Does estimated standard error of the mean says something about the population mean?

Question

The "normal" standard error of the mean (SEM) is the population standard deviation divided by the square root of the sample size. Wikipedia states that the SEM is an estimate of how far the sample mean is likely to be from the population mean.

In practice you don't know about the population standard deviation and use the sample standard deviation instead. The sample standard deviation, however, is only an estimate of the population standard deviation with some unknown error... Despite this unknown error, does the estimated SEM still tell how for the sample mean is likely to be from the population mean?

Richter65 · Answer

The answer is yes: both the actual standard deviation of the mean $sigma_{bar{x}}$ (calculated from the actual population standard deviation $sigma$) and the estimated standard deviation of the mean $s_{bar{x}}$ (calculated from the estimated population standard deviation $s$) tell you, in a sense, how far the sample mean is likely to be from the estimated mean.

What's the difference, then? It turns out that the estimated standard deviation $s_{bar{x}}$ is a biased estimator, and tends to underestimate the actual standard deviation.

What's the difference, then?  It turns out that the estimated standard deviation $s_{bar{x}}$ is a biased estimator, and tends to underestimate the actual standard deviation.

Does estimated standard error of the mean says something about the population mean?

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