Statstics

Rank Correlation – Definition, Real-Life Example, Formula

Published on August 21, 2023
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Table of Contents

1 Definition
2 Real-Life Example
3 Interpretation:
4 Conclusion

Definition

Spearman’s rank correlation coefficient is calculated as:

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

where $d_i$ is the difference between the ranks of corresponding data points and $n$ is the number of data points.

Real-Life Example

Sure, let’s create a hypothetical example to illustrate the concept of Spearman’s rank correlation coefficient. In this example, we’ll consider a small dataset of individuals’ exercise times and their corresponding fitness levels. We will rank the data and then calculate the Spearman’s rank correlation coefficient to analyze the relationship between exercise times and fitness levels.

Here’s the example data:

Exercise Time	Fitness Level
2.5	4
1.8	2
3.2	5
2	3
1.5	1

First, let’s rank the data for exercise times and fitness levels:

Individual	Rank (Exercise Time)	Rank (Fitness Level)
1	3	3
2	2	1
3	5	5
4	1	2
5	4	4

Now, we will calculate the Spearman’s rank correlation coefficient using the formula:

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

where $d_i$ is the difference between the ranks of corresponding data points and $n$ is the number of data points.

For the given example:

$n = 5$

Calculating the squared rank differences ( $d_i^2$ ) and summing them:

$d_1^2 &= (3 - 1)^2 = 4$

$d_2^2 &= (2 - 2)^2 = 0$

$d_3^2 &= (5 - 5)^2 = 0$

$d_4^2 &= (1 - 3)^2 = 4$

$d_5^2 &= (4 - 4)^2 = 0$

The sum of squared rank differences:

$\sum d_i^2 = 4 + 0 + 0 + 4 + 0 = 8$

Now, plug the values into the formula:

$\rho = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)} = 1 - \frac{6 \times 8}{5(5^2 - 1)} = 1 - \frac{48}{100} = 0.52$

The calculated Spearman’s rank correlation coefficient ( $\rho$ ) is approximately 0.52.

Interpretation:

A positive value of Spearman’s rank correlation coefficient ( $\rho$ ) indicates a positive monotonic relationship between the variables. In this example, as the exercise times increase (higher ranks), the fitness levels also tend to increase (higher ranks). The calculated $\rho$ value of 0.52 indicates a moderately strong positive correlation between exercise times and fitness levels.

This suggests that individuals who spend more time exercising tend to have higher fitness levels in this hypothetical dataset.

Keep in mind that this is a simplified example for illustration purposes, and real-world data analysis may involve more complex scenarios and statistical considerations.

Conclusion

Bivariate data analysis provides valuable insights into relationships between two variables. Scatter diagrams help visualize patterns, while correlation coefficients, including rank correlation, provide quantitative measures of associations. Analysts and researchers can make informed decisions based on data-driven insights by understanding these concepts.