A good example of this is the genetic difference within Africa, which are larger, than the genetic differences between Africa and Europe.
From what I understand, on a mathematical level it means that if you take the averages of two groups and compare them to each other, the difference will be less, than the difference between the two extremes within a given group. This generally sounds logical. Since most people on earth are fairly similar, it makes sense that the average African is more similar to the average European, than the two most distant Africans compared to each other.
However, does this also mean, that two randomly selected Africans have a higher chance of being more different than a randomly selected European from a randomly selected African?
How is this law of variation between and within groups to be understood exactly? Does it still apply, when comparing the likelyhood of differnce between individuals, as opposed to the avarages or are those the same thing?
If we imagine a very simple example: two groups with 10 individuals, each assigned a value of 1 to 10.
First group has the values: 1,2,3,4,5,6,7,8,9,10
Second group has the values: 10,20,30,40,50,60,70,80,90,100
If you compare the averages you will get: 5 and 50, a difference of 45.
If you compare the outliers within the second group you will get 10 and 100, a difference of 100.
If this is the case, than what exactly does it mean, that the differences between Africans are greater than between Africans and Europeans? Does it simply mean, that in Africa, there is a higher genetic variance? Does it mean that comparing any group to any other group is inherently useless, as the difference within the group is almost always going to be more significant, than the variance between the groups?
Best Answer
From a purely statistical standpoint, there is no reason at all why the variance within a group should be larger or smaller than between groups.
Whether the within-group or between-group variance is larger is something that is to be observed and tested, not a mathematical fact. In fact comparing within-group and between-group variance to each other is part of the so called Analysis of Variance (ANOVA). Loosely speaking, when an ANOVA is used in inferential statistics, the within-group and between-group variances are compared to each other (into an F-Score), and then it is tested which one is greater (F-Test). If the within-group variance is larger than the between-group variance, this simply means, there is no evidence for any difference between the groups.
This also is the point of the genetic example you gave. There just is no evidence that people from Africa are genetically different from Europeans.
On the other hand, for placebo controlled pharmaceutical trials it should be the opposite. People who receive the medicine should be more similar to each other (in terms of recovery ratio) and the people who receive a placebo should be more similar to each other (again in terms of recovery ratio). Most of the differences should be between participants who receive the true treatment and those who only get a placebo (between-group variance larger than within-group variance).
EDIT:
Another example, which is closely related to the genetic example you gave, can be used to show that this kind of within-group / between-group variance observation is not about the kind of groups which are defined but also depend as much on the type of measurement.
While genetic variation is larger within-group than between groups, IQ testing regularly finds scores for Africa to be about two SDs below the mean for Europeans and Americans. Thus, for IQ the within-group variance in Africa and Europe is actually smaller than the between-group variance. This shows that for IQ there is a difference between these groups, and the causes are highly debated (tests might not be culture-fair, malnutrition, and health-related causes are some of the debated explanations).