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In a group of 1000 people, where each of them is either A, B, C, D or E (no one can be more than one thing). If I have exactly 200 people with each trait (20% per trait), when I take a random sample of 5 people, 1000 times with replacement (a bootstrap-like approach), I would imagine the percentage of these samples where at least one of the traits is not represented, is rather high, as it would be quite fortuitous to have randomly selected one of each. But what if I took a sample of 10 people? or 100?

Even when the distribution of traits is not homogeneous (e.g. A = 40%, B = 30%, C = 10%, D = 10%, E = 10%) or the traits distribution of the whole population is not available (eg. poll), how can we test what sample size is significantly representative?

My broader question is how big should my sample be to insure it has 95% chance of exhibiting the same proportions of traits in the group?

I have the impression that there is a "statistical/mathematical metric" between the number of traits, the size of our target population and how increasing sample sizes become more representative, but I'm missing the keywords to find it...

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  • $\begingroup$ I don`t understand what statistical significance or testing has to do with this question, what would be your $H_0$?. The sample size you need depends on the confidence level you want. And as your population is distributed multinomially you can apply the CLT to obtain confidence intervals. $\endgroup$
    – Sebastian
    Commented Aug 9, 2018 at 11:26
  • $\begingroup$ Could you explain what you mean by "representative of the full diversity"? Does that mean the sample has a high chance of containing at least one of each group? Or would it mean it has a high chance of exhibiting nearly the same proportions of all groups as exist in the population? $\endgroup$
    – whuber ♦
    Commented Aug 9, 2018 at 14:42
  • $\begingroup$ Whuber: It would be your second suggestion. Representative, would in this context mean that a sample of group e.g. A = 40%, B = 30%, C = 10%, D = 10%, E = 10%, would have a high chance of exhibiting nearly the same proportions. I edited the question to make it clearer. $\endgroup$
    – Sergio Henriques
    Commented Aug 10, 2018 at 15:09

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It's not 100% clear exactly what you're looking for, but this "representativeness" is already baked into the process of random sampling. If you take a population and sample randomly, you can compute class proportions from your sample. You can also compute confidence intervals on those class proportions. 95 times out of 100, your true class proportions will fall within your 95% confidence interval of your sample estimate of the class proportions.

What additional sampling gets you is a narrower confidence interval. With only 10 samples, for example, you might be 95% confident that the true proportion of class A is between 35% and 85%. With 1 million samples, you might be 95% confident that the true poportion of class A lies between 59% and 61%. Both samples are representative of your population, but the larger sample characterizes it better with a narrower confidence interval.

In order to completely specify your problem, you need to decide what width of confidence interval you're comfortable with. Do you want to be able to confidently estimate class proportions within 10%? Within 1%? Within 0.1%? Once you know that, you can compute the sample size required to get a confidence interval of a specified width. It's highly unlikely that your sample will exhibit the exact same proportions as your population, all you can do is narrow your confidence interval around your best guess (which is your sample proportions).

See Margin of Error.

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  • $\begingroup$ I guess that in an election scenario, the CI we are comfortable with would the ones which still predict the correct outcome of the election. In my example, it would be the sample size that has a 95% probability of detecting A>B (when A=40%, B=30%), but when we don't know the truth apriori: can we increase the size of random samples incrementally, measure the proportions of the traits for ever larger sample sizes, so that after a certain sample size threshold we are 95% certain we have detected the true proportions of the whole group? What would this test be called? $\endgroup$
    – Sergio Henriques
    Commented Aug 16, 2018 at 15:31
  • $\begingroup$ @SergioHenriques You could do something like that with a pilot study - collect a small sample to get a very rough idea of your study population, and then calculate the sample size required to be able to reliably detect differences on the order of what you saw in the pilot. $\endgroup$
    – Nuclear Hoagie
    Commented Aug 16, 2018 at 16:17
  • $\begingroup$ I have rephrased my question and placed it here: stats.stackexchange.com/questions/362726/… $\endgroup$
    – Sergio Henriques
    Commented Aug 17, 2018 at 19:29

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