Correlation Coefficient Correlation Coefficient

Correlation Coefficient

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The correlation coefficient is a statistical analysis method that is used to measure the strength and the direction of the relationship between two variables. Or, it can also be said that correlation analysis in research helps us to measure the change in one variable caused by the change in other variables. 

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Characteristics of Correlation Coefficient:

  • The values of the correlation coefficient (r) range from –1.0 to + 1.0. 
  • The value of the correlation coefficient describes the strength of the relationship between the two variables. 
  • When values are closer to “r = +/- 1.0” it indicates a stronger relationship between the two variables. 
  • When the correlation coefficient is close to “zero” it indicates that the relationship between the variables is weak.
  • The direction of the relationship between the two variables is described as positive or negative. 
  • The positive “+” sign indicates that the values for both variables change in the same direction
  • The negative “- “ sign shows that the values change in opposite directions
  • The closer the data is to a regression line, the stronger is the relationship between the two variables. 
Correlation Coefficient Correlation Coefficient

A Positive correlation means that when the value of one variable increases the value of the second variable also increases. 

A negative correlation means that when the value of one variable increases, the value of the second variable decreases. 

A zero correlation indicates that there is no relationship between the two variables.

Visualizing Correlation Coefficient

You can use a scatter plot to graphically display the strength and direction of the relationship between two variables. The pair of values are plotted along the axes – x and y – to study the pattern that emerges. 

The relation between the values is determined by how far the data points fall from the regression line. The correlation coefficient indicates how closely the data fit on the line. 

The regression line is the best fitting line in a scatter plot, which takes all the data points into account. When your data points are closer to the straight line, the absolute value is higher and the linear relationship is stronger. 

Perfect correlation: all the data points are on the regression line

High correlation coefficient: all data points are closer to the straight line

Correlation Coefficient Correlation Coefficient

Low correlation coefficient: all data points are spread far away from the line

Correlation Coefficient Correlation Coefficient

Interpretation of Correlation Coefficient

There are many approaches suggested for the interpretation of the correlation coefficient. Descriptors like “strong”, “moderate”, or “weak” are used to translate the relationship. As a guideline, you can use the table to interpret the strength of the relationship from the value of the correlation coefficient. 

Correlation Coefficient

Strength

Type

0.7 to 1.0 

Very Strong

Positive

0.5 to 0.7

Strong

Positive

0.3 to 0.5

Moderate

Positive

0 to 0.3

Weak

Positive

0

None

Zero 

0 to – 0.3

Weak

Negative

-0.3 to -0.5

Moderate

Negative

-0.5 to -0.7

Strong

Negative

-0.7 to -1.0

Very Strong

Negative

 

The value of the correlation coefficient ranges between +1.0 to – 1.0. The value is an indicator of the strength of the relationship between two variables. 

The sign – positive and negative – indicates whether the change in the variables is in the same or opposite direction. 

Absolute value: is the number without its sign. It reflects the magnitude of correlation. If the absolute is greater, then the correlation is stronger. 

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Types of Correlation Coefficient

There are several correlation coefficients you can choose from depending on the linearity of the relationship, the level of measurement, and the distribution of data. 

 

Correlation Coefficient

Relationship

Levels of Measurement

Distribution

Pearson’s r

Linear

Two Quantitative variables – interval or ratio

Normal distribution

Spearman’s rho

Non-linear

Two Ordinal Variables – interval or ratio

Any distribution

Cramer’s V

Non-linear

Two Nominal Variables

Any distribution

Kendall’s tau

Non-linear

Two Ordinal Variables

Any distribution

 

The most common correlation coefficient used in research is Pearson’s r. It is parametric, allows for strong inferences, and measures linear correlation. However, there are certain assumptions in Pearson’s r. The data for your research needs to meet these assumptions, and in case it doesn’t you need to use a non-parametric test. 

Spearman’s rho or Kendall’s tau can be used for non-parametric tests. Kendall’s tau is a preferred choice for small samples. Spearman’s rho is used for wide samples. 

Pearson’s r

Pearson’s r is used to interpret the relationship between two quantitative variables. It cannot be used if your variables have a nonlinear relationship. 

