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Spearman correlation coefficient Spearman correlation coefficient

Spearman correlation coefficient

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What is Spearman correlation coefficient?

Named after Charles Spearman and denoted by a Greek letter ‘ρ’, Spearman correlation coefficient is a nonparametric data analysis technique. It is a measure the strength and direction of the statistical dependence of ranking between two variables. 

It is an appropriate measure to use when the variables are being measure on a least ordinal scale.

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Assumptions for Spearman correlation coefficient

Spearman correlation does not have any assumption about the data distribution, but it is based on Pearson’s correlation assumptions. 

A Pearson’s correlations is a statistical measure of strength between a linear relationship between the variables and its assumptions include:

  • Interval or ratio level
  • Linearly related 
  • Bivariate normally distributed

As for the Spearman correlation coefficient, you can use it when your data does not have the above assumptions.

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Monotonic function in Spearman correlation coefficient

Let us first understand what monotonic function is. A monotonic function, irrespective of the increase in the independent variable, never increases or decreases itself. The graph shown below is the best representation of the monotonic function:

Spearman correlation coefficient Spearman correlation coefficient
  1. Monotonically increasing – as the x variable increases, the y variable never decreases. 
  2. Monotonically decreasing – as the x variable increases, the y variable never increases.
  3. Not monotonic – as the x variable increases, the y variable might increase or decrease.
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Spearman correlation coefficient formula

Spearman correlation coefficient Spearman correlation coefficient

Where,

n is the number of data points of the two variables

di is the difference in ranks of the “ith” element

The Spearman Coefficient (⍴) can take a value between +1 to -1 

Where,

  • A ⍴ value of +1 means a perfect association of rank
  • A ⍴ value of 0 means no association of rank
  • A ⍴ value of -1 means a perfect negative association between ranks.

As closer as the ⍴ value comes to 0, the weaker the association between the two ranks gets.

It is important to rank the data before carrying out the Spearman correlation coefficient, just to observe that the increase in one variable is being followed by the other variable’s monotonous relation. 

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How to calculate Spearman correlation coefficient

In order to understand the working of the Spearman correlation, let’s take an example and discuss the process step-by-step. 

Example: the marks of students in the subjects English and Maths are given as data. 

Step 1: Create a table for the given data

Step 2: Rank the two datasets from highest to lowest. 

In our case, the highest mark obtained will be ranked as “1” and so on. The smallest marks will be given the lowest ranking. This should be done for both English and Maths marks. 

Step 3: Add an extra column for ‘d’ which denotes the difference between the ranks and another column as ‘d square’ which has the square of ‘d’. 

Spearman correlation coefficient Spearman correlation coefficient

Step 4: Add your ‘d square’ values. In our case, ∑d = 12

Step 5: Substitute the values in the Spearman correlation coefficient formula.

=1-(6*12)/(9(81-1))

=1-72/720

=1-01

=0.9

The Spearman rank correlation for our data is 0.9

As mentioned above, the ⍴ value is nearing the +1, they have the perfect association of rank in the data. 

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