page_title: Quantitative genetics
section_title: Dispersion and the genotypic variance
hierarchical_section_title: Quantitative genetics / Extended principles / Dispersion and the genotypic variance
caption_attribution_description: English: Dispersion and components of the Genotypic variance.
context_page_description: Quantitative genetics deals with phenotypes that vary continuously —as opposed to discretely identifiable phenotypes and gene-products.
  Both branches use the frequencies of different alleles of a gene in breeding populations, and combine them with concepts from simple Mendelian inheritance to analyze inheritance patterns across generations and descendant lines. While population genetics can focus on particular genes and their subsequent metabolic products, quantitative genetics focuses more on the outward phenotypes, and makes summaries only of the underlying genetics.
  Due to the continuous distribution of phenotypic values, quantitative genetics must employ many other statistical methods to link phenotypes to genotypes. Some phenotypes may be analyzed either as discrete categories or as continuous phenotypes, depending on the definition of cut-off points, or on the metric used to quantify them. Mendel himself had to discuss this matter in his famous paper, especially with respect to his peas attribute tall/dwarf, which actually was "length of stem".
context_section_description: In the section on genetic drift, and in other sections that discuss inbreeding, a major outcome from allele frequency sampling has been the  dispersion of progeny means. This collection of means has its own average, and also has a variance: the amongst-line variance. (This is a variance of the attribute itself, not of allele frequencies.) As dispersion develops further over succeeding generations, this amongst-line variance would be expected to increase. Conversely, as homozygosity rises, the within-lines variance would be expected to decrease. The question arises therefore as to whether the total variance is changing—and, if so, in what direction. To date, these issues have been presented in terms of the genic (σ ²A ) and quasi-dominance (σ ²D ) variances rather than the gene-model components. This will be done herein as well.
  The crucial overview equation comes from Sewall Wright,   and is the outline of the inbred genotypic variance based on a weighted average of its extremes, the weights being quadratic with respect to the inbreeding coefficient 
  . This equation is:
  where  is the inbreeding coefficient,  is the genotypic variance at 
  f=0,  is the genotypic variance at f=1, 
   is the population mean at f=0, and  is the population mean at f=1.
  The  component [in the equation above] outlines the reduction of variance within progeny lines. The   component addresses the increase in variance amongst progeny lines. Lastly, the  component is seen (in the next line) to address the quasi-dominance variance.  These components can be expanded further thereby revealing additional insight. Thus:-
  Firstly, σ²G(0) [in the equation above] has been expanded to show its two sub-components [see section on "Genotypic variance"]. Next, the σ²G(1) has been converted to 4pqa² , and is derived in a section following. The third component's substitution is the difference between the two "inbreeding extremes" of the population mean [see section on the "Population Mean"].
  Summarising: the within-line components are  and ; and the amongst-line components are  and .
  Rearranging gives the following:
   
  The version in the last line is discussed further in a subsequent section.
  Similarly,
  Graphs to the left show these three genic variances, together with the three quasi-dominance variances,  across all values of f, for p = 0.5 (at which the quasi-dominance variance is at a maximum). Graphs to the right show the Genotypic variance partitions (being the sums of the respective genic and quasi-dominance partitions) changing over ten generations with an example f = 0.10.
  Answering, firstly, the questions posed at the beginning about the total variances [the Σ in the graphs] : the genic variance rises linearly with the inbreeding coefficient, maximizing at twice its starting level. The quasi-dominance variance declines at the rate of (1 − f² ) until it finishes at zero. At low levels of f, the decline is very gradual, but it accelerates with higher levels of f.
  Secondly, notice the other trends. It is probably intuitive that the within line variances decline to zero with continued inbreeding, and this is seen to be the case (both at the same linear rate (1-f) ). The amongst line variances both increase with inbreeding up to f = 0.5, the genic variance at the rate of 2f, and the quasi-dominance variance at the rate of (f − f²). At f > 0.5, however, the trends change. The amongst line genic variance continues its linear increase until it equals the total genic variance. But, the amongst line quasi-dominance variance now declines towards zero,  because (f − f²) also declines with f > 0.5.
caption: Dispersion and components of the genotypic variance
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