Please help me figure out the considerations

Question

Consider the following set of observations on X and Y.

 

X	Y
2	3
3	4
4	8

The correlation coefficient between X and Y is

Question 49 options:

	.43
	.82
	.57
	.76
	.94

Answer 1

Hi Ariana!

These are the steps to calculate the correlation coefficient r :

1. Calculate the following values (these values will be used for the calculations in the next steps) :
2. Calculate x̄, the mean of all of the first coordinates of the data x_i.
3. Calculate ȳ, the mean of all of the second coordinates of the data y_i.
4. Calculate s _x the sample standard deviation of all of the first coordinates of the data x_i.
5. Calculate s _y the sample standard deviation of all of the second coordinates of the data y_i.
6. Use the formula (z_x)_i = (x_i – x̄) / s _x and calculate a standardized value for each x_i.
7. Use the formula (z_y)_i = (y_i – ȳ) / s _y and calculate a standardized value for each y_i.
8. Multiply corresponding standardized values: (z_x)_i(z_y)_i
9. Add the products from the last step together.
10. Divide the sum from the previous step by n – 1, where n is the total number of points in our set of paired data. The result of all of this is the correlation coefficient r.

In your example, here is how r is calculated:

x̄ = (2 + 3 + 4) / 3 = 3

ȳ = (3 + 4 + 8) / 3 = 5

s_x = [(2 - 3)² + (3 - 3)² + (4 - 3)²] / (3 - 1) = 1 --> sqrt(1) = 1

s_y = [(3 - 5)² + (4 - 5)² + (8 - 5)²] / (3 - 1) = 7 --> sqrt(7) ~ 2.65

z_x standardized value for data point (2,3): -1

z_x standardized value for data point (3,4): 0

z_x standardized value for data point (4,8): 1

z_y standardized value for data point (2,3): -0.75

z_y standardized value for data point (3,4): -0.34

z_y standardized value for data point (4,8): 1.13

Multiplication of corresponding standardized values (z_x)_i(z_y)_:

1. For data point (2,3): 0.75
2. For data point (3,4): 0
3. For data point (4,8): 1.13

_____________________________________________________________

CORRELATION COEFFICIENT (r): (0.75 + 0 + 1.13) / (3 - 1) = 0.94

P.S you can also use Google to look up a correlation coefficient calculator, where you can plug in your three data points and it gives you the correlation coefficient. I tried doing this and it also gave me r = 0.94 :)