Understanding Rank Correlation

Rank correlation is a measure used to assess the relationship between the rankings of two variables or sets of data. It helps determine if the rankings are similar (positive correlation) or dissimilar (negative correlation), or if there is no relationship. One of the most common methods for calculating rank correlation is Spearman's rank correlation coefficient (\(\rho\)).

In this problem, we are given the rankings of ten students in two subjects: Mathematics and Statistics. We need to calculate the coefficient of rank correlation between these rankings.

Steps to Calculate Rank Correlation Coefficient

To calculate the coefficient of rank correlation using Spearman's method, we follow these steps:

1. List the ranks for each student in both subjects. Let R1 be the rank in Mathematics and R2 be the rank in Statistics.
2. Calculate the difference (d) between the ranks for each student: \(d = R1 - R2\).
3. Square each difference: \(d^2\).
4. Sum the squared differences: \(\sum d^2\).
5. Use the formula for Spearman's rank correlation coefficient:

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

   Where \(n\) is the number of pairs of observations (number of students in this case).

Calculating Rank Differences and Squared Differences

Let's list the given rankings and perform the calculations for \(d\) and \(d^2\):

Now, we sum the squared differences (\(\sum d^2\)):

\(\sum d^2 = 9 + 1 + 1 + 16 + 36 + 81 + 64 + 1 + 1 + 4 = 214\)

The number of students is \(n = 10\).

Applying the Spearman's Rank Correlation Formula

Now, we substitute the values of \(\sum d^2\) and \(n\) into the formula:

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

\(\rho = 1 - \frac{6 \times 214}{10(10^2 - 1)}\)

\(\rho = 1 - \frac{1284}{10(100 - 1)}\)

\(\rho = 1 - \frac{1284}{10(99)}\)

\(\rho = 1 - \frac{1284}{990}\)

\(\rho = 1 - 1.296969...\)

\(\rho \approx -0.296969...\)

Result of Rank Correlation Calculation

Rounding the calculated rank correlation coefficient to one decimal place, we get:

\(\rho \approx -0.3\)

The coefficient of rank correlation between the rankings of students in Mathematics and Statistics is approximately -0.3.

Interpreting the Rank Correlation Coefficient

The value of the rank correlation coefficient (\(\rho\)) ranges from -1 to +1:

• A value close to +1 indicates a strong positive relationship (high rank in one subject corresponds to high rank in the other).
• A value close to -1 indicates a strong negative relationship (high rank in one subject corresponds to low rank in the other).
• A value close to 0 indicates a weak or no relationship between the rankings.

In this case, \(\rho \approx -0.3\) suggests a weak negative relationship between the rankings in Mathematics and Statistics. This means students who rank higher in Mathematics tend to rank slightly lower in Statistics, and vice versa, but the relationship is not very strong.

Revision Table: Key Formulas

Additional Information on Rank Correlation

Spearman's rank correlation is a non-parametric measure, meaning it does not assume that the data follows a specific distribution (like a normal distribution). It is suitable for ordinal data or when the assumptions for parametric correlation (like Pearson correlation) are not met.

If there are tied ranks (two or more students have the same rank), a slight adjustment to the formula for Spearman's rank correlation is sometimes used, although the simple formula used here is often considered acceptable for a small number of ties.

The significance of the rank correlation coefficient can be tested using statistical tests to determine if the observed correlation is likely due to chance or represents a true relationship in the population.