A Comprehensive Guide to Correlation Coefficient Analysis

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. 

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. 

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. 

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: 

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

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

Population correlation coefficient

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

“ρ” is used for population coefficient

“rs “ is 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