What is the method to find the correlation coefficient?

To find the correlation coefficient, use the formula involving the covariance of the variables and their standard deviations.

The correlation coefficient, often denoted as \( r \), measures the strength and direction of the linear relationship between two variables. To calculate it, you first need to find the covariance of the two variables, which measures how much the variables change together. The formula for covariance is:

\[ \text{Cov}(X, Y) = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{n - 1} \]

where \( X_i \) and \( Y_i \) are the individual data points, \( \bar{X} \) and \( \bar{Y} \) are the means of the variables, and \( n \) is the number of data points.

Next, you need to calculate the standard deviations of both variables, which measure the amount of variation or dispersion of the data points. The formula for standard deviation is:

\[ \sigma_X = \sqrt{\frac{\sum (X_i - \bar{X})^2}{n - 1}} \]
\[ \sigma_Y = \sqrt{\frac{\sum (Y_i - \bar{Y})^2}{n - 1}} \]

Finally, the correlation coefficient \( r \) is found by dividing the covariance by the product of the standard deviations of the two variables:

\[ r = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} \]

The value of \( r \) ranges from -1 to 1. A value of 1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship. This method helps you understand how closely two variables are related in a linear fashion.