probability of rank of a number Suppose I have 10 sample means. I want to find the probability of rank of the population means using sample means. Therefore, I want to perform two experiments.  
First experiment: I pick one of the sample means and compute the probability of being rank from 1 to 10 of the population mean correspond to the sample mean, i.e, what is the probability that this population mean's rank is 1 or 2 or 3....or 10. Note that I used only one sample mean here.
Second experiment:I sort all 10 sample means in the descending order so that the maximum number is in the first and the minimum number is in the last. Then compute the probability of the corresponding population means to be rank 1 if the sample mean's rank is 1, to be rank 2 if the sample mean's rank is 2, and so on. Here I used all sample means.
My question is, are these two experiments equivalent? Note that I assume that I have the tools to compute the probability of the rank. I am are trying to find the ranking probabilities of the population means using sample means. In practice, we estimate population mean by sample mean but we never know the actual population mean.
Thank you
 A: Comment on  related issues.
In a balanced one-factor analysis of variance, one has the model
$$Y_{ij} = \mu_i + e_{ij},$$
for groups (independent samples) $i = 1, \dots, g,$ 
replications $j = 1, \dots, n$ observation within each group,
and $e_{ij} \stackrel{indep}{\sim} Norm(0, \sigma).$
Notice that all treatment groups are assumed to produce normal
data, and with the same standard deviation $\sigma.$ 'Balanced'
means that the number $n$ of replications in each group is the same.
An F-test is used to test the null hypothesis $H_0$ that
all $n$ of the $\mu_i$ are equal against the alternative
that there is at least one difference among them.
If $H_0$ is rejected, then one supposes there is a pattern
of differences among the $\mu_i$ and various procedures
are used to try to  discern that pattern. Names of some of these
'multiple comparison' procedures are Fisher's LSD, Tukey's HSD,
Student-Newman-Kuels, Bonferroni, and so on. These procedures
are based fundamentally on looking at sample means $\bar X_i.$ 
Generally speaking, it is easier to distinguish between
$\mu_i$ and $\mu_{i^\prime}$ if the difference is several
times the standard error $\sigma/\sqrt{n}$, and almost impossible
to make a distinction if the difference is a fraction of the standard
error. In practice, it is not usually possible to put, say 6,
group population means exactly their proper order, often
because people try to save money on the experiment and $n$
is too small.
I think you need to be able to distinguish such differences between
population means
order to make any real sense of 'ranking.'
Your question is a bit vague. I mention this relevant ANOVA
model for two reasons. (1) Browsing explanations of one-way ANOVA
designs and multiple comparison procedures may give you ideas
how to formulate your question more precisely. (2) If your
question is closely related, some part of your problem may
already have been solved.
If you want to give me examples of differences between
neighboring means, sample sizes, and the standard deviation, I may
be able to give you a more specific answer to your question. 
