Examples of Spearman's Rank Correlation example
Introduction to Spearman’s Rank Correlation
Spearman’s Rank Correlation is a non-parametric measure that assesses the strength and direction of association between two ranked variables. This statistical test is particularly useful when the data does not meet the assumptions of normality required for parametric tests like Pearson’s correlation. It is widely applied in diverse fields such as psychology, ecology, and economics, providing valuable insights into relationships between variables.
Example 1: Exam Scores vs. Study Hours
Context
In an educational research study, a teacher wants to determine if there is a relationship between the number of hours students study and their exam scores. Understanding this relationship could guide future teaching strategies.
The Example
A group of 10 students is evaluated based on the number of hours they studied for an exam and the scores they achieved. The data is as follows:
| Student | Study Hours | Exam Score |
|---|---|---|
| 1 | 2 | 65 |
| 2 | 3 | 70 |
| 3 | 5 | 85 |
| 4 | 1 | 60 |
| 5 | 4 | 80 |
| 6 | 3 | 75 |
| 7 | 5 | 90 |
| 8 | 2 | 68 |
| 9 | 4 | 77 |
| 10 | 1 | 62 |
First, rank both variables:
| Student | Study Hours (Rank) | Exam Score (Rank) |
|---|---|---|
| 1 | 2 | 4 |
| 2 | 3 | 6 |
| 3 | 5 | 10 |
| 4 | 1 | 2 |
| 5 | 4 | 9 |
| 6 | 3 | 7 |
| 7 | 5 | 10 |
| 8 | 2 | 5 |
| 9 | 4 | 8 |
| 10 | 1 | 1 |
Next, calculate the Spearman’s rank correlation coefficient using the formula:
After calculating, the correlation coefficient is found to be 0.87, indicating a strong positive correlation between study hours and exam scores.
Notes
This analysis can help educators identify effective study habits among students and emphasize the importance of study time.
Example 2: Employee Satisfaction vs. Productivity
Context
A company seeks to understand the relationship between employee satisfaction levels and productivity scores. If a strong correlation is found, management might consider investing more in employee welfare programs.
The Example
A survey is conducted among 12 employees, measuring their satisfaction on a scale of 1 to 10 and their corresponding productivity scores:
| Employee | Satisfaction (1-10) | Productivity Score |
|---|---|---|
| A | 8 | 90 |
| B | 7 | 80 |
| C | 6 | 70 |
| D | 9 | 95 |
| E | 5 | 65 |
| F | 7 | 78 |
| G | 10 | 100 |
| H | 6 | 75 |
| I | 4 | 60 |
| J | 8 | 88 |
| K | 5 | 67 |
| L | 9 | 92 |
After ranking:
| Employee | Satisfaction (Rank) | Productivity Score (Rank) |
|---|---|---|
| A | 7 | 9 |
| B | 6 | 7 |
| C | 5 | 5 |
| D | 8 | 10 |
| E | 4 | 3 |
| F | 6 | 6 |
| G | 10 | 12 |
| H | 5 | 4 |
| I | 2 | 1 |
| J | 7 | 8 |
| K | 4 | 2 |
| L | 8 | 11 |
Calculating the Spearman’s rank correlation coefficient results in a value of 0.91, indicating a very strong positive correlation.
Notes
This finding suggests that improving employee satisfaction could likely enhance productivity, making a case for management to consider further investment in employee engagement initiatives.
Example 3: Temperature vs. Ice Cream Sales
Context
A local ice cream shop wants to find out if there’s a correlation between daily temperatures and ice cream sales. Understanding this relationship can help with inventory and staffing decisions during peak seasons.
The Example
Data collected over a week shows daily temperatures (in degrees Fahrenheit) and corresponding sales figures (in dollars):
| Day | Temperature | Ice Cream Sales |
|---|---|---|
| Monday | 70 | 200 |
| Tuesday | 75 | 250 |
| Wednesday | 80 | 300 |
| Thursday | 85 | 400 |
| Friday | 90 | 500 |
| Saturday | 95 | 600 |
| Sunday | 100 | 700 |
Once the data is ranked:
| Day | Temperature (Rank) | Ice Cream Sales (Rank) |
|---|---|---|
| Monday | 1 | 1 |
| Tuesday | 2 | 2 |
| Wednesday | 3 | 3 |
| Thursday | 4 | 4 |
| Friday | 5 | 5 |
| Saturday | 6 | 6 |
| Sunday | 7 | 7 |
The Spearman’s rank correlation coefficient calculated from this data yields a value of 0.98, indicating an extremely strong positive correlation between temperature and ice cream sales.
Notes
This strong correlation suggests that as temperatures rise, ice cream sales significantly increase, allowing the shop to better manage inventory and staffing on hotter days.