For the following set of scores,
X Y
4 5
6 5
3 2
9 4
6 5
2 3
a. Compute the Pearson correlation.
b. Add two 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?

a) The Pearson correlation coefficient for the original set of scores is -0.2.

b) The Pearson correlation coefficient for the modified set of scores is -0.2.

c) The Pearson correlation coefficient for the modified set of scores is -0.6071.

To compute the Pearson correlation coefficient, we need to calculate the covariance and the standard deviations of the X and Y variables. Let's calculate each step:

X: 4, 6, 3, 9, 6, 2

Y: 5, 5, 2, 4, 5, 3

a. Compute the Pearson correlation:

Step 1: Calculate the means of X ([tex]\bar{x}[/tex]) and Y ([tex]\bar{y}[/tex]):

[tex]\bar{x}[/tex] = (4 + 6 + 3 + 9 + 6 + 2) / 6 = 5

[tex]\bar{y}[/tex] = (5 + 5 + 2 + 4 + 5 + 3) / 6 = 4.6667

Step 2: Calculate the deviations from the mean for X (dx) and Y (dy):

dx = X - [tex]\bar{x}[/tex]: (-1, 1, -2, 4, 1, -3)

dy = Y - [tex]\bar{y}[/tex]: (0.3333, 0.3333, -2.6667, -0.6667, 0.3333, -1.6667)

Step 3: Calculate the covariance (cov) and the standard deviations (σx and σy):

cov = (dx * dy) / (n - 1)

   = (-1 * 0.3333 + 1 * 0.3333 + -2 * -2.6667 + 4 * -0.6667 + 1 * 0.3333 + -3 * -1.6667) / (6 - 1)

   = -1.2

σx = √((dx * dx) / (n - 1))

   = √(((-1)² + 1² + (-2)² + 4² + 1² + (-3)²) / (6 - 1))

   = √(30 / 5)

   = √(6)

σy = √((dy * dy) / (n - 1))

   = √((0.3333²+0.3333²+(-2.6667)²+(-0.6667)²+0.3333² + (-1.6667)²)/(6- 1))

   = √(6)

Step 4: Calculate the Pearson correlation coefficient (r):

r = cov / (σx * σy)

 = -1.2 / (√(6) * √(6))

 = -1.2 / 6

 = -0.2

Therefore, the Pearson correlation coefficient for the original set of scores is -0.2.

b. Adding two points to each X value and computing the correlation for the modified scores:

Modified X: 6, 8, 5, 11, 8, 4

To compute the correlation, we follow the same steps as in part a:

Step 1: Calculate the means of the modified X ([tex]\bar{x}[/tex]) and Y ([tex]\bar{y}[/tex]):

[tex]\bar{x}[/tex]= (6 + 8 + 5 + 11 + 8 + 4) / 6 = 7

[tex]\bar{y}[/tex] = (5 + 5 + 2 + 4 + 5 + 3) / 6 = 4.6667

Step 2: Calculate the deviations from the mean for the modified X (dx) and Y (dy):

dx = Modified X - [tex]\bar{x}[/tex]: (-1, 1, -2, 4, 1, -3)

dy = Y - [tex]\bar{y}[/tex]: (0.3333, 0.3333, -2.6667, -0.6667, 0.3333, -1.6667)

Step 3: Calculate the covariance (cov) and the standard deviations (σx and σy):

cov = (dx * dy) / (n - 1)

   = (-1 * 0.3333 + 1 * 0.3333 + -2 * -2.6667 + 4 * -0.6667 + 1 * 0.3333 + -3 * -1.6667) / (6 - 1)

   = -1.2

σx = √((dx * dx) / (n - 1))

   = √(((-1)² + 1² + (-2)² + 4² + 1² + (-3)²) / (6 - 1))

   = √(30 / 5)

   = √(6)

σy = √((dy * dy) / (n - 1))

   = √((0.3333² + 0.3333² + (-2.6667)² + (-0.6667)² + 0.3333² + (-1.6667)²) / (6 - 1))

   = √(6)

Step 4: Calculate the Pearson correlation coefficient (r):

r = cov / (σx * σy)

 = -1.2 / (√(6) * √(6))

 = -1.2 / 6

 = -0.2

Adding a constant to every score does not affect the value of the correlation. The correlation remains the same at -0.2.

c. To compute the correlation coefficient after multiplying each of the original X values by 2, let's follow the steps:

Modified X: 8, 12, 6, 18, 12, 4

Step 1: Calculate the means of the modified X ([tex]\bar{x}[/tex]) and Y ([tex]\bar{y}[/tex]):

[tex]\bar{x}[/tex] = (8 + 12 + 6 + 18 + 12 + 4) / 6 = 10

[tex]\bar{y}[/tex] = (5 + 5 + 2 + 4 + 5 + 3) / 6 = 4.6667

Step 2: Calculate the deviations from the mean for the modified X (dx) and Y (dy):

dx = Modified X - [tex]\bar{x}[/tex]: (-2, 2, -4, 8, 2, -6)

dy = Y - [tex]\bar{y}[/tex]: (0.3333, 0.3333, -2.6667, -0.6667, 0.3333, -1.6667)

Step 3: Calculate the covariance (cov) and the standard deviations (σx and σy):

cov = (dx * dy) / (n - 1)

   = (-2 * 0.3333 + 2 * 0.3333 + -4 * -2.6667 + 8 * -0.6667 + 2 * 0.3333 + -6 * -1.6667) / (6 - 1)

   = -3.4667

σx = √((dx * dx) / (n - 1))

   = √(((-2)² + 2² + (-4)² + 8² + 2² + (-6)²) / (6 - 1))

   = √(100 / 5)

   = √(20)

   ≈ 4.4721

σy = √((dy * dy) / (n - 1))

= √((0.3333² + 0.3333²+(-2.6667)²+(-0.6667)²+0.3333² + (-1.6667)²)/(6 - 1))

=√(6)

Step 4: Calculate the Pearson correlation coefficient (r):

r = cov / (σx * σy)

= -3.4667 / (4.4721 * √(6))

≈ -0.6071

Multiplying each score by a constant affects the value of the correlation coefficient. In this case, multiplying each original X value by 2 resulted in a correlation coefficient of approximately -0.6071. It shows a stronger negative correlation compared to the original correlation coefficient of -0.2. The correlation coefficient became closer to -1, indicating a stronger linear relationship between the modified X and Y variables.

Learn more about Pearson correlation here

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