# Statistically Ranking Data

We are sampling the rates of $$n$$ elevators rising up floors in order to see which elevators are the fastest, and which are the slowest. Say we have $$k$$ time samples from each elevator, $$S_1, S_2, ... , S_n$$, where $$S_j$$ is a data set of samples for the $$j$$th elevator $$S_j = (x_1,x_2,...,x_k)$$

Now we know how to test say if $$\mu_1 < \mu_2$$ using a t-test, however we are not sure how to come up with the most likely ordering of means given the data. From there can we calculate our confidence in the estimate?

Any help would be appreciated

If you are looking for the most likely ordering of means then you are going to need to calculate probabilities of orderings of the true means, and those are "posterior probabilities" in the Bayesian sense. That means that you would need to do this analysis within the Bayesian framework, where you set a prior distribution on the mean parameters and then derive the corresponding posterior distribution given the observed data. Once you have your posterior distribution you will be able to calculate the posterior probabilities of each of the $$n!$$ possible orders for the mean (so long as this is not so large as to be computationally infeasible). You will then be able to identify the most probably ordering of the means.