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If one calculates a standard error of the mean from the sample mean and sample standard deviation of n observations drawn from a normally distributed population, does it make sense to call that SEM an estimate of some "true" SEM?

Thus, if I increase n, is it true that I not only reduce the size of my SEM, but also increase the accuracy of my estimate (because the sample standard deviation is better estimate)?

Ian Sudbery
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2 Answers2

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The standard error of the mean is an estimate. Specifically, it is an estimate of the standard deviation of the sample mean, seen as a random variable.

And of course, as you increase your sample size $n$, your sample mean will become less and less variable - which is exactly reflected in the SEM getting smaller.

Stephan Kolassa
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  • Yes, I understand that as you increase your n your sample mean becomes less and less variable - with more data your estimate of the mean becomes better. But does the confidence of your estimate of your confidence in the mean also increase. That is, can you calcualte a standard error of the standard error of the mean, and would this get smaller with increasing n. – Ian Sudbery Nov 02 '21 at 15:11
  • Well, the parameter you are estimating is not the SEM - the parameter is the standard deviation of the sample mean. This you estimate using the SEM. And yes, of course we can also discuss the standard deviation of this estimate, which would be the standard error of the standard deviation of the mean. I believe we have at least one question on this before, but a quick search in the [tag:standard-error] tag didn't turn up anything. Maybe with a little more digging... – Stephan Kolassa Nov 02 '21 at 15:27
  • Perhaps a better wording would be "Does it make sense to describe the SEM as an estimator" – Ian Sudbery Nov 02 '21 at 15:35
  • You need to supply an authoritative reference to back up this answer, because it appears to contradict standard definitions such as (*e.g.*) the one in Snedecor & Cochran, who state "The standard deviation of $\bar X,$ $\sigma/\sqrt{n},$ is often called, alternatively, the *standard error* of $\bar X.$" (*Statistical Methods* 8th Ed. Section 4.4.) Kendall, Stuart, & Ord similarly define the standard error as "the standard deviation of the sampling distribution of the statistic" (*Advanced Theory of Statistics* 5th Ed. section 9.31) and later refer to an *estimated* standard error. – whuber Nov 02 '21 at 15:53
  • @whuber: interesting, thank you. Those quotes, in turn, contradict the usage of "SEM" I have invariably seen - where it is not a parameter, but an estimate. I also can't quite reconcile these quotes with [your comment above](https://stats.stackexchange.com/questions/550658/does-it-make-sense-to-describe-the-sem-as-an-estimate/550659?noredirect=1#comment1011949_550658), where it seems to me you are indeed saying the SEM is an estimate. Can you clarify, or better yet, post an answer? – Stephan Kolassa Nov 02 '21 at 16:02
  • I am saying just the opposite: these textbooks insist the SEM is not an estimate and they help make that distinction by referring to an "estimated" SEM. I have already answered this question in my post at https://stats.stackexchange.com/a/18609/919. – whuber Nov 02 '21 at 16:34
  • Is this just a matter of nomenclature - whether $\widehat{\sigma}/\sqrt{n}$ is the SEM which is an estimate of the standard deviation of the sample mean, or whether $\sigma/\sqrt{n}$ is the SEM, but we usually calcualte $\widehat{\sigma}/\sqrt{n}$ which is an estimate of it? – Ian Sudbery Nov 02 '21 at 17:13
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Consider: $s_x$ (or $\widehat{\sigma}_{x}$, if you prefer) estimates $\sigma$ (with bias), and in the same fashion $s_{\overline{x}}$ (or $\widehat{\sigma}_{\overline{x}}$) estimates $\sigma_{\overline{x}}$. Right? $\frac{s_{\overline{x}}}{\sqrt{n}}$ estimates $\frac{\sigma_{\overline{x}}}{\sqrt{n}}$?

Alexis
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  • That was my thinking, yes. So you agree that it makes sense to say increasing n not just increase the accuracy of our esimate of the mean, but also increases how accurately we can measure the accuracy. – Ian Sudbery Nov 02 '21 at 15:18
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    @IanSudbery Yes, I would agree with that. – Alexis Nov 02 '21 at 15:20