins (m = 4 shared ancestors): When two siblings marry two siblings from another family, their children share all four grandparents (m = 4), giving r = 4 × (1/2)^4 = 0.25 (25.0%).

When two cousins marry, the Coefficient of Inbreeding (F) of their child (Φ_parents) is half the parents’ Coefficient of Relationship:

Coefficient of Inbreeding of Child:   F = Φ_parents = r_parents ÷ 2

First-cousin parents (r = 0.125) produce F = 0.0625 (6.25%); second-cousin parents (r = 0.03125) produce F = 0.015625 (1.5625%).

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Expected Shared Autosomal DNA (E[cM_HIR])

Across the 22 human autosomes, the sex-averaged genetic length of one haploid chromosome set is L = 3,485 cM (2L = 6,970 cM diploid, or ~6,800 cM effective callable length on consumer chips). Using Cotterman’s IBD coefficients—where k_1 is the probability of sharing 1 allele (Half-Identical Region, HIR) and k_2 is the probability of sharing 2 alleles (Fully Identical Region, FIR)—Wright’s coefficient satisfies r = (k_1 ÷ 2) + k_2.

Because consumer DNA match lists report Half-Identical Region (HIR) length without doubling fully identical regions:

Expected Shared Autosomal DNA:
E[cM_HIR] = (k_1 + k_2) × 3,485 cM

• Unilineal Relatives (k_2 = 0): For all standard cousins, half-siblings, aunts/uncles, and grandparents, k_2 = 0 and k_1 = 2r, so E[cM_HIR] = 2 × r × 3,485 cM = r × 6,970 cM ≈ r × 6,800 cM. For a First Cousin (r = 0.125), E[cM_HIR] = 0.25 × 3,485 = 871.25 cM (Shared cM v4 empirical mean: 866 cM).
• Full Siblings (k_1 = 0.50, k_2 = 0.25): At any locus, full siblings have a 50% chance of sharing 1 parental allele (k_1 = 0.50) and a 25% chance of sharing both (k_2 = 0.25). Thus E[cM_HIR] = (0.50 + 0.25) × 3,485 = 0.75 × 3,485 = 2,613.75 cM, matching the Shared cM Project v4 empirical mean of 2,613 cM.
• Double First Cousins (k_1 = 6/16, k_2 = 1/16): Unmerged diploid sharing is 2 × 0.25 × 3,485 = 1,742.5 cM (matching the Shared cM v4 mean of 1,697 cM), while strict non-doubled HIR (k_1 + k_2 = 7/16 = 0.4375) equals 1,524.7 cM.

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Donnelly–Coop Poisson Math: Why Distant Cousins Can Share 0 cM

Under the Donnelly (1983) and Graham Coop Poisson recombination model across 22 autosomes (34 Morgans), d = g_A + g_B meioses from m common ancestors (m = 2 for full cousins, m = 1 for half cousins) yield an expected shared segment count lambda and zero-DNA probability P(0 cM):

Expected IBD Segments (d ≥ 2):   lambda = m × (22 + 34 × d) ÷ 2^(d - 1)
Probability of 0 cM Shared DNA:  P(0 cM) = e^(-lambda)

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Complete Cousin Chart & Shared cM Project v4 Reference Table

Relationship (Abbrev.)	Steps (g_A, g_B)	Meioses d (m)	Wright’s r (%)	Theoretical E[cM_HIR]	Shared cM v4 Mean	Shared cM v4 99% Range	Poisson Segments lambda	Theoretical P(0 cM)
Parent / Child (P/C)	(1, 0)	1 (m=1)	0.5000 (50.0%)	3,485.0 cM	3,485 cM	2,376 – 3,720 cM	22.0 (intact)	0.00%
Full Sibling (FS)	(1, 1)	2 (m=2)	0.5000 (50.0%)	2,613.8 cM	2,613 cM	1,613 – 3,488 cM	90.0 (~58 HIR)	0.00%
Grandparent / Grandchild (GP)	(2, 0)	2 (m=1)	0.2500 (25.0%)	1,742.5 cM	1,754 cM	984 – 2,462 cM	28.0 (1 recomb.)	0.00%
Aunt / Uncle (AU)	(1, 2)	3 (m=2)	0.2500 (25.0%)	1,742.5 cM	1,741 cM	1,201 – 2,282 cM	62.0	0.00%
Half-Sibling (HS)	(1, 1)	2 (m=1)	0.2500 (25.0%)	1,742.5 cM	1,759 cM	1,160 – 2,436 cM	45.0	0.00%
Double 1st Cousin (2×1C)	(2, 2)	4 (m=4)	0.2500 (25.0%)	1,742.5 cM (1,524.7 HIR)	1,697 cM	1,025 – 2,389 cM	79.0 (~46.5 HIR)	0.00%
1st Cousin (1C)	(2, 2)	4 (m=2)	0.1250 (12.5%)	871.3 cM	866 cM	396 – 1,397 cM	39.5	0.00%
Half 1st Cousin (Half 1C)	(2, 2)	4 (m=1)	0.0625 (6.25%)	435.6 cM	449 cM	156 – 979 cM	19.8	0.00%
1st Cousin Once Removed (1C1R)	(2, 3)	5 (m=2)	0.0625 (6.25%)	435.6 cM	433 cM	102 – 980 cM	24.0	0.00%
1st Cousin Twice Removed (1C2R)	(2, 4)	6 (m=2)	0.03125 (3.13%)	217.8 cM	221 cM	33 – 471 cM	14.1	under 0.01%
2nd Cousin (2C)	(3, 3)	6 (m=2)	0.03125 (3.13%)	217.8 cM	229 cM	41 – 592 cM	14.1	under 0.01%
Half 2nd Cousin (Half 2C)	(3, 3)	6 (m=1)	0.01563 (1.56%)	108.9 cM	120 cM	10 – 325 cM	7.1	0.09%
2nd Cousin Once Removed (2C1R)	(3, 4)	7 (m=2)	0.01563 (1.56%)	108.9 cM	122 cM	14 – 353 cM	8.1	0.03%
3rd Cousin (3C)	(4, 4)	8 (m=2)	0.00781 (0.78%)	54.5 cM	73 cM	0 – 234 cM	4.59	1.01%
3rd Cousin Once Removed (3C1R)	(4, 5)	9 (m=2)	0.00391 (0.39%)	27.2 cM	48 cM	0 – 192 cM	2.56	7.71%
4th Cousin (4C)	(5, 5)	10 (m=2)	0.00195 (0.20%)	13.6 cM	35 cM	0 – 139 cM	1.41	24.3% (~50% under 8 cM)
5th Cousin (5C)	(6, 6)	12 (m=2)	0.00049 (0.05%)	3.4 cM	25 cM	0 – 117 cM	0.42	65.7% (~85% under 8 cM)

To separate relationships that share the same average cM (such as 1C1R, Half 1C, and Great-Aunt at ~435 cM), compare birth windows using US Federal Census Dates (1790–1950) and Birth Year Math and Tombstone Birth-Date Math: The 8870 Formula vs. Exact Calendar Days, trace surname spellings with How the NARA Soundex Phonetic Code Works for Surname Variants, and scan family reunion photos with our Photo, Slide & Home Movie Digitization Planner.