From discussion with John:

With regard to standard error of the mean, it may be useful to clarify an underlying implicit assumption: homogeneity.

By taking an average of some data set, the underlying assumption is that the data all belong together (represent a single composition). That is, they are implicitly homogeneous.

And by taking the standard error of the mean, further replication reduces the perceived uncertainty. (Further replication continues to reduce random error, but not systematic error - which will eventually set a limit on such replication).

However, in a natural sample of wide compositional variation, the data are clearly heterogeneous (and should not be averaged together). In this case, the standard deviation of the entire data set is a proxy for the range of the data. And the standard deviation of an individual point represents the X-ray counting statistics that generated that point.

Cheers, Andrew