Confidence when "sorting" multiple samples by mean? In a one-way ANOVA, we test if the mean is different amongst multiple groups.  The results of this might not be super useful when testing a large number of groups, because we only get a confidence related to "someone being out of place" rather than a confidence for "which is biggest", etc.  OTOH, it seems that I can sort my sample means from smallest to largest, and that this provides me with some evidence of which population has the smallest mean, which has the largest mean, which population is in the middle, etc.  However, ANOVA doesn't allow me to quantify evidence for my sorting procedure.
One option here is to compute an X% confidence interval for each mean, use a multiple-comparisons correction (how many comparisons are there?), and make statements about my sorting based on this.  e.g. if the largest sample mean has a confidence interval that doesn't overlap any other CIs, then I have good evidence that this is, in fact, the population with the largest mean.  This seems like it will be extremely conservative.
How can I find the e.g. largest, smallest, median population mean in a group of $M$ populations?

This is related to what the Newman-Keuls method does: "Prior to any mean comparison, all sample means are rank-ordered in ascending or descending order." It starts by comparing the lowest and highest means based on the Studentized range.
If they differ significantly, then it proceeds with sets of further comparisons between successively more closely spaced mean values--lowest versus next-to-highest, highest versus next-to-lowest, etc. If two mean values at the extremes of a range aren't found to differ significantly this way, then "all the null hypotheses within that range would be retained and no further comparisons within smaller ranges are necessary."
So this way you would get the set of other means that can't be distinguished significantly from the highest or the lowest or the median of your ranking of means. It doesn't provide confidence intervals or correction for multiple comparisons other than the use of the Studentized range.
