"You know nothing about a person's intelligence on the basis of his or her skin color. That is just a fact.
There is much more variance among individuals in any racial group than there is between groups..."
The first two sentences are wrong. The final sentence is true. It is presumably motivated by the idea that we should not infer properties of a group to the individual especially where such inference will result in discrimination.
As an empirical point, racial and national differences in intelligence test scores are well established in the literature (for a review see https://en.wikipedia.org/wiki/Intelligence:_Knowns_and_Unknowns). There is plenty of debate that can be had about what this means (i.e., to what extent does it reflect genetic differences versus environmental conditions? in what ways do standard IQ tests fail to measure aspects of intelligence that might be of interest? To what extent are differences declining with reduced inequality and improved living conditions? etc.). But as an empirical point, I've often seen studies with many thousands of people find differences between racial groups in the vicinity of one standard deviation. By conventional rules of thumb, this is a large difference.
Statistical perspective: If groups differ on a variable, then you can use knowledge of group membership to improve your prediction on that variable. For example, occupation predicts income. If I know someone is a doctor, I can predict that they earn more money than other people. As group differences get larger, so will be the improvement in prediction that is provided by knowing group membership.
Here is an example where groups differ by one standard deviation (i.e., one group has mean IQ of 100, and the other has mean IQ of 85):
# Create data: 2 groups differing by 1 SD
> group1 <- data.frame(group = 1, iq = rnorm(1000, mean = 100, sd = 15))
> group2 <- data.frame(group = 2, iq = rnorm(1000, mean = 85, sd = 15))
> x <- rbind(group1, group2)
> summary(lm(iq ~ group, x))
lm(formula = iq ~ group, data = x)
Min 1Q Median 3Q Max
-47.168 -10.344 0.013 10.132 52.188
Estimate Std. Error t value Pr(>|t|)
(Intercept) 114.344 1.064 107.46 <2e-16 ***
group -14.279 0.673 -21.22 <2e-16 ***
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 15.05 on 1998 degrees of freedom
Multiple R-squared: 0.1839, Adjusted R-squared: 0.1835
F-statistic: 450.2 on 1 and 1998 DF, p-value: < 2.2e-16
I then ran a multiple regression. The R-squared represents the percentage of variance explained by group (i.e., 18.4%). Thus, the within-group variance is 100% - 18.4% = 81.6%. So that is the basis for the claim. So one possible estimate based on the literature is that within-group variance is around four times larger than between-group variance. Of course the exact numbers would depend on which groups were being compared and which empirical literature was used to form the estimate.
This raises the question of how big group differences need to be before you could say that "you know something about a person based on group membership".
Presumably, such inference are partially based on practical and ethics issues related to over generalization.
In general, group differences need to be extremely large (perhaps 2 or 3 standard deviations) before the overlap between group distributions gets small.
Legal/ethical perspective: More importantly, generalizing from groups to individuals is illegal in many contexts. For example, many countries have legislation that aims to prevent the use of age, race, gender, religion, etc. as the basis for making employment, education, and other defined decisions. Thus, even if membership of these categories is predictive of something relevant, decision makers are forbidden from using this information. Rather, if they want, for example, to use intelligence scores to influence hiring decisions, then they need to actually measure intelligence, rather than rely on a meta-analysis that shows that the group to which the applicant is a member tends to score higher or lower.
There are many good reason why such a legal regime is ethical and desirable. From a practical perspective, measurement will be far superior if you actually measure the thing of interest rather than inferring it from group membership. Furthermore, in many cases, people base their beliefs about group differences on stereotypes with no empirical backing. But more importantly, it reduces the perpetuation of social inequality. It focuses attention on the skills, competencies, and capacities of the individual.