Introduction to Correlation
In statistics, the Pearson correlation coefficient ($r$) is a measure of the linear correlation between two variables, $x$ and $y$. It gives a value between -1 and +1, where +1 is a total positive linear correlation, 0 is no linear correlation, and -1 is a total negative linear correlation.
The Formula
The formula to calculate the correlation coefficient $r$ is:
$$r = \frac{Cov(x, y)}{\sigma_x \cdot \sigma_y}$$
Where:
- $Cov(x, y)$ is the covariance between $x$ and $y$.
- $\sigma_x$ is the standard deviation of $x$.
- $\sigma_y$ is the standard deviation of $y$.
Note that variance is the square of the standard deviation ($Var(x) = \sigma_x^2$), so we must take the square root of the given variances to find the standard deviations.
Step-by-Step Solution
Given:
- $Cov(x, y) = 18$
- $Var(x) = 16 \implies \sigma_x = \sqrt{16} = 4$
- $Var(y) = 81 \implies \sigma_y = \sqrt{81} = 9$
Calculation:
- Identify the values: $Cov(x, y) = 18$, $\sigma_x = 4$, $\sigma_y = 9$.
- Plug the values into the formula: $$r = \frac{18}{4 \times 9}$$
- Simplify the denominator: $$r = \frac{18}{36}$$
- Final result: $$r = 0.5$$
Conclusion
The coefficient of correlation between the variables is 0.5, indicating a moderate positive linear relationship.