Python Programming for Biology: Bioinformatics and Beyond

Correlation coefficient

As the covariance calculation derives from products of the two variables, the magnitude of

the  measure  is  dependent  on  the  scaling  of  both  X  and  Y.  Correspondingly,  the  units  of

covariance will be the product of the X and Y units. We can remove the effect of scaling by

dividing the covariance by the standard deviations of the two random variables and thus

obtain a dimensionless value that lies between −1 and 1. A value of −1 will mean perfectly

anti-correlated,  that  the  quantities  change  together  but  in  opposite  directions;  a  value  of

---

zero  will  mean  completely  uncorrelated  and  1  means  fully  correlated.  This  is  what  is

called Pearson’s correlation coefficient and it gives us a handy measure of how well two

quantities  are  linearly  correlated,  irrespective  of  any  different  scales  or  units  they  may

have. We can write the equation for this as:

16

Although we have written this for the true standard deviations of the random variables

the calculation is the same for the sample correlation coefficient, except that the unbiased

sample standard deviation is used. If we apply this to various data sets, plotted as graphs

in

Figure 22.7

, we can see how the value of the correlation coefficient corresponds to the

degree of linear correlation.