[MUSIC] In this section of our lecture, we introduce the covariance and the correlation as numerical ways to describe a linear relationship between two variables. The covariance measures the direction of this linear relationship. A positive value indicates an increasing linear relationship, while a negative value indicates a decreasing linear relationship. The population covariance is given by the sum of the differences between the observation, x and the population mean, times the differences between the observation, y and the population mean, divided by the population size, N. A sample covariance is given by the sum of the differences between the observation, x and its mean, times the differences between the observation, y and its mean, divided by the sample size n. Notice that the covariance between two variables behaves in the following way. If the covariance is higher than zero, then x and y tend to move in the same direction. If the covariance between x and y is less than zero, then x and y tend to move in the opposite directions. If the covariance between x and y is equal to 0, x and y are independent. The value of the covariance has a draw back however. The covariance does not provide a measure of the strength of the relationship between the two variables. And this is why, to overcome this drawback, we use the Pearson's Correlation Coefficient. The correlation coefficient provides us with a standardized measure of the linear relationship between the two variables. It is a useful measure because it provides both the direction and the strength of the relationship. The correlation coefficient is computed by dividing the covariance by the product of the standard deviations of the two variables. The population correlation coefficient indicated with the greek letter rho, is computed by dividing the covariance by the product of the population standard deviations. The sample correlation coefficient indicated with the letter r is computed by dividing the covariance by the product of the standard deviations of the two variables. It can be shown that the correlation coefficient, r, ranges from -1 to +1. The closer r to the +1, the closer the data to an increasing straight line. This indicates a positive linear relationship. The closer r to the -1, the closer the data to decreasing straight line. And this indicates a negative linear relationship. When r = 0, there is no linear relationship between x and y, however, this does not mean a lack of relationship. [MUSIC]