The essays of 10 candidates were ranked by three examiners as shown in the table. candidates A B C D E F G H I J Examiner I 1st 3rd 6th 2nd 10th 9th 7th 4th...
Question 1 Report
The essays of 10 candidates were ranked by three examiners as shown in the table.
| candidates | A | B | C | D | E | F | G | H | I | J |
| Examiner I | 1st | 3rd | 6th | 2nd | 10th | 9th | 7th | 4th | 8th | 5th |
| Examiner II | 2nd | 1st | 3rd | 9th | 7th | 4th | 8th | 10th | 5th | 6th |
| Examiner III | 3rd | 2nd | 1st | 6th | 9th | 8th | 7th | 5th | 4th | 10th |
a) Calculate the Spearman's rank correlation coefficient of the ranks assigned by:
(i) Examiners I and lI;
(ii) Examiners I and III
(iii) Examiners II and II.
(b) Using the results in (a), state which two examiners agree most.
Answer Details
Convert each ordinal rank to a number and apply \(r_s=1-\dfrac{6\sum d^2}{n(n^2-1)}\) with \(n=10\), so \(n(n^2-1)=990\).
| Cand. | I | II | III |
|---|---|---|---|
| A | 1 | 2 | 3 |
| B | 3 | 1 | 2 |
| C | 6 | 3 | 1 |
| D | 2 | 9 | 6 |
| E | 10 | 7 | 9 |
| F | 9 | 4 | 8 |
| G | 7 | 8 | 7 |
| H | 4 | 10 | 5 |
| I | 8 | 5 | 4 |
| J | 5 | 6 | 10 |
(a)(i) Examiners I and II. \(d=I-II:\ -1,2,3,-7,3,5,-1,-6,3,-1\); \(\sum d^2=1+4+9+49+9+25+1+36+9+1=144\).\[r_s=1-\frac{6(144)}{990}=1-\frac{864}{990}=0.127\]
(a)(ii) Examiners I and III. \(d=I-III:\ -2,1,5,-4,1,1,0,-1,4,-5\); \(\sum d^2=4+1+25+16+1+1+0+1+16+25=90\).\[r_s=1-\frac{6(90)}{990}=1-\frac{540}{990}=0.455\]
(a)(iii) Examiners II and III. \(d=II-III:\ -1,-1,2,3,-2,-4,1,5,1,-4\); \(\sum d^2=1+1+4+9+4+16+1+25+1+16=78\).\[r_s=1-\frac{6(78)}{990}=1-\frac{468}{990}=0.527\]
(b) Agreement. The largest coefficient is \(0.527\), so Examiners II and III agree most.
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