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.
42.
In the same way if " y " always " decreases " when " x " " increases ", the rank correlation coefficients will be " 1, while the Pearson product-moment correlation coefficient may or may not be close to " 1, depending on how close the points are to a straight line.
43.
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 ), a rank correlation measures the relationship between income and educational level.
44.
For example, a test based on Spearman's rank correlation coefficient would be called non-parametric since the statistic is computed from the rank-order of the data disregarding their actual values ( and thus regardless of the distribution they were sampled from ), whereas those based on the Pearson product-moment correlation coefficient are parametric tests since it is computed directly from the data values and thus estimates the parameter known as the population correlation.
45.
Thus the society's preferences show cycling : A is preferred over B which is preferred over C which is preferred over A . A paradoxical feature of relations between the voters preferences described above is that although the majority of voters agree that A is preferable to B, B to C, and C to A, all three coefficients of rank correlations between the voters preferences are negative ( namely,-. 5 ), as calculated with Spearman's rank correlation coefficient formula designed by Charles Spearman much later.
46.
Thus the society's preferences show cycling : A is preferred over B which is preferred over C which is preferred over A . A paradoxical feature of relations between the voters preferences described above is that although the majority of voters agree that A is preferable to B, B to C, and C to A, all three coefficients of rank correlations between the voters preferences are negative ( namely,-. 5 ), as calculated with Spearman's rank correlation coefficient formula designed by Charles Spearman much later.
47.
Aside from the existence of the reference values, the data aren't expected to follow any given distribution, or necessarily match the variance or mean of the reference values . ( Theoretically there should be a simple transformation function from the data values to the reference values, but we can't specify " a priori " what the function should be-the only constraint that we have at the moment is that the transformation function would be monotonic and non-decreasing . ) As I understand it, the way to handle such a situation is with non-parametric statistics, and the way to tell " how well " the data match the reference values ( rank-order-wise ) is probably by using the Kendall tau rank correlation coefficient ( though I'm still a little unclear on the difference between it and the Kendall tau distance or Kendall's W, and where one is used in preference to the others ).
