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Correlation Coefficient Correlation Coefficient

Correlation Coefficient: The Key to Variable Insights

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What is a Correlation Coefficient (r)?

A correlation coefficient (r) is a statistical measure of the strength of the relationship between two variables; x and y. There are many different types of correlation coefficients, however, Pearson’s correlation coefficient is most widely used in research. 

Pearson’s correlation coefficient, also known as Pearson’s R, is a correlation coefficient that is generally used in linear regression to find the strength and direction of the linear relationship between two variables.

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Correlation Coefficient (r) Formula

Pearson’s r can be calculated using the following correlation coefficient 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
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How to Interpret the Correlation Coefficient (r)?

The value of r always lies between +1 and -1. Values above 0 indicate a positive relationship; when one value increases, the other increases as well. Values below 0 indicate a negative relationship; when one value increases, the other decreases. When r is 0, there is no correlation between the two variables. 

More precise interpretations can be made by seeing which one of the following values is closest to your correlation coefficient r: 

  • Exactly -1 : Perfect downward sloping (negative) linear relationship 
  • -0.7 : Strong downward sloping (negative) linear relationship 
  • -0.5 : Moderate downward sloping (negative) linear relationship 
  • -0.3 :  Weak downward sloping (negative) linear relationship 
  • 0 : No linear relationship
  • 0.3 : Weak upward sloping (positive) linear relationship
  • 0.5 : Moderate upward sloping (positive) linear relationship
  • 0.7 : Strong upward sloping (positive) linear relationship
  • Exactly +1 : Perfect upward sloping (positive) linear relationship

The following image reflects how different r values are reflected on a scatter plot:

Correlation Coefficient Correlation Coefficient
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Limitations to Pearson’s Correlation Coefficient

The limitations of pearson’s r are:

  • A key limitation of Pearson’s r is that it cannot distinguish between independent and dependent variables. Therefore, also if a relationship between two variables is found, Pearson’s r does not indicate which variable was ‘the cause’ and which was ‘the effect’. 
  • Pearson’s r cannot be used to determine nonlinear relationships. 
  • Pearson’s r does not provide information on the slope of the line, and only indicates the existence and type of relationship. The slope must be found by creating a scatter plot. 

Variables that can be used for Pearson’s Correlation Coefficient

It is important to keep in mind that Pearson’s correlation coefficient connot be used with all types of variables and the two variables must be measured either on the interaval scale or the ratio scale. The variables do not, however, need to be measured on the same scale; Pearson’s r can still be used when one variable is on the interval scale while the other is on the ratio scale. 

Additionally, the two variables do not have to be measured in the same units. For instance, the correlation coefficient r could be used to correlate a person’s height to their food intake, although these are completely different units of measurement (height is measured in feet while food intake is measured in calories).

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FAQs on Correlation Coefficient (r)

A correlation coefficient (r) is a statistical measure that reflects the strength of the relationship between two variables; x and y.  

 r values range between -1 and +1. Values above 0 indicate a positive, or direct, relationship while values below 0 indicate a negative, or indirect, relationship. When r is 0, it indicates that there is no relationship between the two variables.

The two main limitations of Pearson’s R are;

  • It cannot determine the nonlinear relationships between variables
  • It does not distinguish between dependent and independent variables

The two main types of correlation coefficients are Pearson’s correlation coefficient (Pearson’s R) and Spearman’s correlation coefficient (Spearman’s p). Pearson’s R indicates the strength and direction of the linear relationship between two variables while Spearman’s p indicates the strength and direction of the monotic relationship between two variables.

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