Question

Two persons taste 10 red Italian wines, grading them by an ordinal scale between 1 and 5. The results
are as follows:
Grades Grades
Wine No. Person 1 Person 2
1 1 2
2 2 3
3 4 5
4 5 4
5 2 2
6 2 2
7 4 3
8 3 4
9 1 3
10 4 2
(a) Find the Spearman’s rank correlation by
i. the definition;
ii. using the formula
rs = 1 − 6(sum of i to n di^2) / n(n^2-1)
Note that this only gives an approximation when there are tied data.
(b) Test whether Spearman’s rho is significantly different from zero.

Answer 1

In order to calculate Spearman's rank correlation coefficient we first have to sort the grades for each person. The resulting rank numbers are averaged for tied observations:

                                Grades     Rank  Rank
                    Wine No.    Person 1        (ties)
                       1           1         1    1.5
                       9           1         2    1.5
                       6           2         3     4
                       5           2         4     4
                       2           2         5     4
                       8           3         6     6
                      10           4         7     8
                       7           4         8     8
                       3           4         9     8
                       4           5        10    10


                                 Grades    Rank  Rank
                    Wine No.    Person 2        (ties)
                       5           2         1    2.5
                       1           2         2    2.5
                      10           2         3    2.5
                       6           2         4    2.5
                       2           3         5     6
                       7           3         6     6
                       9           3         7     6
                       8           4         8    8.5
                       4           4         9    8.5
                       3           5        10    10

The final table of ranks includes the differences of the ranks as well as the squared differences:

Wine No.    Person 1  Rank       Person 2  Rank      Rank      squared
                                                   Difference   Diff.
   1           1       1.5         2       2.5        -1          1
   2           2        4          3        6         -2          4
   3           4        8          5       10         -2          4
   4           5       10          4       8.5        1.5        2.25
   5           2        4          2       2.5        1.5        2.25
   6           2        4          2       2.5        1.5        2.25
   7           4        8          3        6          2          4
   8           3        6          4       8.5       -2.5        6.25
   9           1       1.5         3        6        -4.5       20.25
  10           4        8          2       2.5        5.5       30.25
------------------------------------------------------------------------
                                               sum    0.0       76.50

The sum of the squared differences is used to calculate Spearman's rank correlation coefficient as follows:

r_s = 1 - (6*76.5/(10*(100-1)) = 0.5364.