Editing Quantitative genetics (section)

==== The mean after random fertilization====
The contribution of '''AA''' is <math display="inline">p^2 (+)a</math>, that of '''Aa''' is <math display="inline">2pq d</math>, and that of '''aa''' is <math display="inline">q^2 (-)a</math>. Gathering together the two '''a''' terms and accumulating over all, the result is: <math display="inline"> a(p^2-q^2) + 2pq d</math>. Simplification is achieved by noting that <math display="inline"> (p^2-q^2) = (p-q)(p+q)</math>, and by recalling that <math display="inline"> (p+q) = 1</math>, thereby reducing the right-hand term to <math display="inline">(p-q)</math>.

The succinct result is therefore <math display="inline"> G = a(p-q) + 2pqd</math>.<ref name="Falconer 1996"/> {{rp|110}}

This defines the population mean as an "offset" from the homozygote midpoint (recall '''a''' and '''d''' are defined as ''deviations'' from that midpoint). The Figure depicts '''G''' across all values of '''p''' for several values of '''d''', including one case of slight over-dominance. Notice that '''G''' is often negative, thereby emphasizing that it is itself a ''deviation'' (from '''mp''').

Finally, to obtain the ''actual'' Population Mean in "phenotypic space", the midpoint value is added to this offset: <math display="inline"> P = G + mp</math>.

An example arises from data on ear length in maize.<ref name="Sinnott Dunn & Dobzhansky">{{cite book|last1=Sinnott|first1=Edmund W.|last2=Dunn|first2=L. C.|last3=Dobzhansky|first3=Theodosius|title=Principles of genetics.|url=https://archive.org/details/principlesofgene00sinn|url-access=registration|date=1958|publisher=McGraw-Hill|location=New York}}</ref>{{rp|103}} Assuming for now that one gene only is represented, '''a''' = 5.45&nbsp;cm, '''d''' = 0.12&nbsp;cm [virtually "0", really], '''mp''' = 12.05&nbsp;cm. Further assuming that '''p''' = 0.6 and '''q''' = 0.4 in this example population, then:

'''G''' = 5.45 (0.6 − 0.4) + (0.48)0.12 = '''1.15&nbsp;cm''' (rounded); and

'''P''' = 1.15 + 12.05 = '''13.20&nbsp;cm''' (rounded).