v Pearson's Correlation Coefficient - Statistics

Pearson's Correlation Coefficient

Based on this StackOverflow answer by cbare.

Preliminaries

import statistics as stats

Create Data

x = [1,2,3,4,5,6,7,8,9]
y = [2,1,2,4.5,7,6.5,6,9,9.5]

Calculate Pearson's Correlation Coefficient

There are a number of equivalent expression ways to calculate Pearson's correlation coefficient (also called Pearson's r). Here is one.

$$r={\frac {1}{n-1}}\sum _{i=1}^{n}\left({\frac {x_{i}-{\bar {x}}}{s_{x}}}\right)\left({\frac {y_{i}-{\bar {y}}}{s_{y}}}\right)$$

where \(s_{x}\) and \(s_{y}\) are the sample standard deviation for \(x\) and \(y\), and \(\left({\frac {x_{i}-{\bar {x}}}{s_{x}}}\right)\) is the standard score for \(x\) and \(y\).

# Create a function
def pearson(x,y):

    # Create n, the number of observations in the data
    n = len(x)

    # Create lists to store the standard scores
    standard_score_x = []
    standard_score_y = []

    # Calculate the mean of x
    mean_x = stats.mean(x)

    # Calculate the standard deviation of x
    standard_deviation_x = stats.stdev(x)

    # Calculate the mean of y
    mean_y = stats.mean(y)

    # Calculate the standard deviation of y
    standard_deviation_y = stats.stdev(y)

    # For each observation in x
    for observation in x: 

        # Calculate the standard score of x
        standard_score_x.append((observation - mean_x)/standard_deviation_x) 

    # For each observation in y
    for observation in y:

        # Calculate the standard score of y
        standard_score_y.append((observation - mean_y)/standard_deviation_y)

    # Multiple the standard scores together, sum them, then divide by n-1, return that value
    return (sum([i*j for i,j in zip(standard_score_x, standard_score_y)]))/(n-1)
# Show Pearson's Correlation Coefficient
pearson(x,y)
0.9412443251336238