Question on Spearman's Rank Correlation Coefficient

Question

I'm doing some practice questions in my statistics book, and started doing this one:

Find Spearman's rank correlation coefficient between X and Y for this set of data:

$X$  $13$  $20$  $22$  $18$  $19$  $11$  $10$  $15$  
$Y$  $17$  $19$  $23$  $16$  $20$  $10$  $11$  $18$

I set out the data in a table, and found the difference between each value.

$d$      $4$  $1$  $1$  $2$  $1$  $1$  $1$  $3$  
$d^2$  $16$  $1$  $1$  $4$  $1$  $1$  $1$  $9$  

From this we can see that $\Sigma d^2 = 34$

I then calculated $r_s$ using the formula $r_s = 1- \frac{6\Sigma d^2}{n(n^2 -1)}$:

$r_s = 1- \frac{6\Sigma d^2}{n(n^2 -1)}$

     $=1 - \frac{6*34}{8*63}$

     $= 1- \frac{204}{504}$

     $\approx 0.5952$

However, in the book the answer is given as $0.881$. So, am I wrong or is the book wrong?

Answer 1

To calculate spearman's rank correlation coefficient, you need to first convert the values of X and Y into ranks. For example in the X values, you should replace the lowest value (10) with a 1, then the second lowest (11) with a 2 until the largest (22) is replaced with 8.

Once you have done this to both the X and Y values, you can proceed with the method as above.

Hope that helps!

Answer 2

It looks like you have looked at the differences in the tabulated values to work out the $d$ values. But you're supposed to rank the values and work out the differences in the rankings.

When you do this, you will find that the $d$ values are $$1,1,0,2,1, 1,1,1. $$

This will give the correct value for $r_s$