A rank correlation coefficient measures the degree of similarity between two rankings, and can be used to assess the significance of the relation between them.
For example, two common nonparametric methods of significance that use rank correlation are the Mann–Whitney U test and the Wilcoxon signed-rank test.
A rank correlation coefficient can measure that relationship, and the measure of significance of the rank correlation coefficient can show whether the measured relationship is small enough to likely be a coincidence.
If there is only one variable, the identity of a college football program, but it is subject to two different poll rankings (say, one by coaches and one by sportswriters), then the similarity of the two different polls' rankings can be measured with a rank correlation coefficient.
As another example, in a contingency table with low income, medium income, and high income in the row variable and educational level—no high school, high school, university—in the column variable),[1] a rank correlation measures the relationship between income and educational level.
The coefficient is inside the interval [−1, 1] and assumes the value: Following Diaconis (1988), a ranking can be seen as a permutation of a set of objects.
Thus we can look at observed rankings as data obtained when the sample space is (identified with) a symmetric group.
(rho) are particular cases of a general correlation coefficient.
objects, which are being considered in relation to two properties, represented by
is defined as Equivalently, if all coefficients are collected into matrices
In particular, the general correlation coefficient is the cosine of the angle between the matrices
is the number of concordant pairs minus the number of discordant pairs (see Kendall tau rank correlation coefficient).
be a uniformly distributed discrete random variables on
Using basic summation results from discrete mathematics, it is easy to see that for the uniformly distributed random variable,
Gene Glass (1965) noted that the rank-biserial can be derived from Spearman's
The rank-biserial correlation had been introduced nine years before by Edward Cureton (1956) as a measure of rank correlation when the ranks are in two groups.
Dave Kerby (2014) recommended the rank-biserial as the measure to introduce students to rank correlation, because the general logic can be explained at an introductory level.
The rank-biserial is the correlation used with the Mann–Whitney U test, a method commonly covered in introductory college courses on statistics.
Kerby showed that this rank correlation can be expressed in terms of two concepts: the percent of data that support a stated hypothesis, and the percent of data that do not support it.
The Kerby simple difference formula states that the rank correlation can be expressed as the difference between the proportion of favorable evidence (f) minus the proportion of unfavorable evidence (u).
To illustrate the computation, suppose a coach trains long-distance runners for one month using two methods.
The stated hypothesis is that method A produces faster runners.
The race to assess the results finds that the runners from Group A do indeed run faster, with the following ranks: 1, 2, 3, 4, and 6.
For example, the fastest runner in the study is a member of four pairs: (1,5), (1,7), (1,8), and (1,9).
All four of these pairs support the hypothesis, because in each pair the runner from Group A is faster than the runner from Group B.
The only pair that does not support the hypothesis are the two runners with ranks 5 and 6, because in this pair, the runner from Group B had the faster time.
By the Kerby simple difference formula, 95% of the data support the hypothesis (19 of 20 pairs), and 5% do not support (1 of 20 pairs), so the rank correlation is r = .95 − .05 = .90.
The maximum value for the correlation is r = 1, which means that 100% of the pairs favor the hypothesis.
A correlation of r = 0 indicates that half the pairs favor the hypothesis and half do not; in other words, the sample groups do not differ in ranks, so there is no evidence that they come from two different populations.
An effect size of r = 0 can be said to describe no relationship between group membership and the members' ranks.