Editing Quantitative genetics (section)

====Standardized selection – the normal distribution====

The entire ''base population'' is outlined by the normal curve<ref name="Balaam 1972"/>{{rp|78–89}} to the right. Along the '''Z axis''' is every value of the attribute from least to greatest, and the height from this axis to the curve itself is the frequency of the value at the axis below. The equation for finding these frequencies for the "normal" curve (the curve of "common experience") is given in the ellipse. Notice it includes the mean ('''μ''') and the variance ('''σ<sup>2</sup>'''). Moving infinitesimally along the z-axis, the frequencies of neighbouring values can be "stacked" beside the previous, thereby accumulating an area that represents the '''probability''' of obtaining all values within the stack. [That's '''integration''' from calculus.] Selection focuses on such a probability area, being the shaded-in one from the ''selection threshold (z)'' to the end of the superior tail of the curve. This is the ''selection pressure''. The selected group (the effective parents of the next generation) include all phenotype values from '''z''' to the "end" of the tail.<ref>Theoretically, the tail is '''''infinite''''', but in practice there is a ''quasi-end''.</ref> The mean of the '''''selected group''''' is '''μ<sub>s</sub>''', and the difference between it and the base mean ('''μ''') represents the '''selection differential (S)'''.  By taking partial integrations over curve-sections of interest, and some rearranging of the algebra, it can be shown that the "selection differential" is '''S = [ y (σ / Prob.)] ''', where '''y''' is the ''frequency'' of the value at the "selection threshold" '''z''' (the ''ordinate'' of ''z'').<ref name="Crow & Kimura"/>{{rp|226–230}} Rearranging this relationship gives '''S / σ = y / Prob.''', the left-hand side of which is, in fact, ''selection differential divided by standard deviation''—that is the ''standardized selection differential (i)''. The right-side of the relationship provides an "estimator" for '''i'''—the ordinate of the ''selection threshold'' divided by the ''selection pressure''. Tables of the Normal Distribution<ref name="Snedecor & Cochran"/> {{rp|547–548}} can be used, but tabulations of '''i''' itself are available also.<ref name="Becker 1967">{{cite book|last1=Becker|first1=Walter A.|title=Manual of procedures in quantitative genetics.|date=1967|publisher=Washington State University|location=Pullman|edition=Second}}</ref>{{rp|123–124}} The latter reference also gives values of '''i''' adjusted for small populations (400 and less),<ref name="Becker 1967"/>{{rp|111–122}} where "quasi-infinity" cannot be assumed (but ''was'' presumed in the "Normal Distribution" outline above). The ''standardized selection differential ('''i''')'' is known also as the '''''intensity of selection'''''.<ref name="Falconer 1996"/>{{rp|174; 186}}

Finally, a cross-link with the differing terminology in the previous sub-section may be useful: '''μ''' (here) = "P<sub>0</sub>" (there), '''μ<sub>S</sub>''' = "P<sub>S</sub>" and '''σ<sup>2</sup>''' = "σ<sup>2</sup><sub>P</sub>".