Correlation Coefficient Example

This below example illustrates how to calculate correlation coefficient of two given data sets.

The formula for calculating Correlation Coefficient is given in the image below


Example
Find Correlation coefficient for X and Y values are given below
X= (1, 2, 3, 4, 5)
Y= {11, 22, 34, 43, 56}

Solution :
Step 1:
Find Mean for X and Y
X=15/5=3
Y=166/5=33.2

Step 2:
Calculate Standard Deviation for Y inputs:
σ_x = &radic(1/(N - 1)*((x₁ - x_m)² + (x₂ - x_m)² = .... + (x_n - x_m)²))

= &radic(1/(5 - 1)((11 - 33.2)² + (22 - 33.2)² + (34-33.2)² + (43-33.2)² + (56-33.2)²))

= &radic(1/4((-22.2)² + (-11.2)² + (0.8)² + (9.8)² + (22.8)²))

= &radic(1/4((492.84) + (125.44) + (0.64) + (96.04) + (519.84)))

= &radic(308.7)

= 17.5699


Step 3:
Standard Deviation for X Inputs:

σ_x = &radic(1/(N - 1) x ((x₁ - x_m)² + (x₂ - x_m)² + .... + (x_n - x_m)²))

= &radic(1/(5 - 1)((1 - 3)² + (2 - 3)² + (3 - 3)² + (4 - 3)² + (5 - 3)²))

= &radic(1/4((-2)² + (-1)² + (0)² + (1)² + (2)²))

= &radic(1/4((4) +(1) + (0) + (1) + (4)))

= &radic(2.5)

= 1.5811

Σ((X-X) * (Y-Y)
= (1 - 3) * (11 - 33.2) + (2 - 3)(22 - 33.2) + (3 - 3)(34 - 33.2) + (4 - 3)(43 - 33.2) + (5 - 3)(56 - 33.2)
= -2 * -22.2 + -1 * -11.2 + 0 * 0.8 + 1 * 9.8 + 2 * 22.8
= 44.4 + 11.2 + 0 + 9.8 + 45.6
= 111
Correlation Coefficient=111/((5 - 1) * 1.5811 * 17.5699)
Correlation Coefficient (r) = 0.9989

Hence the Correlation coefficient of the above data set is 0.9989

When you do this calculations yourself, this online Correlation coefficient Calculator can be used to verify your results.