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We use the SE in order to tell, how far away are we to the true mean.

Why does this formula use the standard deviation of a sample to compute the mean of the sample means ? (with other words, true population mean)

Wouldn't it somehow be more intuitive it the formula for computing the mean of the sample means would have the mean of the actual sample inside it ?

$$SE = \frac{s}{\sqrt{n}}$$

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    $\begingroup$ I don't follow your question. The SE is the standard error, what does this have to do with the "mean of means"? (I guess you meant the expected value of the mean) $\endgroup$
    – Firebug
    Mar 22 '17 at 0:11
  • $\begingroup$ @Firebug I just edited it $\endgroup$
    – Oleg
    Mar 22 '17 at 0:22
  • $\begingroup$ Central limit theorem? $\endgroup$
    – SmallChess
    Mar 22 '17 at 1:24
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    $\begingroup$ The standard error doesn't tell you how far you are from the true mean - you can't use it to work out where the true mean is, relative to your sample mean - but rather it measures the uncertainty in your estimate in the mean. The standard error gives you some idea of how far out your sample mean might be from the true mean, but not eg in which direction it is wrong, so you can't use it to improve your estimate. $\endgroup$
    – Silverfish
    Mar 22 '17 at 11:42
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You appear to be operating under some misunderstanding - your question seems to rely on a false premise.

If you're trying to give a point estimate of the mean of the distribution of sample means, that would indeed (often) be the sample mean.

The standard error of the mean is used to give some idea of the associated uncertainty in the location of the mean of that distribution. The standard error of the mean is the standard deviation of the distribution of sample means. It tells you something about how far away from the thing you're trying to estimate your sample mean will typically be.

e.g. if you wanted to give a confidence interval for that population mean of the distribution of sample means, it would typically be based on that sample mean $\pm$ a margin of error (interval halfwidth) that is a multiple of that standard error.

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