Editing Correlation (section)

==Rank correlation coefficients==
{{Main|Spearman's rank correlation coefficient|Kendall tau rank correlation coefficient}}

[[Rank correlation]] coefficients, such as [[Spearman's rank correlation coefficient]] and [[Kendall's tau|Kendall's rank correlation coefficient (τ)]] measure the extent to which, as one variable increases, the other variable tends to increase, without requiring that increase to be represented by a linear relationship. If, as the one variable increases, the other ''decreases'', the rank correlation coefficients will be negative. It is common to regard these rank correlation coefficients as alternatives to Pearson's coefficient, used either to reduce the amount of calculation or to make the coefficient less sensitive to non-normality in distributions. However, this view has little mathematical basis, as rank correlation coefficients measure a different type of relationship than the [[Pearson product-moment correlation coefficient]], and are best seen as measures of a different type of association, rather than as an alternative measure of the population correlation coefficient.<ref name="Yule and Kendall">Yule, G.U and Kendall, M.G. (1950), "An Introduction to the Theory of Statistics", 14th Edition (5th Impression 1968). Charles Griffin & Co. pp 258–270</ref><ref name="Kendall Rank Correlation Methods">Kendall, M. G. (1955) "Rank Correlation Methods", Charles Griffin & Co.</ref>

To illustrate the nature of rank correlation, and its difference from linear correlation, consider the following four pairs of numbers <math>(x,y)</math>:

:(0,&nbsp;1), (10,&nbsp;100), (101,&nbsp;500), (102,&nbsp;2000).

As we go from each pair to the next pair <math>x</math> increases, and so does <math>y</math>. This relationship is perfect, in the sense that an increase in <math>x</math> is ''always'' accompanied by an increase in <math>y</math>. This means that we have a perfect rank correlation, and both Spearman's and Kendall's correlation coefficients are 1, whereas in this example Pearson product-moment correlation coefficient is 0.7544, indicating that the points are far from lying on a straight line. In the same way if <math>y</math> always ''decreases'' when <math>x</math> ''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.  Although in the extreme cases of perfect rank correlation the two coefficients are both equal (being both +1 or both −1), this is not generally the case, and so values of the two coefficients cannot meaningfully be compared.<ref name="Yule and Kendall"/> For example, for the three pairs (1,&nbsp;1) (2,&nbsp;3) (3,&nbsp;2) Spearman's coefficient is 1/2, while Kendall's coefficient is&nbsp;1/3.