Correlation Coefficient Formula Correlation Coefficient

Correlation Coefficient Formula

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What is a Correlation Coefficient?

A correlation coefficient is a statistical measure of the strength of the linear relationship between two variables, x and y. A linear correlation coefficient that has a value greater than 0 denotes a positive relationship; as one variable increases, the other increases as well. A value less than 0 denotes a negative relationship; as one value increases, the other decreases (or vice versa). A value of 0 denotes that the two variables have no relationship and, therefore, one does not influence the other.

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Types of Correlation Coefficients

There are two main types of correlation coefficients: 

  • Pearson’s Correlation Coefficient: Pearson’s correlation coefficient is the most commonly used correlation coefficient. It measures the strength and direction of the linear relationship between two variables.
  • Spearman’s Rank-Order Correlation Coefficient: Spearman’s rank-order correlation is a nonparametric measure of rank correlation. It uses a monotonic function to describe the strength of the relationship between two variables.

Pearson’s Correlation Coefficient

Pearson’s correlation coefficient, also referred to as Pearson’s r, measures the strength of the linear correlation between two variables. It can take on values ranging from +1 to – 1 where values more than 0 indicate a positive linear correlation, values below 0 indicate a negative linear correlation, and  a value of 0 indicates no correlation. 

The following image demonstrates how Pearson’s r varies as the strength and relationship between the two variables change: 

Correlation Coefficient Formula Correlation Coefficient

Pearson’s Correlation Coefficient Formula

Pearson’s r can be calculated using the following formula:

pxy =Cov(x,y)xy

Where, 

  • pxy : Pearson product-moment correlation coefficient
  • Cov(x,y): Covariance of variables x and y
  • x : Standard deviation of x
  • y : Standard deviation of y

Limitations of Pearson’s Correlation Coefficient

Let’s take a look at some of the limitations of Pearson’s R:

  • It cannot be used to identify nonlinear relationships between two variables.
  • It does not differentiate between dependent and independent variables. Therefore, if a linear relationship is identified between x and y, the correlation coefficient does not indicate which variable is ‘the cause’ and which is ‘the effect’.

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Spearman’s Rank Correlation Coefficient

Spearman’s rank-order correlation coefficient, also referred to as Spearman’s p, is a nonparametric measure of rank correlation. Like Pearson’s r, Spearman’s p is also used to identify the strength and direction of the linear relationship between two variables. However, the key difference between the two is that Spearman’s correlation coefficient determines the strength and direction of the monotonic relationship between two variables while Pearson’s correlation coefficient determines the strength and direction of the linear relationship between two variables. 

A monotonic relationship is a relationship that does one of the following:

  1. As the value of one variable increases, the value of the other increases as well.
  2. As the value of one variable increases, the value of the other decreases.

The following image illustrates how monotonic and non-monotonic relationships are presented: 

Correlation Coefficient Formula Correlation Coefficient

Spearman’s Correlation Coefficient Formula

Spearman’s p can be calculated using the following formula:

p=1-6di2n(n2-1)

Where, 

  • p : Spearman’s rank correlation coefficient
  • di : Difference between the two ranks of each observation
  • n : Number of observations

Limitations of Spearman’s Correlation Coefficient

Let’s take a look at some of the limitations of Spearman’s p:

  • Less sensitive to strong outliers relative to Pearson’s r.
  • Uses rank observations rather than the actual values of the observations, making its outcomes less reliable.

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FAQs on Correlation Coefficient Formula

A correlation coefficient is a statistical measure of the strength of the linear relationship between two variables, x and y.

There are two main types of correlation coefficients; Pearson’s correlation coefficient and Spearman’s correlation coefficient. 

Pearson’s correlation coefficient is a statistical measure of the strength of the linear relationship between two variables. It also indicates the direction of the relationship (positive/negative) between the variables. 

 A few limitations of Pearson’s correlation coefficient are;

  • Cannot be used to identify nonlinear relationships between two variables.
  • Does not differentiate between dependent and independent variables.

Spearman’s rank correlation coefficient is a nonparametric measure of rank correlation. Unlike Pearson’s correlation coefficient, which measures the strength and direction of linear relationships, Spearman’s correlation coefficient measures the strength and direction of monotonic relationships.

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