Statistics How To

Spearman Rank Correlation (Spearman’s Rho): Definition and How to Calculate it

Correlation Coefficients >
Spearman Rank Correlation / Spearman’s Rho

What is Spearman Rank Correlation / Spearman’s Rho?

The Spearman rank correlation coefficient, rs, is the nonparametric version of the Pearson correlation coefficient. Your data must be ordinal, interval or ratio. Spearman’s returns a value from -1 to 1, where:
+1 = a perfect positive correlation between ranks
-1 = a perfect negative correlation between ranks
0 = no correlation between ranks.

Contents:

  1. No Tied ranks example.
  2. What to do with tied ranks.

Spearman Rank Correlation: Worked Example (No Tied Ranks)

The formula for the Spearman rank correlation coefficient when there are no tied ranks is:
rank correlation coefficient formula


Sample Question:
The scores for nine students in physics and math are as follows:
Physics: 35, 23, 47, 17, 10, 43, 9, 6, 28
Mathematics: 30, 33, 45, 23, 8, 49, 12, 4, 31
Compute the student’s ranks in the two subjects and compute the Spearman rank correlation.

Step 1: Find the ranks for each individual subject. I used the Excel rank function to find the ranks. If you want to rank by hand, order the scores from greatest to smallest; assign the rank 1 to the highest score, 2 to the next highest and so on:
spearman rank correlation 1

Step 2: Add a third column, d, to your data. The d is the difference between ranks. For example, the first student’s physics rank is 3 and math rank is 5, so the difference is 3 points. In a fourth column, square your d values.
spreaman 2

Step 4: Sum (add up) all of your d-squared values.
4 + 4 + 1 + 0 + 1 + 1 + 1 + 0 + 0 = 12. You’ll need this for the formula (the Σ d2 is just “the sum of d-squared values”).

Step 5: Insert the values into the formula. These ranks are not tied, so use the first formula:
rank correlation coefficient formula

= 1 – (6*12)/(9(81-1))
= 1 – 72/720
= 1-0.1
= 0.9
The Spearman Rank Correlation for this set of data is 0.9.

Spearman Rank Correlation: What to do with Tied Ranks

Tied ranks are where two items in a column have the same rank. Let’s say two items in the above example tied for ranks 5 and 6. The following image shows each tied data point assigned a mean rank of 5.5:
tied-ranks

When this happens, you have a couple of options. You could also use the easier formula for tied ranks *if* you only have one or two tied ranks here and there. The image above shows the workings for the ties and the d-squared values you’ll need to input into the simple version of the formula above. However, that option may leave you with little confidence in any p-values you produce (Kinnear and Gray, 1999). A better option may be to calculate correlation with another method, like Kendall’s Tau.

Another option is simply to use the full version of Spearman’s formula (actually a slightly modified Pearson’s r), which will deal with tied ranks:

Full Spearman’s r formula (Cleff, 2013. p. 4)


Where:

  • R(x) and R(y) are the ranks,
  • R(x)bar and R(y)bar are the mean ranks.

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References

Clef, T. (2013). Exploratory Data Analysis in Business and Economics: An Introduction Using SPSS, Stata, and Excel. Springer Science and Business Media.
Kinnear and Gray (1999). SPSS for Windows Made Simple. Taylor and Francis.
Rees, D. (2000). Essential Statistics. CRC Press.

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