The most famous result in the study of structured populations come from Sewall Wright. He showed that in an island model, where each subpopulation is of size $N$ and the migration rate is $m$, then the pairwise $F_{ST}$ is

$$F_{ST} = frac{1}{4Nm+1}$$

This equation gives the expected $F_{ST}$. Because populations are finite in size ($N$), genetic drift yield this value to vary.

What is the variance in $F_{ST}$ in the infinite island model?

*References*

evolution in mendelian population is the original paper who derived this result from Sewall Wright.

Indirect measures of gene flow and migration: FST≠$frac{1}{4Nm+1}$ is an influential paper in the field.

GENE FLOW IN NATURAL POPULATIONS is a famous review as well.

From Lewontin and Krakauer 1973, the ratio

$$frac{F_{ST}(d-1)}{ar F_{ST}}$$

approximatively follows $chi^2$ distribution of degree $k=d-1$. Here $d$ is the number of demes (number of islands), $F_{ST}$ is the random variable of the $chi^2$ distribution and $ar F_{ST}$ is the average $F_{ST}$ that is $ar F_{ST} = frac{sum F_{ST}}{n}$, where $n$ is the number of $F_{ST}$ values.

The variance of a $chi^2$ distribution is $2k$, therefore

$$varleft(frac{F_{ST}(d-1)}{ar F_{ST}} ight) = 2d-2$$

Taking $frac{d-1}{ar F_{ST}}$ out of the ratio, the variance of $F_{ST}$ becomes

$$var(F_{ST})=left(frac{d-1}{ar F_{ST}} ight)^2(2d-2)$$

, which simplifies into

$$var(F_{ST}) = frac{2(d-1)^3}{ar F_{ST}^2}$$

The above expression is probably the most interesting result but one could go further and express the variance independently from the mean (by replacing $ar F_{ST}$ by Slatkin 1991 expectation for $ar F_{ST}$ in a finite island. It yields to

$$var(F_{ST}) = frac{2(d-1)^3}{left(frac{1}{1+4Nm(frac{d}{d-1})^2} ight)^2}$$

, which again "simplifies" into

$$var(F_{ST}) = frac{2 left(4 d^2 m N+d^2-2 d+1 ight)^2}{d-1}$$