The important point, I think, is that SDs and SEs measure two different things. The standard deviation is a measure of variation within a sample, and is an estimate of the amount of variation in the population from which the sample was drawn. The standard error is a measure of uncertainty (sampling variation) in some statistic, such as the sample mean, and is used to derive confidence intervals and such. As sample size increases, the standard deviation converges (almost) on its "true" value, while the standard error converges on zero.
of course, to have a real confusing mess ... we could have an SE of an SD, right?
take hundreds of samples of size n ... from a population ... find the estimate of the population sd in each case (s) ... and see what the standard deviation would be for that set of s estimates ...
for example ... what if i took 500 samples of n=25 from a population where mu = 100 but, we don't know the population sd? (well, i do but, i am not revealing it yet) ...
MTB > dotp c26
Dotplot: estsd
Each dot represents up to 2 points . : . :. : ::.:::: . : ::::::::::.: :.. :::::::::::::.::::.. :::::::::::::::::::::: . . ..:::::::::::::::::::::::: : . ...:::::::::::::::::::::::::::::::.:... . ---+---------+---------+---------+---------+---------+---estsd 4.5 6.0 7.5 9.0 10.5 12.0
MTB > desc c26
Descriptive Statistics: estsd
Variable N Mean Median TrMean StDev estsd 500 7.9853 7.9136 7.9805 1.1310
we see that across 500 samples ... the "typical" estimate of the sd is 7.98 ... or nearly 8 ... so, as our best single guess ... we might "infer" that the population sigma is about 8 ... and, in fact it was ... BUT of course, this is hindsight having 500 sd values from 500 samples ... usually, all you would have is ONE sd value from your ONE sample
but, note that in the dotplot ... you get varying sample estimates for sigma ... they range from around 5 to 11 ... and, the standard deviation of this SAMPLING DISTRIBUTION OF STANDARD DEVIATIONS is about 1.13 ...
the value of 1.13 here is our SE .... or standard error of the standard deviation ...
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