Linear Correlation Coefficient Linear Correlation Coefficient

Linear Correlation Coefficient

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What is a Linear Correlation Coefficient?

The linear correlation coefficient, also known as Pearson’s correlation coefficient or Pearson’s r, is a value that reflects the strength and direction of the linear relationship between two variables, x and y. This value is calculated by finding the ratio of the covariance between the two variables and the product of their standard deviations. It is a normalised measurement of covariances where the result (value of r) always lies between -1 and 1). 

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Assumptions of the Pearson’s Correlation Coefficient

Pearson’s correlation coefficient makes certain assumptions. If even one of these assumptions is not met, the test cannot be accurately conducted and an alternative correlation test should be used instead. 

These are the assumptions made by Pearson’s Correlation Coefficient: 

  • The sample is selected randomly and is representative of the target population.
  • Both the variables are measured on a continuous scale; the interval or ratio scale.
  • The data contains paired samples and therefore each subject has both x and y variable values. 
  • There must be independence of observations; values of variables between subjects should have no relationship.
  • There is a linear association between the variables.
  • There are no outliers present in the data.
  • The x and y variables follow an (approximately) normal distribution.

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Types of Correlations

The correlation coefficient is reflected by Pearson’s r. The value of r can range between -1 and 1, where;

  • Positive values indicate a Positive Correlation (0<r1)
  • Negative values indicate a Negative Correlation  (-1r<1)
  • A Value of 0 indicates No Correlation  (r=0)

Positive Correlation

A positive correlation exists between two variables when they both move in the same direction; if one increases, the other does too and if one decreases, the other decreases too. When r takes a positive value, it indicates a positive correlation. The closer the value is to 0, the weaker the positive relationship between the variables. The closer the value is to 1, the stronger the positive relationship between the variables.  

Negative Correlation

A negative correlation exists between two variables when they both move in the opposite directions; if one increases, the other decreases too and if one decreases, the other increases. When r takes a negative value, it indicates a negative correlation. The closer the value is to 0, the weaker the negative relationship between the variables. The closer the value is to -1, the stronger the negative relationship between the variables.  

No Correlation

When two variables have no statistical association, they are said to have no correlation. In this case, their correlation coefficient (also known as r) is 0. 

Correlation Correlation Coefficient Formula

The most widely used correlation coefficient formula in the study of statistics is Pearson’s correlation coefficient formula because it is easy to use and interpret. The formula allows us to calculate the value of which, as we’ve understood, denotes the strength and correlation of the linear relationship between two variables. 

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

Where, 

  • n = Total number of samples
  • i=1nxi = Total of the x values
  • i=1nyi = Total of the y values
  • i=1nxi yi = Sum of the product of x and y
  • i=1nxi2= Sum of the squares of x
  • i=1nyi2 = Sum of the squares of y

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Interpreting the Strength of r

The following table reflects broad cut-offs that can be used to interpret strength the value of r, whether negative or positive:

Value of r (Correlation Coefficient)

Interpretation

0.00 – 0.10

No Correlation

0.10 – 0.39

Weak Correlation

0.40 – 0.69

Moderate Correlation

0.70 – 0.89

Strong Correlation

0.90 – 1.00

Very Strong Correlation

FAQs on Linear Correlation Coefficient

The linear correlation coefficient, also referred to as Pearson’s correlation coefficient, is a value that indicates the strength and direction of the linear relationship between two variables; x and y. 

The most widely used correlation coefficient in statistics is the Pearson’s correlation coefficient formula, which is;

The correlation coefficient between two variables is denoted by r. R can range between the values -1 and 1, where positive values indicate a positive correlation, negative values indicate a negative correlation, and 0 indicates no correlation.

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