Calculating Spearman's Correlation Coefficient for X and Y Data

This solution demonstrates the step-by-step calculation of the Spearman's rank correlation coefficient ($ \rho $) for the provided paired observations of variables X and Y.

What is Spearman's Rank Correlation Coefficient?

Spearman's rank correlation coefficient is a statistical measure used to evaluate the strength and direction of association between two ranked variables. It assesses how well the relationship between two variables can be described using a monotonic function (a function that is either entirely non-increasing or entirely non-decreasing). It's useful even when the relationship isn't strictly linear.

Key Steps in Calculation

The calculation involves the following steps:

1. Ranking Data: Assign ranks to the observations for each variable (X and Y) independently.
2. Handling Ties: If there are repeated values (ties) in the data for a variable, assign the average of the ranks they would occupy.
3. Calculating Rank Differences: Find the difference ($ d_i $) between the ranks of corresponding pairs of X and Y observations.
4. Squaring Differences: Square each of the rank differences ($ d_i^2 $).
5. Summing Squared Differences: Calculate the sum of all the squared differences ($ \sum d_i^2 $).
6. Applying the Formula: Use the Spearman's correlation formula to find $ \rho $.

Data Analysis and Ranking

The given observations are:

• Variable X: 3, 4, 4
• Variable Y: 10, 10, 9

The total number of observations is $ n = 3 $.

Ranking Variable X

The values for X are 3, 4, 4. Sorting these gives 3, 4, 4.

• The value 3 is the smallest, so it gets rank 1.
• The two values of 4 are tied for the 2nd and 3rd positions. To handle this tie, we calculate the average rank: $ \frac{2 + 3}{2} = 2.5 $. Both 4s receive the rank 2.5.
• The ranks for X are: 1, 2.5, 2.5.

Ranking Variable Y

The values for Y are 10, 10, 9. Sorting these gives 9, 10, 10.

• The value 9 is the smallest, so it gets rank 1.
• The two values of 10 are tied for the 2nd and 3rd positions. The average rank is calculated as: $ \frac{2 + 3}{2} = 2.5 $. Both 10s receive the rank 2.5.
• The ranks for Y are: 2.5, 2.5, 1.

Calculating Rank Differences ($ d_i $) and Squared Differences ($ d_i^2 $)

We create a table to track the ranks, their differences, and the squared differences:

Spearman's Correlation Formula Application

The formula for Spearman's rank correlation coefficient ($ \rho $) is:

$ \rho = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)} $

Using the values we calculated:

• Number of observations, $ n = 3 $.
• Sum of squared differences in ranks, $ \sum d_i^2 = 4.5 $.

Now, substitute these values into the formula:

$ \rho = 1 - \frac{6 \times 4.5}{3(3^2 - 1)} $

First, calculate the numerator:

$ 6 \times 4.5 = 27 $

Next, calculate the denominator:

$ 3(3^2 - 1) = 3(9 - 1) = 3 \times 8 = 24 $

Now, complete the calculation for $ \rho $:

$ \rho = 1 - \frac{27}{24} $

$ \rho = 1 - 1.125 $

$ \rho = -0.125 $

Final Result

The Spearman's correlation coefficient for the given data is $ -0.125 $. This suggests a very weak negative monotonic relationship between the variables X and Y.