Standard Error vs. Standard Deviation of Sample Mean I'm having difficulty figuring out the difference between (typically referred to as S.E):
A. The Standard Error
And
B. The Standard Deviation of the Sample Mean (typically referred to as s)
Are they the same thing?
OR


*

*S.E is the standard deviation of the the mean of the sampling distribution similar formula as the standard deviation except you use n-1 instead of n in the denominator

*s is the standard deviation of a single random sample -- same formula as the standard deviation 

 A: The standard deviation of the mean is usually unknown. We would write it as $$  \sigma_{\bar x } ={\sigma \over \sqrt n} $$
The standard error of the mean is an estimate of the standard deviation of the mean. $$ \hat \sigma_{\bar x}  = {s \over \sqrt n}.  $$ 
A: A standard error can be computed for almost any parameter you compute from data, not just the mean. The phrase "the standard error" is therefore ambiguous. I assume you are asking about the standard error of the mean.
Here are the key differences between the standard deviation (SD) and the standard error of the mean (SEM)


*

*The SD quantifies scatter — how much the values vary from one
another.

*The SEM quantifies how precisely you have determined the true mean of the
population. It takes into account both the value of the SD and the
sample size.

*Both SD and SEM are in the same units -- the units of the data.

*The SEM, by definition, is always smaller than the SD.

*The SEM gets smaller as your samples get larger. This makes sense,
because the mean of a large sample is likely to be closer to the true
population mean than is the mean of a small sample. With a huge
sample, you'll know the value of the mean with a lot of precision
even if the data are very scattered.

*The SD does not change predictably as you acquire more data. The SD
you compute from a sample is the best possible estimate of the SD of
the overall population. As you collect more data, you'll assess the
SD of the population with more precision. But you can't predict
whether the SD from a larger sample will be bigger or smaller than
the SD from a small sample. (This is not strictly true. It is the
variance -- the SD squared -- that doesn't change predictably, but
the change in SD is trivial and much much smaller than the change in
the SEM.)

*The SEM is hard to define conceptually. The only real "purpose" of an SEM is as an "ingredient" to compute the confidence interval of the mean.

*The SEM is computed from the SD and sample size (n) as $$SEM ={SD \over \sqrt n}.  $$


(From the GraphPad statistics guide that I wrote.)
A: The official term for the dispersion measure (of a distribution, of a sample etc) is "standard deviation" - the square root of the variance. 
The tern "standard error" is more often used in the context of a regression model, and you can find it as "the standard error of regression". It is the square root of the sum of squared residuals from the regression - divided sometimes by sample size $n$ (and then it is the maximum likelihood estimator of the standard deviation of the error term), or by $n-k$ ($k$ being the number of regressors), and then it is the ordinary least squares (OLS) estimator of the standard deviation of the error term.
So you see that they are closely related, but not the same thing.
A: For normally distributed data, the SE = s, as the mean is an explicit parameter of the normal distribution. In general, standard error arises in Likelihood theory, where you are forming inferences from a likelihood function as opposed to the true sampling distribution. For example, if you are modeling some data as iid Exponential then you would form the likelihood function for your data $L(X|\lambda)= \prod L_{exp}(x_i|\lambda)$, with unknown $\lambda$ and then optimize L(X|$\lambda$) for $\lambda^*$ (i.e, maximum likelihood estimator). The standard error is defined as the curvature of the quadratic approximation to log(L(X|$\lambda^*$))at the MLE, which will equal the standard deviation for normal data. the only difficulty is that for non-normal data, you will need to do a second step to transform the actual parameters of your distribution (e.g., $\lambda$) into an estimate of the sample mean. Here, you would map $\frac{1}{\lambda} \rightarrow\mu$, so the likelihood of the latter equals that of the former, then take the log of that likelihood and get a standard error of that transformed likelihood function. Sorry for the long answer, but its not super clear cut in all cases. Sometime, its even used loosely, so yo need to read the documentation to really know.
