Rank Correlation Coefficient

In statistics, the Rank Correlation Coefficient measures the strength and direction of the relationship between two sets of ranked data. It is denoted by the symbol 'r' and ranges from -1 to +1, where -1 indicates a perfect negative relationship, +1 indicates a perfect positive relationship, and 0 indicates no relationship.

Calculation of Rank Correlation Coefficient

To calculate the Rank Correlation Coefficient, we need two sets of ranked data. Let's assume the two sets of data are X and Y, where X represents the ranks given by one examiner and Y represents the ranks given by another examiner. The steps to calculate the Rank Correlation Coefficient are as follows:

1. Assign ranks to each data point in both sets. If there are any ties, assign the average rank to the tied data points.
2. Calculate the difference between the ranks of each pair of corresponding data points. Let's denote this as 'd'.
3. Calculate the sum of the squared differences, denoted as 'd^2'.
4. Calculate the Rank Correlation Coefficient using the formula:

r = 1 - ((6 * d^2) / (n * (n^2 - 1)))

where 'n' is the number of data points.

Example

Let's consider the marks awarded by two examiners for 10 students:

Examiner 1 (X) : 80, 70, 90, 60, 75, 85, 95, 65, 80, 75
Examiner 2 (Y) : 70, 75, 80, 65, 70, 90, 85, 60, 80, 75

Step 1: Assign ranks

Examiner 1 (X) : 6, 3, 9, 1, 4, 7, 10, 2, 6, 4
Examiner 2 (Y) : 4, 6, 8, 2, 4, 9, 7, 1, 6, 4

Step 2: Calculate differences

d = (6-4), (3-6), (9-8), (1-2), (4-4), (7-9), (10-7), (2-1), (6-6), (4-4)
= 2, -3, 1, -1, 0, -2, 3, 1, 0, 0

Step 3: Calculate sum of squared differences

d^2 = 2^2 + (-3)^2 + 1^2 + (-1)^2 + 0^2 + (-2)^2 + 3^2 + 1^2 + 0^2 + 0^2
= 4 + 9 + 1 + 1 + 0 + 4 + 9 + 1 + 0 + 0
= 29

Step 4: Calculate Rank Correlation Coefficient

n = 10

r = 1 - ((6 * 29) / (10 * (10^2 - 1)))