Do students who study more hours tend to earn higher exam scores?

• We will explore this question using correlation.
• We will visualize the pattern and calculate a correlation coefficient.
• The dataset contains 15 fictional students, created only to demonstrate the statistical idea.

Correlation describes the direction and strength of a linear relationship between two numerical variables.

• Positive: as one variable increases, the other tends to increase.
• Negative: as one increases, the other tends to decrease.
• Near zero: little linear relationship is visible.

Correlation does not prove cause and effect.

Pearson’s sample correlation coefficient is

\[r=\frac{\sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i=1}^n(x_i-\bar{x})^2\sum_{i=1}^n(y_i-\bar{y})^2}}\]

• \(x_i\) = study hours and \(y_i\) = exam score.
• \(\bar{x}\) and \(\bar{y}\) are the sample averages.
• \(-1\leq r\leq 1\). Values near \(1\) indicate a strong positive linear association.

We can also summarize the overall trend with a simple regression line:

\[\widehat{y}=b_0+b_1x\]

For these example data, the fitted line is approximately

\[\widehat{\text{score}}=46+5.2\times \text{hours}\]

• The slope is the expected score difference for one additional study hour in this example.
• A fitted line describes association, not necessarily a causal effect.

The code below calculates the correlation and fits a line:

r <- cor(students$hours, students$score)
model <- lm(score ~ hours, data = students)
summary(model)

Result: Pearson’s \(r\) is approximately 0.97, a strong positive correlation in this fictional dataset.

• The fictional sample has a strong positive linear association (\(r\approx\) 0.97).
• Both the scatterplot and the numerical coefficient tell the same story.
• The histogram describes scores alone, while the scatterplot shows the relationship.
• We cannot conclude that extra study hours caused higher scores.

Limitations: The data are invented for practice; the sample is small and other factors such as previous knowledge and sleep are not included.

Conclusion: Correlation is a useful first step for understanding how two numerical variables move together. A responsible analysis also considers context and possible confounding factors.