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I understand the rationale that underpins representative sampling (i.e. to avoid bias so that the sample 'broadly' represents the target population).

Suppose the size of the target population is 600 and the population comprises of four types of units as follows:

  • 100 white units
  • 250 black units
  • 50 green units
  • 200 yellow units

Questions

  1. What would be a representative sample (overall and subgroup) in this case?
  2. Is the sample in (1) appropriate for only descriptive analysis or inferential analysis or both?
  3. If you have a representative sample, is a priori power analysis needed?
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  • $\begingroup$ At least part of the answer will depend on whether you wish only to identify the relative proportions, or if you wish to ask questions/take measures of the individual units and try to determine if the results are significantly different based on unit colour. For the latter, multidimensional cell sizes can influence the minimum sample numbers. $\endgroup$ – Michelle Jan 24 '12 at 5:54
  • $\begingroup$ Be careful with a term "representative sample". Kruskal and Mosteller (1979a, 1979b and 1979c) give nine different meanings of the term. See stats.stackexchange.com/a/43044/3330 for the references. $\endgroup$ – djhurio Nov 8 '12 at 6:06
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One. Any sample for which every population member has a known, non-zero (but not necessarily equal) probability of being selected. So long as you know the probability, you can use post-stratification weighting to help your inference to population.

Two. Both. Arguably, descriptive analysis is a special case of inferential analysis anyway.

An interesting additional distinction is between inference about a finite population (your 600 units) and a more metaphysical infinite size "data generating process" of which even the 600 units in the population are just a sample. In either case, you will be able to analyse it with the sample generated in response to question 1, but the techniques will differ slightly, particularly with regard to what is called the "finite population correction".

Three. No. But any inference you do should certainly do power analysis at some stage. It could be useful to do power analysis before determining the sample size, but this is difficult if you don't know the probability distribution of the variables of interest. Often, sample sizes in my experience are chosen with the aim of a 95% confidence interval for a particular parameter estimated from the resulting sample being within a particular range set by the key stakeholders. This is a form of power analysis and is very helpful to do before setting the sample size. But not essential if you are prepared to just live with what you get.

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Adding to @PeterEllis excellent reply, there are many types of sample that are representative, and they each have strengths and weaknesses. However, a good start is simple random sampling without replacement. This would entail randomly choosing numbers between 1 and 600 and using the corresponding units in your sample.

However, other methods might be better, depending on what you are trying to do.

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