# 3.3.6: $$r^2$$, The Correlation of Determination

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The Regression ANOVA hypothesis test can be used to determine if there is a significant correlation between the independent variable ($$X$$) and the dependent variable ($$Y$$). We now want to investigate the strength of correlation.

In the earlier chapter on descriptive statistics, we introduced the correlation coefficient ($$r$$), a value between ‐1 and 1. Values of $$r$$ close to 0 meant there was little correlation between the variables, while values closer to 1 or ‐1 represented stronger correlations.

In practice, most statisticians and researchers prefer to use $$r^{2}$$, the coefficient of determination as a measure of strength as it represents the proportion or percentage of the variability of $$Y$$ that is explained by the variability of $$X$$. 87

##### $$r^2$$

$$r^{2}=\dfrac{S S_{\text{regression}}{S S_{\text {Total }}} \dquad 0 \% \leq r^{2} \leq 100 \%$$

$$r^{2}$$ represents the percentage of the variability of $$Y$$ that is explained by the variability of $$X$$. We can also calculate the correlation coefficient ($$r$$) by taking the appropriate square root of $$r^{2}$$, depending on whether the estimate of the slope ($$b_1$$) is positive or negative:

If $$b_{1}>0, r=\sqrt{r^{2}}$$

If $$b_{1}<0, r=-\sqrt{r^{2}}$$

##### Example: Rainfall and sales of sunglasses

For the rainfall data, the coefficient of determination is:

$$r^{2}=\dfrac{341.422}{380}=89.85 \%$$

89.85% of the variability of sales of sunglasses is explained by rainfall.

We can calculate the correlation coefficient ($$r$$) by taking the appropriate square root of $$r^{2}$$:

$$r=-\sqrt{.8985}=-0.9479$$

Here we take the negative square root since the slope of the regression line is negative. This shows that there is a strong, negative correlation between sales of sunglasses and rainfall.

3.3.6: $$r^2$$, The Correlation of Determination is shared under a CC BY-SA license and was authored, remixed, and/or curated by LibreTexts.