Statistics Definitions > Spearman Rank Correlation / Spearman’s Rho

## What is Spearman Rank Correlation / Spearman’s Rho?

The Spearman rank correlation coefficient, r_{s}, 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**:

## Spearman Rank Correlation: Worked Example (No Tied Ranks)

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

**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:

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.

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 d^{2} 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:

= 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:

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:

**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.

If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

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Thank you for the step by step guidelines. I’m beginning to enjoy my subject because of your assistance. VerY stunning, correct an prompt.

I’m beginning to enjoy my subject because of your assistance. VerY stunning, correct an prompt.

thanks for the help If three judges are judging contestants, using spearmans rank correlaton how do u find the best Or the top three winners in the contest of 7 people

This website has helped me so so much with my work. Amazingly straight forward

Also advice how to calculate number of observations (n) when we have the value of Co-relation (R)

If you have just r, n could be anything. For example, a set of 100, 1000, or 10000 numbers could all have the same r value. I think you’d need a little more info to come up with n.

I successfully worked through the above example, using your step by step. However, I have tied ranks in my dataset, and am unable to decipher the formula for that. Do you have a step-by-step for data with tied ranks?

Penny,

Not yet — it’s in the plan for the future though.

The formula is very similar to how you work the Pearson Correlation (with summations of differences). It’s very cumbersome — good luck :)

Thanks but how I wish there was a worked out example with tied scores!

hi i have one question regarding to spearman rank correlation .how can i compute for the given three rank using spearman rank correlation coefficient in order to find the nearest approach

Hello, Semir,

Could you rephrase your question, please? I don’t understand quite what you mean when you say “find the nearest approach”.

Zippie: I added an example. Hope it helps :)

Thnk u soooo much it’s really very helpful

thankyou so much for the information.it helps a lot..but would you mind explain further on how to get the /d/, for an instance the X variable is 5 and Y variable is 7 why is that the d squared is positive not negative.?

Do you mean d, the difference between ranks? D squared will always result in a positive number because a negative times a negative is a positive.

The formula used for tied ranks is not correct. The correct formula is :. 1-6* summation D^2/N^3-N + 1/2(m^3-m)+………+….n no. of times rank repeated + all the repeated ranks, where m is combined repeated rank.

Thanks for your comment, Sarfaraz. Could you link me to a reference/source? Thank you.