Editing Correlation ratio (section)

==Definition==

Suppose each observation is ''y<sub>xi</sub>'' where ''x'' indicates the category that observation is in and ''i'' is the label of the particular observation.  Let ''n<sub>x</sub>'' be the number of observations in category ''x'' and
 
:<math>\overline{y}_x=\frac{\sum_i y_{xi}}{n_x}</math>   and   <math>\overline{y}=\frac{\sum_x n_x \overline{y}_x}{\sum_x n_x},</math>

where <math>\overline{y}_x</math> is the mean of the category ''x'' and <math>\overline{y}</math> is the mean of the whole population. The correlation ratio η ([[eta (letter)|eta]]) is defined as to satisfy

:<math>\eta^2 = \frac{\sum_x n_x (\overline{y}_x-\overline{y})^2}{\sum_{x,i} (y_{xi}-\overline{y})^2}</math>

which can be written as
:<math>\eta^2 = \frac{{\sigma_{\overline{y}}}^2}{{\sigma_{y}}^2}, \text{ where }{\sigma_{\overline{y}}}^2 = \frac{\sum_x n_x (\overline{y}_x-\overline{y})^2}{\sum_x n_x} \text{ and } {\sigma_{y}}^2 = \frac{\sum_{x,i} (y_{xi}-\overline{y})^2}{n},</math>
i.e. the weighted variance of the category means divided by the variance of all samples.
 
If the relationship between values of <math>x </math> and values of <math>\overline{y}_x</math> is linear (which is certainly true when there are only two possibilities for ''x'') this will give the same result as the square of Pearson's [[Pearson product-moment correlation coefficient|correlation coefficient]]; otherwise the correlation ratio will be larger in magnitude. It can therefore be used for judging non-linear relationships.