Comparison of subpopulations: do I need normalisation? I have a population of people. Each person has one of three characteristics, say X, Y or Z. I want to compare other characteristics of these people, using the characteristics X, Y and Z as a independent variable. 
My hypothesis would be something like: "The mean of the salary for people with characteristic X is bigger than the mean of the salary for people with characteristic Y and Z" or "The mean of the number of children of people with characteristic X equals the mean of the number of children of people with characteristic Y and Z". 
What I know beforehand is that these 3 subpopulations differ greatly in size and that they're independent. My question is: do I need to normalize/standardize/use z-score of the numerical variables, for instance the salary/number of children, so I can do a fair comparison? Would making a random sample of the entire population make anything easier?
 A: Some of this is easy: 


*

*If you want compare group X with groups Y and Z combined, the distinction between Y and Z is irrelevant. Conversely, if you want to compare three groups X, Y, Z, then that is a different problem. 

*Differences in group size are not in themselves an insuperable problem and it's part of the job of whatever test procedure you use to take that variation into account. Sampling your data to get (more nearly) equal sample sizes is not needed and just discards information. It remains true that a small sample size implies a small amount of information which may make differences harder to establish. 

*Similarly, there is no need for any kind of standardization. It's not clear what you have in mind here, but it's not needed. 
Your problems otherwise sound standard, inviting (e.g.) t-tests or ANOVA or Wilcoxon-Mann-Whitney or Kruskal-Wallis according to whether you are comparing two or three groups and what assumptions you can make in the light of the usual preliminary graphical analysis and summary statistics. Usually it is easiest to set up testing as testing a null hypothesis that groups do not differ. 
