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It is easy to quantify and measure genotypic variation, because genes are digital, like the 1's and 0's of a computer. You just check how many places the gene "letters" A,C,G,and,T differ. But how does one quantify and measure phenotypic variation? The reason I am asking this question is because there are several species, like dogs and also us humans, where their genotypic variation is very small, but, at least intuitively, the phenotypic variation is large compared to their genotypic variation. Humans of different races look very different from one another, and different breeds of dogs look even more different, despite, in both cases, being extremely genetically similar. Has any book or paper or text proposed a rigorous definition to quantify and measure phenotypic variation?

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It depends on the phenotype you're quantifying. If you want to quantify height, for example, just measure it and record the number. However, you seem to be implying "overall" phenotypic variation of all phenotypes. You may be interested in looking up "PheWASs," or phenome-wide association studies. You may also be interested in the correlation of every phenotype to every other phenotype and every phenotype's variance.

Also, phenotype extends beyond how something "looks" to disease susceptibility, protein expression, behavior, and internal organ structure. You'd have to take all of this into account, which would be difficult. Perhaps something analogous would be studies of protein expression: search for "transcriptome -wide association study."

Lastly, you have a misconception about the genotypic variance of humans and dogs. It's correct that small genotypic variation can lead to large phenotypic variation. However, humans have a large degree of genetic diversity between us, and dogs also have a decent amount. You are not alone in asking this question. One authors said "The domestic dog displays greater levels of morphological and behavioral diversity than have been recorded for any land mammal" (See also: "Behavioral Phenotyping" section of that link, and a related paper). Here are FST charts from Google showing the impressive genetic diversity of our and their species:

(Of course, FST is a poor measure and much of the variation exists outside the way it is shown/defined in these graphs.)

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