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Wikipedia has 3 definitions of standard error:

The first is ${\sigma }_{\bar {x}}\ ={\frac {\sigma }{\sqrt {n}}}$ where $\sigma$ is known standard deviation of the population,

the second is ${\sigma }_{\bar {x}}\ \approx {\frac {s}{\sqrt {n}}}$ where $s$ is the standard deviation of the sample, and

the third which is least clear for me is the following ${\text{s}}_{\bar {x}}\ ={\frac {s}{\sqrt {n}}}$

The latter is accompanied with words

In those contexts where standard error of the mean is defined not as the standard deviation of the sample mean, but as its estimate, this is the estimate typically given as its value.

So I'm trying to understand what ${\text{s}}_{\bar {x}}$ is? Is this sample standard error that is equal to sample standard deviation or this is the estimate of one of them? As for me I don't see any difference with the second definition

James Flash
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    It may help to understand that none of these is a definition: they are formulas, each applicable in its own special cases. For a *characterization* of what standard errors are generally, see https://stats.stackexchange.com/questions/18603. As far as your specific question goes, have you noticed that the first formula is in terms of a *population* property and the second is in terms of a *sample* property? – whuber Apr 29 '20 at 12:21
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    Yes, I noticed that. Thanks for the useful link – James Flash Apr 29 '20 at 12:54

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