## Question 539:

1

The correlation coefficient, denoted  r is a measure of association between two variables. It can take on values between -1.0 and 1.0. The higher or lower the number, the stronger the association (correlation). To compute the correlation, we use the following formula which computes the covariance.

 Covariance = Sum of (X-Xbar)(Y-Ybar)   ---------------------------- N-1

 Correlation (r) = Covariance   ---------------------------- sx sy
• Xbar and Ybar are the means of X and Y.
• sx is the standard deviation of X and sy is the standard deviation of Y.

First find the mean and standard deviation for X and Y.
1. Find the means: X= 4.875 and Y = 10.625
2. Find the standard deviations sx = 1.7268 and sy = 3.378
3. Subtract each value of x from the mean of x (4.875) , and each value of y from the mean of y (10.625). These are deviation scores.
4. Multiply each X deviation score with each Y deviation score to get the product deviation scores.
5. Add up all the product deviation scores.
6. Divide by one less than the number being correlated (8-1) =7 to get the covariance. We get -5.196.
7. Divide the covariance by the product of the standard deviations we found in step 2. 1.7268*3.378 = 5.8333. So the correlation is -5.196/5.833 = -.8908
The coefficient of determination is a way of expressing how one variable explains the variation in the other variable. It is found by squaring the correlation coefficient and is denoted as R2 .  Squaring r -.89082 =.7935 so the coefficient of determination is .7935 or 79.35%.  So we would say that X explains 79.35% of the variance of Y.