# 2.3: Correlation

As before (in §§4 and 5), when we moved from describing histograms with words (like symmetric) to describing them with numbers (like the mean), we now will build a numeric measure of the strength and direction of a linear association in a scatterplot.

[def:corrcoeff] Given bivariate quantitative data $$\{(x_1,y_1), \dots , (x_n,y_n)\}$$ the [Pearson] correlation coefficient of this dataset is $r=\frac{1}{n-1}\sum \frac{(x_i-\overline{x})}{s_x}\frac{(y_i-\overline{y})}{s_y}$ where $$s_x$$ and $$s_y$$ are the standard deviations of the $$x$$ and $$y$$, respectively, datasets by themselves.

We collect some basic information about the correlation coefficient in the following

[fact:corrcoefff] For any bivariate quantitative dataset $$\{(x_1,y_1), \dots ,(x_n,y_n)\}$$ with correlation coefficient $$r$$, we have

1. $$-1\le r\le 1$$ is always true;

2. if $$|r|$$ is near $$1$$ – meaning that $$r$$ is near $$\pm 1$$ – then the linear association between $$x$$ and $$y$$ is strong

3. if $$r$$ is near $$0$$ – meaning that $$r$$ is positive or negative, but near $$0$$ – then the linear association between $$x$$ and $$y$$ is weak

4. if $$r>0$$ then the linear association between $$x$$ and $$y$$ is positive, while if $$r<0$$ then the linear association between $$x$$ and $$y$$ is negative

5. $$r$$ is the same no matter what units are used for the variables $$x$$ and $$y$$ – meaning that if we change the units in either variable, $$r$$ will not change

6. $$r$$ is the same no matter which variable is begin used as the explanatory and which as the response variable – meaning that if we switch the roles of the $$x$$ and the $$y$$ in our dataset, $$r$$ will not change.

It is also nice to have some examples of correlation coefficients, such as Many electronic tools which compute the correlation coefficient $$r$$ of a dataset also report its square, $$r^2$$. There reason is explained in the following

[fact:rsquared] If $$r$$ is the correlation coefficient between two variables $$x$$ and $$y$$ in some quantitative dataset, then its square $$r^2$$ it the fraction (often described as a percentage) of the variation of $$y$$ which is associated with variation in $$x$$.

[eg:rsquared] If the square of the correlation coefficient between the independent variable how many hours a week a student studies statistics and the dependent variable how many points the student gets on the statistics final exam is $$.64$$, then 64% of the variation in scores for that class is cause by variation in how much the students study. The remaining 36% of the variation in scores is due to other random factors like whether a student was coming down with a cold on the day of the final, or happened to sleep poorly the night before the final because of neighbors having a party, or some other issues different just from studying time.