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.
   

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 formula helps measure the strength and direction of the relationship between two variables. Several forms of the formula are used depending on the type of data. The main correlation coefficient formulas are given below.

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.
   
• \(Σx^2\) is the sum of squares of the first value.
   
• \(Σy^2\) is the sum of squares of the second value. 

Sample Correlation Coefficient Formula
 

\(r_{xy} = \frac{s_{xy}}{s_x \cdot s_y} \)

\(S_x, S_y \) represent the standard deviations
 

\(S_{xy}\) is the sample covariance

Population Correlation Coefficient Formula

\(\rho_{xy} = \frac{\sigma_{xy}}{\sigma_x \, \sigma_y} \)
 

Where:\(\sigma_x, \sigma_y \) is the population standard deviation

\(Σxy \) is the population covariance

 Linear Correlation Coefficient Formula
 

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

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:
   

\(\rho_{xy} = \frac{\text{Cov}(x, y)}{\sigma_x \, \sigma_y} \)
         
  Here: 

\(\rho_{xy}\) represents Pearson’s product-moment correlation coefficient
 

Cov(x, y) is the covariance of variables x and y

\(\sigma_x, \sigma_y \) are the standard deviations for variables x and y.

Suppose a teacher wants to check whether students who study more hours tend to score higher marks. The data for 5 students is given below: 

To calculate the correlation coefficient using the formula: 

\(\rho_{xy} = \frac{\sigma_{xy}}{\sigma_x \, \sigma_y} \)
 

Find the mean of x and y:
 
\({\bar x} = {{2 + 3 + 5 + 6 + 4}\over{5 }}= 4 \)
 

\({\bar y} = {{50 + 60 + 80 + 85 + 70}\over {5}} = 69 \)

Finding the covariance: 

\(\text{Cov}(x, y) = \frac{1}{n} \sum (x_i - \bar{x})(y_i - \bar{y}) \)

\(Σ (x - x) (y - y) = 38 + 9 + 11 + 32 + 0 \\ \ \\ = 90\)

Calculating the standard deviations: 
 

For x: \(\sigma_x = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}} \)
 

\( (x_i - \bar{x})^2 = (-2)^2 + (-1)^2 + 1^2 + 2^2 + 0^2 \)

\(= 4 + 1 + 1 + + 4 \\ \ \\ = 10\)

\(= \sqrt{\frac{10}{5}} \)

\(={ \sqrt 2} = 1.414\)

For y: 

\((y - {\bar y})^2 = 361 + 81 + 121 + 256 + 1 = 820 \)
 

\(\sigma_y = {\sqrt {820 \over 5}} \)

\(= \sqrt {164}\)

=12.806

Apply the formula: \(\rho_{xy} = \frac{\sigma_{xy}}{\sigma_x \, \sigma_y} \)
 

\(\rho_{xy} = {{18 \over(1.414) (12.806)}}\)

\(= {18\over 18.10} = 0.994 \)

\(\rho_{xy} = 0.99\)

The correlation coefficient indicates the strength of the relationship between two variables and whether it is positive or negative. Here are a few tips and tricks to master the correlation coefficient.

• Understand the basic concept of the correlation coefficient. The correlation coefficient measures the strength and direction of the relationship between two variables.
   
• Students should understand and memorize the formula related to the correlation coefficient.
   
• Students can organize the data using tables with columns for x, y, x2, y2, and xy to simplify calculations.
   
• Students can use correlation coefficient calculators to verify the answer.
   
• Teachers can use real-life examples to make the correlation meaningful for students.
   
• Parents can encourage students to use tools such as calculators, Excel, or statistical software to help them understand the concept of correlation.

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.