How to Calculate Correlation Coefficient? Correlation Coefficient

How to Calculate Correlation Coefficient?

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Correlation Coefficient is used to understand and establish a relationship between variables. In finances, it is used to analyze and understand the set variables. In business, the correlation coefficient helps determine future sales and market trends.

Correlation Coefficient

A correlation coefficient is a statistical approach that measures the strength and direction of the relationship between the two variables. It is used to measure the dependency of the response variable on the explanatory variable. 

  • The correlation coefficient lies between +1 and -1. 
  • A Positive Correlation Coefficient, a value with a + sign, indicates that one variable is influenced by the other variable. This means that when the value of one variable increases the other increases as well. 
  • In the case of, Negative Correlation Coefficient, the relationship between the variables is inverse. This means if the value of one variable increases the other decreases. 
  • Zero Correlation Coefficient informs us that there is an absence of a relationship between the variables.

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Types of Correlation Coefficient

Pearson Correlation: 

Pearson correlation is used to examine the linear relationship between the two separate variables. The correlation coefficient will show values closer to + 1 and -1 when the relationship is stronger between the variables. 

It is the most used type of correlation coefficient. We will be using the Pearson Correlation formula to explain how you can calculate Correlation Coefficient. 

Spearman Correlation:

The Spearman Correlation helps to determine the monotonic nature of the relationship between the variables. This method does not use the dataset that follows a normal distribution. Instead, it uses ordinal variables. 

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Pearson Correlation Coefficient Formula

Sample Correlation Coefficient: 

ρX,Y = cov(X,Y)σXσY

= ∑ni=1(Xi−¯X) (Yi−¯Y)/ ΣnΣi=1(Xi−¯X)2nΣi=1(Yi−¯Y)2

Cov: covariance

σX: standard deviation of x

σY: standard deviation of y

Population Correlation Coefficient: 

rx,y = ni=1(xi−¯x)(yi−¯y) / √ΣnΣi=1(xi−¯x)2√∑nΣi=1(yi−¯y)2

Linear Correlation Coefficient

r = nxy)−(Σx)(Σy) / [nΣx2−(Σx)2][nΣy2−(Σy)2] 

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Correlation Coefficient Formula

This is the formula that you may have seen the most. It is the Pearson Correlation formula applied to a sample. 

Explaining the terms:

  • “n” is the sample size
  • X(i), y (i) are the sample points
  • x̅, ȳ, are the mean of individual data points

When we rearrange the formula, we get:

Further rearranging the formula gives us a simplified formula for Correlation Coefficient:

Here, 

  • s x: sample standard deviation of x
  • s y: sample standard deviation of y

Now with the simplified formula, we will begin the steps to calculate the correlation coefficient.

  1. Calculate the mean of all the “x” data points
  2. Calculate the mean of all the “y” data points
  3. Calculate the standard deviation of the “x” data points and the “y” data points
  4. Substitute the values in [x (i) – x̅]/ s x and [ y(i) – ȳ] / sy
  5. Now multiply the standardized value together
  6. Add the products together 
  7. Divide the sum by n-1
  8. The result will be your correlation coefficient

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