Question

Ranks obtained by 5 students in two tests are as follows:

Then the Spearman's rank correlation coefficient \( (\rho) \) is:

• A. \( 0.6 \)
• B. \( 0.4 \)
• C. \( 0.2 \)
• D. \( 0 \)

Hint:

Always ensure \( \sum d = 0 \) as a quick check before squaring. If the sum of differences is not zero, there is a calculation error in the individual differences.

Answer 1

The Correct Option is A

Step 1: Treat the ranks as ordinary numbers and use the Pearson correlation formula.
The rank pairs are \( (1,2), (3,1), (2,3), (4,5), (5,4) \). Since Spearman's coefficient is mathematically the same as Pearson's correlation applied to ranks, it can be computed that way instead. The mean of both rank columns is \( 3 \).

Step 2: Work out deviations from the mean and their products.
Deviations in \( X \): \( -2, 0, -1, 1, 2 \). Deviations in \( Y \): \( -1, -2, 0, 2, 1 \). \[ \sum xy = 2+0+0+2+2 = 6, \quad \sum x^2 = 10, \quad \sum y^2 = 10 \]

Step 3: Compute the correlation.
\[ \rho = \frac{\sum xy}{\sqrt{\sum x^2 \cdot \sum y^2}} = \frac{6}{\sqrt{10 \times 10}} = \frac{6}{10} = 0.6 \] \[ \boxed{0.6} \]