The coefficient of rank correlation in a beauty contest of 10 candidates by two judges A & B was found to be 0.5. If it was later discovered that the difference in rank of one candidate was wrongly taken as 3 instead of 7. The corrected coefficient of rank correlation is?
(A) 0.2

Question

The coefficient of rank correlation in a beauty contest of 10 candidates by two judges A & B was found to be 0.5. If it was later discovered that the difference in rank of one candidate was wrongly taken as 3 instead of 7. The corrected coefficient of rank correlation is?
(A) 0.26
(B) 0.32
(C) 0.49
(D) 0.93

Accepted Answer

> [!TLDR]
> Spearman's rank correlation $R = 1 - \dfrac{6\sum d^2}{n(n^2-1)}$; fix $\sum d^2$ for the misread difference and recompute.

### Step 1 — find the original Σd².
With $n = 10$, $n(n^2-1) = 10 	imes 99 = 990$ and $R = 0.5$:
$$0.5 = 1 - \frac{6\sum d^2}{990} \Rightarrow \frac{6\sum d^2}{990} = 0.5 \Rightarrow \sum d^2 = \frac{0.5 	imes 990}{6} = 82.5.$$

### Step 2 — correct the wrong term.
One difference was recorded as 3 instead of 7. Remove $3^2 = 9$ and add $7^2 = 49$:
$$\sum d^2_{	ext{corrected}} = 82.5 - 9 + 49 = 122.5.$$

### Step 3 — recompute R. …

Student	Judge X rank	Judge Y rank	$d$	$d^{2}$
A	1	2	$-1$	1
B	2	1	$1$	1
C	3	4	$-1$	1
D	4	3	$1$	1
E	5	5	$0$	0

Spearman's Rank Correlation

How it works

Interpreting and adjusting

Quick example

Step 1 — find the original Σd².

Step 2 — correct the wrong term.

Step 3 — recompute R. …

Step 1 — find the original Σd².

Step 2 — correct the wrong term.

Step 3 — recompute R. …

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