Do students who report more weekly study time tend to earn higher final mathematics grades?

• Studying is one possible factor associated with achievement.
• A relationship in observed data does not demonstrate cause and effect.
• We will examine study-time categories and final mathematics grades.

• Source: UCI Student Performance dataset, Mathematics records (Cortez, 2008).
• Sample: 395 students from two Portuguese secondary schools.
• Study time (studytime): 1 = <2 hours/week; 2 = 2–5; 3 = 5–10; 4 = >10.
• Final grade (G3): 0–20 points.

Study time is an ordered category, not an exact number of hours. The student records come from school reports and questionnaires.

Pearson’s correlation coefficient measures the strength and direction of a linear association:

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

• Here, \(x\) is the ordered study-time code (1–4); \(y\) is final grade.
• \(r\) ranges from \(-1\) (negative) to \(+1\) (positive).
• Observed Pearson \(r\) = 0.098 (treating the ordered codes as equally spaced for this descriptive calculation).

A simple linear model summarizes the average association:

\[\widehat{G3}=b_0+b_1(\mathrm{studytime})\]

For our dataset:

\[\widehat{G3} = 9.33 + 0.53 \times \mathrm{studytime}\]

• \(b_0\) = model intercept (an extrapolation to code 0, which is not a valid category).
• \(b_1\) = estimated change in grade points per one-category increase in the study-time code.
• Because study-time categories cover unequal hour ranges, this is not a per-hour effect.

This is the R code used for the main statistical calculations:

# The dataset has already been loaded in setup.
cor(study_data$studytime, study_data$G3)
model <- lm(G3 ~ studytime, data = study_data)
summary(model)
aggregate(G3 ~ studytime, data = study_data, FUN = mean)

The model is descriptive; it does not adjust for prior grades, attendance, or other influences.

Interpretation: Compare the center and spread of grades as the reported study-time group increases. Horizontal jitter is only for readability; it does not represent more precise study hours.

Interpretation: The boxes show medians and middle-half ranges; overlap between groups demonstrates that study time alone cannot determine a student’s grade.

Hover over the points to inspect reported group and grade; jitter only prevents overplotting.

• Sample size: 395 students.
• Correlation: \(r = 0.098\).
• Regression slope: 0.534 grade points per study-time category.
• \(R^2\): 0.010 (about 1.0% of the sample’s grade variation explained by this one-code linear model).

Our question: Is reported weekly study time associated with final mathematics grades?

• In this sample, the association is positive (Pearson \(r = 0.098\)).
• Study time alone explains only part of grade variation; group ranges overlap.
• Limits: observational data, self-reported category ranges, unequal category widths, secondary-school sample, and omitted factors such as prior achievement.
• Takeaway: Association is not causation; this model should not be used to promise score gains from extra study time.

Data citation: Cortez, P. (2008). Student Performance [Data set]. UCI Machine Learning Repository. https://doi.org/10.24432/C5TG7T

Background: Cortez, P., & Silva, A. (2008). Using data mining to predict secondary school student performance.