2026-10-18
Research Question:
Is there a relationship between weekly running mileage and average running pace?
This presentation uses simple linear regression to investigate the relationship between training volume and running performance.
A simulated dataset representing 20 weeks of running training was created for educational purposes.
The objective is to demonstrate how statistical methods can describe relationships between variables.
Data Source: Simulated data created for this project.
Observations: 20 hypothetical training weeks.
Variables:
The values were constructed to illustrate a negative relationship between weekly mileage and running pace.
| Week | Mileage | Pace |
|---|---|---|
| 1 | 8 | 9.4 |
| 2 | 10 | 9.2 |
| 3 | 9 | 9.3 |
| 4 | 12 | 9.0 |
| 5 | 11 | 9.1 |
Interpretation:
The simulated data show a negative relationship between weekly mileage and average running pace.
As weekly mileage increases, the average time required to run one mile generally decreases.
This pattern was intentionally built into the simulated dataset.
Simple linear regression models the relationship between weekly mileage and average running pace.
The statistical model is: \[ Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i \]
Where:
Linear regression estimates the intercept and slope by minimizing the sum of squared residuals.
\[ SSE = \sum_{i=1}^{n} \left(y_i - (\hat{\beta}_0 + \hat{\beta}_1 x_i)\right)^2 \] The estimated slope is:
\[ \hat{\beta}_1 = \frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})} {\sum_{i=1}^{n}(x_i-\bar{x})^2} \]
The estimated intercept is:
\[ \hat{\beta}_0 = \bar{y} - \hat{\beta}_1\bar{x} \]
We fit a simple linear regression model using the lm() function in R.
running_model <- lm(Pace ~ Mileage, data = running_data)
| Coefficient | Estimate |
|---|---|
| Intercept | 10.1369 |
| Mileage | -0.0926 |
The fitted regression equation is:
\[ \hat{y} = 10.1369 - 0.0926x \]
Estimated Regression Equation:
\[ \hat{y} = 10.1369 - 0.0926x \]
Slope Interpretation:
For each additional mile of weekly running volume, the predicted running pace decreases by approximately 0.0926 minutes per mile.
This corresponds to approximately 5.56 seconds per mile.
Intercept Interpretation:
The model predicts a pace of 10.1369 minutes per mile at zero weekly mileage. However, zero miles is outside the observed range, so this prediction is not practically meaningful.
Residuals measure the difference between observed and predicted running pace.
Interpretation:
Residuals are small and lie near zero, indicating that the regression line closely fits the simulated observations.
Hover over the points to explore each simulated training week.
Interpretation:
Hover over individual points to view the week, mileage, and pace. The simulated data illustrate a negative relationship between mileage and pace.
Key Findings:
The simulated data show a strong negative relationship between weekly mileage and running pace.
The regression model estimates a decrease of approximately 5.56 seconds per mile in predicted pace for each additional weekly mile.
Small residuals indicate that the linear regression model closely fits the simulated observations.
Final Takeaway:
Linear regression provides a useful statistical method for exploring relationships between training volume and running pace.
Important Considerations:
The dataset contains 20 simulated training weeks, not actual running observations.
The results do not establish that increasing mileage causes faster running performance.
Other factors, including training intensity, recovery, and fitness level, may influence running pace.
Real-world conclusions would require actual running data and further statistical analysis.
Conclusion:
The simulated data demonstrate linear regression, but the results cannot be generalized to real runners.