...the result is never pretty. I made this point in a comment, but I've hoisted it up here because this issue deserves more visibility.

Physics professor Steve Hsu makes this argument:

...you may have read the misleading statistic, spread by the intellectually dishonest Lewontin, that 85% percent of all human genetic variation occurs within groups and only 15% between groups. This neglects the correlations in the genetic data that are revealed in a cluster analysis. See here for a simple example which shows that there can be dramatic group differences in phenotypes even if every version of every gene is found in two groups (i.e., 100% of the variation is found within each group) -- as long as the frequency or probability distributions are distinct. Sadly, understanding this point requires just enough mathematical ability that it has eluded all but a small number of experts.)

This is a misrepresentation of what geneticists are claiming. It is absolutely true that 85-90% of genetic variation is found within populations (see this this recent paper in Science, and this short review in Nature Genetics for more discussion). That number is in fact calculated using differences in allele frequencies. No geneticist, Lewontin included, is claiming that 15% of all different gene versions are only found in single populations.

Hsu is correct in saying that it is the frequencies of alleles among different populations that are important, but he's wrong to claim that geneticists are missing this point. (People who study Higgs bosons aren't the only ones who know how to do principal components analysis.)

But the larger issue is whether these genetic differences add up to significant biological differences. Lewontin says no. Other geneticists don't go quite as far in saying there is 'no biological basis for race', but there are few that I am aware of who think that the between-population genetic variance prodcues major biological differences in our most complex traits.

To be fair, we should say that the jury is still out on how much phenotypic variation this population-specific genetic variation explains. Hsu places his bet:

Rather than proving that race is skin-deep, non-existent, or unimportant, modern genetic science is both proving that it is in fact existent, but also sets the foundation for the study of its true importance, which is probably somewhere in between the indifference of the sociologists and the hyperbole of the racists.

But in fact these genetic studies have not ruled out the idea that race is unimportant or skin deep. Yes, there are genetic differences, but whether these differences add up to anything more than superficial characteristics is still an open question. (The vast majority of genetic variants are very likely to be basically neutral variants.)

Hsu does lay out the question in another post:

Two groups that form distinct clusters are likely to exhibit different frequency distributions over various genes, leading to group differences.

This leads us to two very distinct possibilities in human genetic variation:

Hypothesis 1: (the PC mantra) The only group differences that exist between the clusters (races) are innocuous and superficial, for example related to skin color, hair color, body type, etc.

Hypothesis 2: (the dangerous one) Group differences exist which might affect important (let us say, deep rather than superficial) and measurable characteristics, such as cognitive abilities, personality, athletic prowess, etc.

A standard argument against H2 is that the 50k years during which groups have been separated is not long enough for differential natural selection to cause any group differences in deep characteristics. I find this argument quite naive, given what we know about animal breeding and how evolution has affected the (ever expanding list of) "superficial" characteristics. Many genes are now suspected of having been subject to strong selection over timescales of order 5k years or less.

The argument against hypothesis 2 is not naive, however it is naive to think that the selective pressures on human populations are anywhere close to the highly engineered selective pressures you find in animal breeding. And selection pressure isn't the only issue: there has been a fairly substantial amount of gene flow among human populations over the last 50k years, which tends to act against population differentiation. Given what we know about human population history, and the likelihood that a very complex trait like intelligence is controlled by many genes with pleiotropic effects, I find it implausible that significant differences in intelligence between populations have arisen over the past 50k years.

Hsu ridicules the "ever expanding list" of 'superficial' characteristics, but an expanding list is exactly what we should expect as we start to look at the genetic basis of traits. Superficial is a bad word perhaps; what really matters is not what's externally visible, but whether a trait can be influenced by a single (non-pathological) allele of large effect, which makes it easy to produce rapid and significant changes between human populations.

In the case of athletic prowess, you may have a single genetic variant that causes a major increase in athletic ability (here is an extreme example). Something similar appears to have happened with skin pigmentation in human populations that migrated out of Africa: single variants with a large effect on skin color was able to quickly reach high frequency in populations that moved out of Africa.

This is in contrast with intelligence: single genetic variants with large beneficial effects on intelligence are probably much, much harder to come by, because any large genetic change is likely to screw up something as complex as cognition. To get significant differences in intelligence between populations, my biological intuition is that you would need many small but non-negligible changes that together add up to something large enough to make a difference in the population distribution of intelligence.

In sum:

1. There are genetic differences between human populations, but genetic variance between populations is small compared to variance among populations.

2. We don't know how much of that minor amount of between-population genetic variance has an effect on phenotype.

3. Complicated traits, involving many genes with likely pleiotropic effects are, given human population history, much less likely to differ significantly (at least the genetic component) between different populations.