Editing Correlation (section)

=== Simple linear correlations ===

[[File:Anscombe's quartet 3.svg|thumb|325px|right|[[Anscombe's quartet]]: four sets of data with the same correlation of 0.816]]
The Pearson correlation coefficient indicates the strength of a ''linear'' relationship between two variables, but its value generally does not completely characterize their relationship. In particular, if the [[conditional expectation|conditional mean]] of <math>Y</math> given <math>X</math>, denoted <math>\operatorname{E}(Y \mid X)</math>, is not linear in <math>X</math>, the correlation coefficient will not fully determine the form of <math>\operatorname{E}(Y \mid X)</math>.

The adjacent image shows [[scatter plot]]s of [[Anscombe's quartet]], a set of four different pairs of variables created by [[Francis Anscombe]].<ref>{{cite journal | last=Anscombe | first=Francis J. | year=1973 | title=Graphs in statistical analysis | journal=The American Statistician | volume=27 | issue=1 | pages=17–21 | jstor=2682899 | doi=10.2307/2682899}}</ref> The four <math>y</math> variables have the same mean (7.5), variance (4.12), correlation (0.816) and regression line (<math display="inline">y=3+0.5x</math>). However, as can be seen on the plots, the distribution of the variables is very different. The first one (top left) seems to be distributed normally, and corresponds to what one would expect when considering two variables correlated and following the assumption of normality. The second one (top right) is not distributed normally; while an obvious relationship between the two variables can be observed, it is not linear. In this case the Pearson correlation coefficient does not indicate that there is an exact functional relationship: only the extent to which that relationship can be approximated by a linear relationship. In the third case (bottom left), the linear relationship is perfect, except for one [[outlier]] which exerts enough influence to lower the correlation coefficient from 1 to 0.816. Finally, the fourth example (bottom right) shows another example when one outlier is enough to produce a high correlation coefficient, even though the relationship between the two variables is not linear.

These examples indicate that the correlation coefficient, as a [[summary statistic]], cannot replace visual examination of the data. The examples are sometimes said to demonstrate that the Pearson correlation assumes that the data follow a [[normal distribution]], but this is only partially correct.<ref name="thirteenways"/> The Pearson correlation can be accurately calculated for any distribution that has a finite [[covariance matrix]], which includes most distributions encountered in practice. However, the Pearson correlation coefficient (taken together with the sample mean and variance) is only a [[sufficient statistic]] if the data is drawn from a [[multivariate normal distribution]]. As a result, the Pearson correlation coefficient fully characterizes the relationship between variables if and only if the data are drawn from a multivariate normal distribution.