What is the correlation coefficient of the following data? Is it a strong or weak correlation?
x: 1, 3, 5, 7, 9
y: 13, 43, 51, 76, 81
To calculate the correlation coefficient, we first need to find the mean of both x and y:
Mean of x: (1 + 3 + 5 + 7 + 9) / 5 = 5
Mean of y: (13 + 43 + 51 + 76 + 81) / 5 = 52.8
Next, we need to calculate the numerator of the correlation coefficient formula:
Σ((xi - mean of x) * (yi - mean of y)):
(1-5) * (13-52.8) = -192.4
(3-5) * (43-52.8) = -20.8
(5-5) * (51-52.8) = 3.6
(7-5) * (76-52.8) = 184.8
(9-5) * (81-52.8) = 275.2
Add them all up:
-192.4 + (-20.8) + 3.6 + 184.8 + 275.2 = 250.4
Now, we need to calculate the denominator of the correlation coefficient formula:
√(Σ(xi - mean of x)²) * √(Σ(yi - mean of y)²):
√(4 + 4 + 0 + 4 + 16) * √((38.8)² + (10.8)² + (1.2)² + (23.2)² + (28.2)²)
= √(28) * √(1502.24)
= 5.29 * 38.75
= 206.19
Now, we can calculate the correlation coefficient:
250.4 / 206.19 = 1.215
The correlation coefficient is 1.215. This indicates a strong positive correlation between the variables x and y.