Testing similarity of rank orders Are there any statistical tests for the similarity of rank orderings? E.g. suppose 4 students a, b, c and d take a test 5 times over a period of 5 months, and each month we rank the students' scores:
a: 1, 3, 2, 1, 3
b: 2, 4, 3, 2, 2
c: 3, 2, 4, 4, 4
d: 4, 1, 1, 3, 1
Is there a test that can test if each month's rankings are significantly similar to each other? So the null hypothesis is that each month's rankings are different to each other...I guess the opposite of a Friedman test?
 A: The Friedman test seems appropriate for testing your null hypothesis here.
It's equivalent to a significance test of Kendall's W, which is sometimes used as an index of agreement between rankings that ranges between $0$ (no agreement, essentially random rankings) and $1$ (full agreement, all ranks identical).
In your case you have $m=5$ tests and $n=4$ students, and the Friedman $\chi^2 = 3.96, \space p = .266$. This result can be transformed to find Kendall's W by $\chi^2/(m(n-1))$. In your data, $W = .264$.
You can equivalently think about this in terms of calculating all the Spearman rank correlations between every possible pair of monthly rankings and finding the average. The average Spearman correlation is $(mW-1)/(m-1)$, which for your data yields $.08$. Not a very strong correlation.
With a bit of algebra, you can actually find the average Spearman correlation between the rankings directly from the Friedman $\chi^2$ result
$$\frac{\chi^2-n+1}{(m-1)(n-1)}$$
Note that a similar test using the average Kendall's Tau correlation is provided by Ehrenberg (1952).

References
Ehrenberg, A. S. C. (1952). On sampling from a population of rankers. Biometrika, 39(1/2), 82-87.
Kendall, M. G. and J. D. Gibbons. 1990. Rank Correlation Methods. 5th ed. London: Griffin.
