Suppose each observation is yxi where x indicates the category that observation is in and i is the label of the particular observation. Let nx be the number of observations in category x and
where is the mean of the category x and is the mean of the whole population. The correlation ratio η (eta) is defined as to satisfy
It is worth noting that if the relationship between values of and values of is linear (which is certainly true when there are only two possibilities for x) this will give the same result as the square of the correlation coefficient, otherwise the correlation ratio will be larger in magnitude. It can therefore be used for judging non-linear relationships.
Then the subject averages are 36, 33 and 78, with an overall average of 52.
The sums of squares of the differences from the subject averages are 1952 for Algebra, 308 for Geometry and 600 for Statistics, adding to 2860, while the overall sum of squares of the differences from the overall average is 9640. The difference between these of 6780 is also the weighted sum of the square of the differences between the subject averages and the overall average:
The limit refers to the case without dispersion in the categories contributing to the overall dispersion. The trivial requirement for this extreme is that all category means are the same.
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