There are certain assumptions that the data needs to meet in order to use Pearson’s r: 

  • The two variables must be on Interval or Ratio Level of Measurement
  • Data from the two variables must follow a normal distribution
  • The data must not have outliers
  • The data should be from a representative or random sample
  • There must be a linear relationship between both the variables

The formula of Pearson’s r is: 

 
  • rxy= strength of the correlation between variables x and y
  • n = sample size
  • ∑ = sum of what follows…
  • X = every x-variable value
  • Y = every y-variable value
  • XY = the product of each x-variable score and the corresponding y-variable score

 

Most software can quickly work out the formula and generate the correlation coefficient from your data. 

To calculate the “r”, first the covariance of the variables is determined. Then, the resulting quantity is divided by the product of the standard deviation of those variables. 

Pearson sample and Pearson population:

When you have decided to use the formula of Pearson’s r, you also need to decide upon whether you are working with data from a sample or from the population. Both the sample and population have a different formula with different symbols and inputs. 

“r” is used for the formula of sample correlation coefficient

“rho” or Greek letter “ρ” is used for the population correlation coefficient

Sample correlation coefficient

 
  • rxy= strength of the correlation between variables x and y
  • cov(x,y) = covariance of x and y
  • sx = sample standard deviation of x
  • sy = sample standard deviation of y

 

The formula uses the sample covariance between the variables and the sample standard deviation. 

Population correlation coefficient

 
  • ρXY= strength of the correlation between variables X and Y
  • cov(X,Y) = covariance of X and Y
  • σX = population standard deviation of X
  • σY = population standard deviation of Y

 

It uses the population covariance between the variables and the population standard deviation. 

Spearman’s rho:

Spearman’s rho, also called Spearman’s rank correlation coefficient, is used for non-parametric tests. It is the commonly used alternative of Pearson’s r. 

It is called a rank correlation coefficient because instead of using the raw data, it uses the ranking of the data from each variable. Spearman’s rho is generally used when one of the variables is on an ordinal level of measurement or when the variables do not follow a normal distribution.

Spearman’s rho examines the monotonicity of relationships. It is used when the relationship between the variables is non-linear. The characteristic of a monotonic relationship is that each variable changes in one direction but not at the same rate. 

Positive Monotonic indicates that when one variable increase the second variable also increases

Negative Monotonic indicates that one variable increases the other variable decreases

In the case of Spearman’s rank correlation coefficient

 
  • rs= strength of the rank correlation between variables
  • di = the difference between the x-variable rank and the y-variable rank for each pair of data
  • d2i = sum of the squared differences between x- and y-variable ranks
  • n = sample size

 

“ρ” is used for population coefficient

rsis used for sample coefficient

To calculate Spearman’s rho, you first need to rank the data from each variable in the order of lowest to highest. Next, you need to measure the difference between the ranks of the variables for each pair of data and use that as the main input in the formula. 

Correlation coefficient +1: means all the ranks for each variable match for each data pair

Correlation coefficient -1: means the rankings for one variable are the exact opposite of the rankings of the other variable. 

Correlation coefficient near 0: means there is no monotonic relationship

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Features of Correlation Coefficient

  • Descriptive statistics: It helps you to summarize the sample data. There is no need for you to infer anything about the population. 
  • When you have two variables, it is bivariate statistics, and when you have more than two variables, it is multivariate statistics. 
  • Correlation coefficients can also show the practical significance of the result. 
  • You can compare coefficients between studies since it is unit-free. 

Using Correlation Coefficient for Surveys

Correlation coefficients can be used for surveys such as employee satisfaction/ engagement, customer satisfaction, and other types of surveys. 

In market research, the aim of the researcher is to analyze the quantitative data collected from surveys. The researcher uses a correlation coefficient to identify and understand the relationship and trends between two variables. 

FAQs

A Correlation Coefficient “r” describes the strength and direction of the relationship between two variables. The value always ranges between +1 to – 1.

A Positive correlation means that when the value of one variable increases the value of the second variable also increases. 

A Negative correlation means that when the value of one variable increases, the value of the second variable decreases. 

 Zero correlation reflects that there is no relationship between the two variables

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