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There's a nationwide dataset of the average scores on test, by school (each school has one entry in the dataset). There are significant differences between the means and SDs among the states. A subset is made for Florida. When calculating the standard error of the mean for Florida, do I use the nationwide SD (population SD), or Florida's SD? There are 110 values for FL, and the SD is 25% larger than with the nationwide dataset.

Under the CLT, the sample mean and SD is supposed to approach the population mean and SD upon repeated sampling, but that won't happen when FL is different. So the "population," from FL's perspective, is FL, and with repeated sampling in FL (if possible), the mean and SD would approach a "true" FL.

I'm thinking the issue is with non-random sampling, except studies that compare traits among males and females consider the samples to be random - and FL is a trait, like M/F is.

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There is no reason to believe that the variability of test scores will be the same from one state to another. If fact, you say that there are 'significant differences' among states.

So if you are doing inference for Florida (making confidence intervals or doing t tests), then you should use the data from Florida (sample size, mean, standard deviation, and standard error).

Similarly, if you are using a 2-sample procedure to compare Florida with another state, then you should use the data for the two states.

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  • $\begingroup$ That's what I was thinking, however, this seems to contradict the central limit theorem, unless the "population" isn't the nationwide data. $\endgroup$
    – user179810
    Jun 30 '20 at 22:04
  • $\begingroup$ @MikeSmith How does it contradict the central limit theorem? I also would argue that the population is just Florida, not the whole US, so the variance of the rest of the US is not so important to what you're doing. $\endgroup$
    – Dave
    Jun 30 '20 at 22:14
  • $\begingroup$ The population for Florida is schools in Florida. If you have all the schools in FL then you can describe the scores in that state, but there is no point in making CIs or doing tests. If you have a random sample of schools from FL, then you can make inferences to the state population. If you have some schools from FL getting into your sample in some haphazard, but not carefully randomized way, then you can pretend it's "like" a random sample and do inference, or admit it's a mess and describe what you have, admitting your have partial nonrandom data. $\endgroup$
    – BruceET
    Jun 30 '20 at 23:08
  • $\begingroup$ @BruceET - When is the decision made to disregard national data, based on an attribute? Is there some accepted criteria? If you do a national study on a new medication, when is the decision made to disregard the female set of the data when looking at males? The state is just an attribute of national data. $\endgroup$
    – user179810
    Jul 1 '20 at 22:29
  • $\begingroup$ Well, there is not nearly enough data on differences btw F and M responses to drugs because of under-representation of women in clinical trials. But indisputably differences do exist. // Less controversially, one expects differences in distributions of test scores btw states such as $AR, LA, MS, AL$ and New England. Those differences include variability, and it would be counterproductive to ignore them. $\endgroup$
    – BruceET
    Jul 1 '20 at 22:36

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