Editing Quantitative genetics (section)

==== Derivation of ''σ<sup>2</sup><sub>G(1)</sub>''====
Recall that when ''f=1'', heterozygosity is zero, within-line variance is zero, and all genotypic variance is thus ''amongst-line'' variance and deplete of dominance variance. In other words, '''σ<sup>2</sup><sub>G(1)</sub>''' is the variance amongst fully inbred line means. Recall further [from "The mean after self-fertilization" section] that such means (G<sub>1</sub>'s, in fact) are '''G = a(p-q)'''. Substituting ''(1-q)'' for the ''p'', gives '''G<sub>1</sub> = a (1 − 2q)''' = '''a − 2aq'''.<ref name="Falconer 1996"/>{{rp|265}} Therefore, the '''σ<sup>2</sup><sub>G(1)</sub>''' is the '''σ<sup>2</sup><sub>(a-2aq)</sub>''' actually. Now, in general, the ''variance of a difference (x-y)'' is ''' [ σ<sup>2</sup><sub>x</sub> + σ<sup>2</sup><sub>y</sub> − 2 cov<sub>xy</sub> ]'''.<ref name="Snedecor & Cochran">{{cite book|last1=Snedecor|first1=George W.|last2=Cochran|first2=William G.|title=Statistical methods.|date=1967|publisher=Iowa State University Press|location=Ames|isbn=0-8138-1560-6|edition=Sixth}}</ref>{{rp|100}}<ref name="Kendall & Stuart">{{cite book|last1=Kendall|first1=M. G.|last2=Stuart|first2=A.|title=The advanced theory of statistics. Volume 1.|date=1958|publisher=Charles Griffin|location=London|edition=2nd}}</ref> {{rp|232}} Therefore, '''σ<sup>2</sup><sub>G(1)</sub> = [ σ<sup>2</sup><sub>a</sub> + σ<sup>2</sup><sub>2aq</sub> − 2 cov<sub>(a, 2aq)</sub> ] '''. But '''a''' (an allele ''effect'') and '''q''' (an allele ''frequency'') are ''independent''—so this covariance is zero. Furthermore, '''a''' is a constant from one line to the next, so '''σ<sup>2</sup><sub>a</sub>''' is also zero. Further, '''2a''' is another constant (k), so the '''σ<sup>2</sup><sub>2aq</sub>''' is of the type ''σ<sup>2</sup><sub>k X</sub>''. In general, the variance ''σ<sup>2</sup><sub>k X</sub>'' is equal to '''k<sup>2</sup> σ<sup>2</sup><sub>X</sub> '''.<ref name="Kendall & Stuart"/>{{rp|232}} Putting all this together reveals that ''' σ<sup>2</sup><sub>(a-2aq)</sub> = (2a)<sup>2</sup> σ<sup>2</sup><sub>q</sub> '''. Recall [from the section on "Continued genetic drift"] that ''σ<sup>2</sup><sub>q</sub> = pq f ''. With ''f=1'' here within this present derivation, this becomes ''pq 1'' (that is '''pq'''), and this is substituted into the previous.

The final result is: '''σ<sup>2</sup><sub>G(1)</sub> = σ<sup>2</sup><sub>(a-2aq)</sub> = 4a<sup>2</sup> pq = 2(2pq a<sup>2</sup>) = 2 σ<sup>2</sup><sub>a</sub> '''.

It follows immediately that '''''f'' σ<sup>2</sup><sub>G(1)</sub> = ''f'' 2 σ<sup>2</sup><sub>a</sub> '''. [This last ''f'' comes from the ''initial Sewall Wright equation'' : it is '''''not''''' the ''f '' just set to "1" in the derivation concluded two lines above.]