Pearson Correlation Coefficient

The Pearson correlation coefficient is a measure of the linear correlation between two variables X and Y, giving a value between +1 and -1, inclusive,

where

• 1 is a perfect positive correlation
• 0 is no correlation
• -1 is a perfect negative correlation

The formula for Pearson correlation coefficient is:

where

• cov(X, Y) is the covariance between X and Y.
• σ_X is the standard deviation of X.
• σ_Y is the standard deviation of Y.

The Pearson correlation coefficient can be expressed as:

where

or

Example

X: (1 2 3 4 6 7 9)
Y: (4 5 6 5 6 7 9)

We can organize the data set in the form of table:

From this table, we can find the sample mean:

• x̄ = ΣX / n = 32 / 7 = 4.5714
• ȳ = ΣY / n = 42 / 7 = 6

Here is another table:

From this table we can find SS values:

• Sxx = 196 - 32² / 7 = 49.7143
• Syy = 268 - 42² / 7 = 16
• Sxy = 218 - (32)(42) / 7 = 26

Therefore, the Pearson correlation coefficient will be:

• r = 26 / √(49.7143 × 16) = 0.9219

From this we can find coefficient of determination (r²):

• r² = 0.9219² = 0.8499