Calculate the Pearson Product-Moment Coefficient of Correlation (r) for the following data set. Round your answer to two decimal places.
x y
10 2
8 3
7 5
5 3
r = -0.45
To calculate the Pearson Product-Moment Coefficient of Correlation (r), you will first need to calculate several other values. Here are the steps to get the value of r:
Step 1: Calculate the mean (average) for both x and y.
Do this by adding up all the x values and dividing by the number of data points (which is 4 in this case).
Mean of x = (10 + 8 + 7 + 5) / 4 = 30 / 4 = 7.5
Do the same for the y values:
Mean of y = (2 + 3 + 5 + 3) / 4 = 13 / 4 = 3.25
Step 2: Calculate the deviations for both x and y.
Deviation of x = x - mean of x
Deviation of y = y - mean of y
For the x values:
10 - 7.5 = 2.5
8 - 7.5 = 0.5
7 - 7.5 = -0.5
5 - 7.5 = -2.5
For the y values:
2 - 3.25 = -1.25
3 - 3.25 = -0.25
5 - 3.25 = 1.75
3 - 3.25 = -0.25
Step 3: Calculate the product of the deviations.
Multiply the deviation of x by the deviation of y for each data point and add them together.
(2.5)(-1.25) + (0.5)(-0.25) + (-0.5)(1.75) + (-2.5)(-0.25) = -5.1875
Step 4: Calculate the sum of the squares of the deviations for both x and y.
Square each deviation and add them up.
Sum of squares of deviations of x = (2.5)^2 + (0.5)^2 + (-0.5)^2 + (-2.5)^2 = 6.25 + 0.25 + 0.25 + 6.25 = 13
Sum of squares of deviations of y = (-1.25)^2 + (-0.25)^2 + (1.75)^2 + (-0.25)^2 = 1.5625 + 0.0625 + 3.0625 + 0.0625 = 4.75
Step 5: Calculate the square root of the product of the sum of squares of deviations.
Square root of [(sum of squares of deviations of x) * (sum of squares of deviations of y)]
Square root of (13 * 4.75) = square root of 61.75 ≈ 7.85
Step 6: Calculate the coefficient of correlation (r).
r = (product of deviations) / (square root of [(sum of squares of deviations of x) * (sum of squares of deviations of y)])
r = -5.1875 / 7.85 ≈ -0.66
Therefore, the Pearson Product-Moment Coefficient of Correlation (r) for the given data set is approximately -0.66.