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I know that to construct confidence intervals, standard errors must be calculated, a process which in turn makes use of the CLT (but I am not clear how). I also understand that, very generally, the CLT states that the resulting distribution of a large number of sample means will be approximately normal, regardless of the shape of the underlying distribution from which the sample means originated from.

Naturally if we could take an infinite number of samples means from a population and generate the sampling distribution of these sample mean, then calculating the SE of my sample statistic would be straightforward (i.e. the SD of the sampling distribution). However, generating the sampling distribution of the mean is not possible with only one sample, so we run into issues when generating the standard error.

So, my question is: how exactly does the process of constructing confidence intervals make use of the CLT when one only has one sample to work with?

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  • $\begingroup$ Generally, creating a CI has nothing to do with the CLT. The CLT is sometimes used justify the computation of approximate confidence intervals for statistics that are linear combinations of data, such as sums and means. For instance, most of the replies to a question about what a CI is make no reference to the CLT: see stats.stackexchange.com/questions/2272. $\endgroup$
    – whuber ♦
    Commented Dec 1, 2015 at 22:46
  • $\begingroup$ Thanks for your comment. My presumption was that the CLT came in handy when calculating the SE of the sample statistic, yet I'm unclear on how. My thoughts: how else, then, would estimate the SE of a sampling distribution of say, sample means, when one only has one sample (i.e. one sample mean)? If one had many samples, which in turn would give us many sample means, we would then be able to construct a sampling distribution and finally calculate the SE. What am I missing? $\endgroup$
    – StatsScared
    Commented Dec 2, 2015 at 17:23
  • $\begingroup$ The concept of confidence interval differs from that of standard error. In many applications they are related, but that doesn't change this basic conceptual fact; and the applications in which they are unrelated show why you cannot generally expect to compute an SE from a CI. See, for instance, my illustration at stats.stackexchange.com/a/426994/919. $\endgroup$
    – whuber ♦
    Commented Sep 12, 2019 at 15:01

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