## Spearman's Rank Correlation Coeffcient

The correlation coefficientmeasurements the goodness of fit of a set of pointstoa straight line but two quantities may in fact observe a perfectcorrelation without all the points lying on a straight line. Thequantities x and y have a perfect though non linear relationship asshown on the graph below. The relationship is perfect for the rangesof x and y in that if x increases then so does y.

The correlation betweenvariablesandthathave a non linear relationship may be better expressed by a functionof the variables called the rank correlation coefficient. The rankcorrelation coefficient is found by ordering theinterms of size, labelling themEachhasa correspondingbutwhen theareput into order themaynot have the same order as theSupposethehaveranksLabelthe ranks of theas(thoughwe may in fact take them as).

For eachfindandfind

insome order and each of these is equal toso

insome order and each of these is equal to

and similarly for

Spearman's Rank correlationcoefficient is given by