For the following set of scores:
X Y
6 4
3 1
5 0
6 7
4 2
6 4
a) Compute the Pearson correlation.
b) Add 2 points to each X value and compute the correlation for the modified scores. How does adding a constant to every score affect the value of the correlation?
c)Multiply each of the original X values by 2 and compute the correlation for the modified scores. How does multiplying each score by a constant affect the value of the correlation?
To compute the Pearson correlation for the given set of scores, follow these steps:
a) Compute the Pearson correlation:
Step 1: Calculate the means of X and Y.
- The mean of X = (6 + 3 + 5 + 6 + 4 + 6) / 6 = 5
- The mean of Y = (4 + 1 + 0 + 7 + 2 + 4) / 6 = 3
Step 2: Calculate the deviations from the means for X (X - mean(X)) and for Y (Y - mean(Y)).
- Deviations X: 6-5=1, 3-5=-2, 5-5=0, 6-5=1, 4-5=-1, 6-5=1
- Deviations Y: 4-3=1, 1-3=-2, 0-3=-3, 7-3=4, 2-3=-1, 4-3=1
Step 3: Calculate the squared deviations for X (deviations from mean(X))^2 and Y (deviations from mean(Y))^2.
- Squared deviations X: 1^2 = 1, (-2)^2 = 4, 0^2 = 0, 1^2 = 1, (-1)^2 = 1, 1^2 = 1
- Squared deviations Y: 1^2 = 1, (-2)^2 = 4, (-3)^2 = 9, 4^2 = 16, (-1)^2 = 1, 1^2 = 1
Step 4: Calculate the products of deviations for X and Y (deviations from mean(X) * deviations from mean(Y)).
- Products of deviations: 1 * 1 = 1, (-2) * (-2) = 4, 0 * (-3) = 0, 1 * 4 = 4, (-1) * (-1) = 1, 1 * 1 = 1
Step 5: Sum up the squared deviations for X, squared deviations for Y, and products of deviations.
- Sum of squared deviations X = 1 + 4 + 0 + 1 + 1 + 1 = 8
- Sum of squared deviations Y = 1 + 4 + 9 + 16 + 1 + 1 = 32
- Sum of products of deviations = 1 + 4 + 0 + 4 + 1 + 1 = 11
Step 6: Calculate the Pearson correlation coefficient.
- Pearson correlation coefficient = Sum of products of deviations / sqrt[(Sum of squared deviations X) * (Sum of squared deviations Y)]
- Pearson correlation coefficient = 11 / sqrt(8 * 32) = 11 / sqrt(256) = 11 / 16 ≈ 0.6875
The Pearson correlation for the given set of scores is approximately 0.6875.
b) Adding 2 points to each X value and computing the correlation for the modified scores:
When adding a constant to every score, it does not affect the value of the correlation. The Pearson correlation coefficient remains the same. The correlation measures the linear relationship between the variables, and adding a constant to each score does not change that relationship.
c) Multiplying each of the original X values by 2 and computing the correlation for the modified scores:
When multiplying each score by a constant, it scales the values but does not change the linear relationship between the variables. Thus, the Pearson correlation coefficient will remain the same.