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In statistics, the correlation ratio is a measure of the relationship between the statistical dispersion within individual categories and the dispersion across the whole population or sample.## Range

The correlation ratio $eta$ takes values between 0 and 1. The limit $eta=0$ represents the special case of no dispersion among the means of the different categories, while $eta=1$ refers to no dispersion within the respective categories. Note further, that $eta$ is undefined when all data points of the complete population take the same value.
## Example

Suppose there is a distribution of test scores in three topics (categories):

Suppose each observation is y_{xi} where x indicates the category that observation is in and i is the label of the particular observation. Let n_{x} be the number of observations in category x and

- $overline\{y\}\_x=frac\{sum\_i\; y\_\{xi\}\}\{n\_x\}$ and $overline\{y\}=frac\{sum\_x\; n\_x\; overline\{y\}\_x\}\{sum\_x\; n\_x\},$

where $overline\{y\}\_x$ is the mean of the category x and $overline\{y\}$ is the mean of the whole population. The correlation ratio η (eta) is defined as to satisfy

- $eta^2\; =\; frac\{sum\_x\; n\_x\; (overline\{y\}\_x-overline\{y\})^2\}\{sum\_\{x,i\}\; (y\_\{xi\}-overline\{y\})^2\}.$

It is worth noting that if the relationship between values of $x\; ;$ and values of $overline\{y\}\_x$ 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.

- Algebra: 45, 70, 29, 15 and 21 (5 scores)
- Geometry: 40, 20, 30 and 42 (4 scores)
- Statistics: 65, 95, 80, 70, 85 and 73 (6 scores).

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:

- $5\; (36-52)^2\; +\; 4\; (33-52)^2\; +6\; (78-52)^2\; =\; 6780$

- $eta^2\; =\; frac\{6780\}\{9640\}=0.7033ldots$

- $eta\; =\; sqrt\{frac\{6780\}\{9640\}\}=0.8386ldots$

The limit $eta\; =\; 0$ 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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Last updated on Saturday August 30, 2008 at 09:32:13 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday August 30, 2008 at 09:32:13 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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