Do Longer Study Hours Really Improve Exam Scores?
- Michael Lee, MBA
- 3 days ago
- 3 min read
The Correlation Test: Understanding How Variables Move Together
Continuing Our Inferential Analysis Series
This article continues our ongoing journey into inferential analysis. We began with an introductory article on inferential thinking, then explored:
The 1-sample proportion test
The Chi-Square test
The T-Test
Now, we look at another powerful tool: the Correlation Test, used to understand whether two numerical variables are related—and how strongly.
The Story Behind the Test
The idea of correlation dates back to the late 19th century when Sir Francis Galton, a cousin of Charles Darwin, studied the relationship between parental height and children's height. He introduced the concept of correlation to measure how strongly two variables move together. His work laid the foundation for modern statistical correlation analysis, which is now widely used in economics, psychology, business, and healthcare.
What Is the Objective?
A correlation test determines whether there is a statistical relationship between two numerical variables. It helps answer questions like:
Do advertising expenses impact sales revenue?
Is there a relationship between temperature and ice cream sales?
Do students who study longer score higher on exams?
Important: Correlation does not imply causation—it only shows whether variables move together, not whether one causes the other.
✅ Use this test when: You have two numerical variables and want to test if they are linearly related.
🚫 Don't use it when: Your variables are categorical—in that case, use a Chi-Square test instead.
How It Works
Step 1: Define Hypotheses
Null Hypothesis (H₀): There is no correlation between the two variables.
Alternative Hypothesis (H₁): There is a significant correlation between the two variables.
tep 2: Collect Data
Let’s analyze whether study hours and exam scores are correlated. We collect data from 10 students:
Study Hours (X) | Exam Score (Y) |
2 | 50 |
4 | 55 |
6 | 60 |
8 | 65 |
10 | 70 |
12 | 75 |
14 | 78 |
16 | 85 |
18 | 88 |
20 | 90 |
Step 3: Compute the Correlation Coefficient
The Pearson correlation coefficient (r) is calculated using:
Where:
r ranges from -1 to 1:
r > 0: Positive correlation (both variables increase together)
r < 0: Negative correlation (one variable increases, the other decreases)
r = 0: No correlation
🔍 Note: Pearson’s r measures linear relationships. For non-linear trends, consider Spearman’s or Kendall’s correlation coefficients.
Step 4: Determine Statistical Significance
The p-value determines whether the correlation is statistically significant.
If p < 0.05, we reject the null hypothesis and conclude that the correlation is significant.
Interpreting the Results
After calculating Pearson’s correlation for our dataset, we find:
r = 0.98 indicates a very strong positive correlation between study hours and exam scores.
Since p < 0.05, we conclude that the correlation is statistically significant.
This means that students who study longer tend to score higher, but this does not prove that studying causes higher scores.
Want proof that correlation doesn’t imply causation? Look up spurious correlations like “Ice cream sales vs. shark attacks.” They move together—but for totally unrelated reasons.
Real-World Applications
1. Business & Sales Forecasting
Companies analyze whether social media engagement correlates with sales performance. A strong positive correlation might indicate that increasing social media efforts could boost sales.
2. Healthcare & Risk Factors
Doctors study whether exercise frequency is correlated with lower blood pressure. A strong correlation would support recommendations for exercise to improve cardiovascular health.
3. Climate & Economic Impact
Researchers analyze whether rising temperatures correlate with increased energy consumption. A significant correlation could help plan for energy demands in hotter seasons.
Final Thoughts
The correlation test is an essential tool for identifying relationships between numerical variables. It provides valuable insights for business, healthcare, and scientific research, helping professionals make data-driven decisions.
But what if you want to go beyond identifying relationships—and actually predict outcomes? That’s where regression analysis comes in. Stay tuned for the next article in the series.
If you want to gain hands-on experience with hypothesis testing and other analytical techniques, our 2-day course, Problem Solving Using Data Analytics, provides practical applications and real-world exercises. For those curious about how Generative AI can enhance statistical testing, our Data Analytics in the Age of AI course explores AI-driven analytics and automation.
Think your metrics are moving together? Test it—and know for sure.
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