How to Calculate the Correlation Coefficient

The correlation coefficient is a statistical metric that measures how strongly two variables are linearly related.  The values of the correlation coefficient range from -1 to 1. If the correlation coefficient is -1, the relationship between the variables indicates a negative or inverse correlation. When the correlation coefficient is 1, the variables are in positive correlation and are directly proportional. The correlation coefficient of zero indicates that there is no significant relationship between the variables.
Here are a few key takeaways to help you grasp the concept at a glance:

• Correlation coefficients are applied to measure the strength of the relationship possessed by two variables.
   
• A correlation coefficient of 1 indicates a direct relationship between two variables and a -1 shows the variables are in negative correlation.
   
• If the variables don’t possess a significant connection or a weak correlation, the correlation coefficient will be 0.
   
• The Pearson coefficient or Pearson’s R is the most prominent type of correlation coefficient that measures the strength and direction of linear relationships.
   

We can calculate the correlation coefficient easily by understanding each step listed below:

• To calculate the correlated coefficient, the initial step is to find the covariance of the variables provided.
   
• Now, divide the resultant covariance of the variables by the product of their standard deviation.
   
• To better understand the concept, let’s look at its equation:

            ρxy =  Cov(x, y) / σx σy
            Here: 
            ρxy represents Pearson’s product-moment correlation coefficient
            Cov(x, y) is the covariance of variables x and y
            σx, σy are the standard deviations for variables x and y.

\( r = \frac{n\sum xy - (\sum x)(\sum y)} {\sqrt{\bigl[n\sum x^{2} - (\sum x)^{2}\bigr] \bigl[n\sum y^{2} - (\sum y)^{2}\bigr]}} \),

Where:

• r denoted is the correlation coefficient.
   
• n is the number of data pairs.
   
• Σx is the sum of all values for the first variable
  .
• Σy is the sum of all values for the second variable.
   
• Σxy is the sum of the product of first and second values.
   
• Σx² is the sum of squares of the first value.
   
• Σy² is the sum of squares of the second value. 
   

Correlation and Regression are two related but different concepts. Understanding their differences will help you understand them better. Let’s look at a few key differences between Correlation and Regression:
 

The correlation coefficient formulas vary in different types. We will now look at each of them:

Pearson’s Correlation Coefficient Formula 

\( r = \frac{n\sum xy - (\sum x)(\sum y)} {\sqrt{\bigl[n\sum x^{2} - (\sum x)^{2}\bigr] \bigl[n\sum y^{2} - (\sum y)^{2}\bigr]}} \)

Where:

• n is the number of data pairs
   
• Σx is the sum of all values for the first variable.
   
• Σy is the sum of all values for the second variable.
   
• Σxy is the sum of the product of first and second values.
   
• Σx² is the sum of squares of the first value.
   
• Σy² is the sum of squares of the second value. 

Linear Correlation Coefficient Formula

\(r_{xy} = \frac{n \sum_{i=1}^n x_i y_i - \left( \sum_{i=1}^n x_i \right)\left( \sum_{i=1}^n y_i \right)} {\sqrt{n \sum_{i=1}^n x_i^2 - \left( \sum_{i=1}^n x_i \right)^2} \; \sqrt{n \sum_{i=1}^n y_i^2 - \left( \sum_{i=1}^n y_i \right)^2}} \)

\( r_{xy} = \frac{n \displaystyle\sum_{i=1}^{n} x_i y_i - \left( \displaystyle\sum_{i=1}^{n} x_i \right) \left( \displaystyle\sum_{i=1}^{n} y_i \right)} {\sqrt{\,n \displaystyle\sum_{i=1}^{n} x_i^2 - \left( \displaystyle\sum_{i=1}^{n} x_i \right)^2}\; \sqrt{\,n \displaystyle\sum_{i=1}^{n} y_i^2 - \left( \displaystyle\sum_{i=1}^{n} y_i \right)^2}} \)Sample Correlation Coefficient Formula

rxy= Sxy/ Sx Sy

Sx, Sy represent the standard deviations

Sxy is the sample covariance

Population Correlation Coefficient Formula
 

Ρxy = σxy/ σx σy

Where: 

σx σy is the population standard deviation
Σxy is the population covariance
 

Correlation coefficients are applied in various fields to determine the linear relationship between two different quantities. Let’s learn how they can be applied in various fields:

• Schools apply the correlation coefficient in analyzing how external factors like study hours, and screen time, affect their academic performance.
   
• Businesses plan their marketing strategies using the correlation between consumer behavior and discounts.
   
• Correlation is applied in healthcare to study how sleeping hours or eating habits affect certain diseases.
   
• In social science, correlation is used to determine the relationship between literacy and the scope of job opportunities.
   
• Phone manufacturers analyze the impact of mobile usage on battery life.