Question

Let \( (1, 3), (2, 4), (7, 8) \) be three independent observations. Then the sample Spearman rank correlation coefficient based on the above observations is ________ (rounded off to two decimal places).

Hint:

For Spearman's rank correlation, rank the data and compute the differences in ranks. If the ranks are perfectly correlated, the coefficient will be 1.

Answer 1

Step 1: Rank the data. 
We have three observations: \( (1, 3) \), \( (2, 4) \), and \( (7, 8) \). We rank the \( x \)-values and the \( y \)-values separately:
\( x \)-values: \( 1, 2, 7 \) → ranks: \( 1, 2, 3 \)
\( y \)-values: \( 3, 4, 8 \) → ranks: \( 1, 2, 3 \)
So, the ranked data is: \[ {Ranks for } x: (1, 2, 3), \quad {Ranks for } y: (1, 2, 3). \] Step 2: Compute the differences in ranks. 
The rank differences \( d_i \) for each observation are: \[ d_1 = 1 - 1 = 0, \quad d_2 = 2 - 2 = 0, \quad d_3 = 3 - 3 = 0. \] Step 3: Compute the Spearman rank correlation coefficient. 
The Spearman rank correlation coefficient \( \rho \) is given by: \[ \rho = 1 - \frac{6 \sum_{i=1}^n d_i^2}{n(n^2 - 1)}, \] where \( n = 3 \) is the number of observations. Since all \( d_i = 0 \), the sum of squared rank differences is \( \sum d_i^2 = 0 \). 
Therefore: \[ \rho = 1 - \frac{6 \times 0}{3(9 - 1)} = 1. \] Thus, the sample Spearman rank correlation coefficient is \( \boxed{1.00} \).