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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?

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Some of this is easy:

  1. 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.

  2. 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.

  3. 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.

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  • $\begingroup$ Thank you for the quick answer ! What I meant for the standardization is that, for me, it sounds kind of weird compare a group of 500 individuals with another of 10000. Even if they come from the same population. Am I trying to find problems where there isn't one ? $\endgroup$ Commented Mar 24, 2014 at 17:03
  • $\begingroup$ It's not weird at all. You can compare floods (rare) with other events, cases of unusual diseases with the whole population, Nobel Prize winners with the rest of us. Statistically the small print includes the sample sizes in both cases. $\endgroup$
    – Nick Cox
    Commented Mar 24, 2014 at 17:08
  • $\begingroup$ Just one more question: if now I have two populations, say from different cities and I want to compare the mean of number of children X people from city A have with X people from city B. Then I would need standardisation ? As the means come from different populations ? $\endgroup$ Commented Mar 24, 2014 at 17:25
  • $\begingroup$ No, it's the same question! It's not clear what you mean by standardisation, but whatever it is, consider that tests provide it for you. Also, means coming from different populations is not a problem in any sense, but part of what you test for. $\endgroup$
    – Nick Cox
    Commented Mar 24, 2014 at 17:29
  • $\begingroup$ On a previous research, with similar data, a professor recommended me to do this dataminingblog.com/standardization-vs-normalization as the populations would differ in size. As I did for different sizes of populations, I thought that I would need for different sizes of subpopulations too. $\endgroup$ Commented Mar 24, 2014 at 17:42

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