Correlation: Correlation Coefficient, Types and Formulas

Correlation can be defined as the statistical measure of a relationship between any data involving two variables. It is also known as dependence. Correlation is the most used method to establish a relationship between two variables. Specifically, correlation is a linear relationship between two variables. Two variables can be directly correlated or indirectly correlated. Correlation is of primarily three types, positive correlation, negative correlation and no correlation.

Also Read: Probability and Statistics

Key Terms: Statistics, Correlation Coefficient, Variables, Covariance, Data, Population, Variance, Standard Deviation

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What is Correlation?

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A measurement which is used to quantify a relationship between variables is known as Correlation. 

• Two variables are said to be directly correlated if an increase or decrease in one variable causes a simultaneous increase or decrease in another. 
• If an increase in one causes a decrease in another or vice versa, the variables are known to be indirectly correlated. 
• If a change in an independent variable does not cause any change in the dependent variable, they are said to be uncorrelated.

Correlation measures the direction and the extent of a relationship between variables.

Correlation 

Calculation of Correlation

Correlation can be calculated following the steps given below.

• The first step is to determine the variance of the variables. 
• The second step is to divide that quantity by the product of those variables' standard deviations.

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Correlation Coefficient

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The extent of the statistical relationship, between two ratio level variables is described by the correlation coefficient. It is denoted by r.

• The correlation coefficient is always between -1 and +1. 
• When r is close to 0 it means there is little relationship between the variables 
• When r is away from 0 it is, in either the positive or negative direction.
• The two variables are often given the symbols X and Y. 

Also Read: Probability

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Scatter Diagram

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A diagram that shows the values of two variables X and Y, representing how they relate to each other, is known as a scatter diagram. Scatter diagrams provide a visual representation of the relationship between variables. It displays how closely two variables are related.

The values of variables X and Y are given on the horizontal and vertical axis respectively.

Also Read: Mode Formula

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Types of Correlation

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There are three types of relations between variables and so, three types of correlation are described individually below.

Types of Correlation

Positive Correlation

Positive correlation exists when the values of the two variables move in the same direction such that if there is an increase/decrease in the value of one variable, it is followed by an increase/decrease in the value of the other variable.

Negative Correlation

Negative correlation exists when the values of the two variables move in the opposite direction such that if there is an increase/decrease in the value of one variable, it is followed by a decrease/increase in the value of the other variable.

No Correlation

When two variables are completely independent and there is no relation between them, no correlation is said to exist between them.

Also Read: Mean, Median and Mode 

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Formulas of Correlation

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There are four types of correlation coefficient formulas, namely:

• Pearson Correlation Coefficient Formula
• Linear Correlation Coefficient Formula
• Sample Correlation Coefficient Formula
• Population Correlation Coefficient Formula

Each of the formulas mentioned above is discussed below briefly.

Pearson Correlation Coefficient Formula

Pearson correlation coefficient is the most used formula, especially for linear dependency between the data sets. If we get the value of +1, the data are positively correlated, and -1 has a negative correlation.

The formula for the Pearson correlation coefficient is given as:

r = [n(Σxy) - (Σx)(Σy)]/√[nΣx² - (Σx)²][nΣy² - (Σy)²]

Where 

n = Quantity of Information

Σx = Total of the First Variable Value

Σy = Total of the Second Variable Value

Σxy = Sum of the Product of first & Second Value

Σx² = Sum of the Squares of the First Value

Σy² = Sum of the Squares of the Second Value

Linear Correlation Coefficient Formula

The formula for the linear coefficient is given as:

r_xy = {n Σx_iy_i - Σx_i Σy_i}/√n Σx_i² - ( Σx_i)² √n Σy_i² - ( Σy_i)²

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Sample Correlation Coefficient Formula

The formula for the sample correlation coefficient is given by:

r_xy = S_xy/S_xS_y

Where 

S_xy is the sample covariance

S_x and S_y are the sample standard deviations

Population Correlation Coefficient Formula

The population correlation coefficient formula is given as: 

r_xy = σ_xy/σ_xσ_y

Where

σ_x and σ_y are the population standard deviations 

σ_xy is the population covariance

Also Read:  

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Examples of Correlation

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To understand the concept of the correlation coefficient, an example is illustrated below. 

In the table, Years of Education and Age of Entry to Labour Force gives the number of years of formal education (X) and the age of entry into the labour force (Y ). Both variables are measured in years; all males are aged close to 30.

As we can see that most males enter the labour force soon after they leave formal schooling. Thus, a close relationship between these two variables is expected.

Also Read: Mode of Grouped Data

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Things to Remember

• Correlation, also known as dependence, is the most widely used method to establish a relationship between two variables. 
• It can be defined as a measurement to quantify the relationship between two variables.
• Correlation coefficient tells about the extent of the statistical relationship between two variables.
• Correlation coefficient is always between +1 to -1.
• When there is no correlation between two variables, they are completely independent of each other.