Editing Pearson correlation coefficient (section)

==Mathematical properties==
The values of both the sample and population Pearson correlation coefficients are on or between −1 and 1. Correlations equal to +1 or −1 correspond to data points lying exactly on a line (in the case of the sample correlation), or to a bivariate distribution entirely [[Support (measure theory)|supported]] on a line (in the case of the population correlation). The Pearson correlation coefficient is symmetric: corr(''X'',''Y'')&nbsp;=&nbsp;corr(''Y'',''X'').

A key mathematical property of the Pearson correlation coefficient is that it is [[invariant estimator|invariant]] under separate changes in location and scale in the two variables. That is, we may transform ''X'' to {{math|''a'' + ''bX''}} and transform ''Y'' to {{math|''c'' + ''dY''}}, where ''a'', ''b'', ''c'', and ''d'' are constants with {{math|''b'', ''d'' > 0}}, without changing the correlation coefficient. (This holds for both the population and sample Pearson correlation coefficients.) More general linear transformations do change the correlation: see ''{{section link||Decorrelation of n random variables}}'' for an application of this. In particular, it might be useful to notice that corr(''-X'',''Y'')&nbsp;=&nbsp;-corr(''X'',''Y'')