Exercises: Statistical Foundations for Football Analytics

Difficulty Levels

• Level 1 (Foundational): Basic probability and statistics concepts
• Level 2 (Applied): Applying concepts to football data
• Level 3 (Intermediate): Multi-step analysis and interpretation
• Level 4 (Advanced): Building models and complex inference
• Level 5 (Expert): Research-level statistical analysis

Section 1: Probability (Level 1-2)

Exercise 1.1: Basic Probability

Level 1 | Probability

A kicker has the following field goal success rates by distance: - 0-29 yards: 97% - 30-39 yards: 88% - 40-49 yards: 78% - 50+ yards: 63%

Calculate: 1. Expected points from a 35-yard attempt (make = 3 points) 2. Expected points from a 52-yard attempt 3. At what success rate does attempting a FG have lower expected value than punting (assume punt = 0 points)?

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Exercise 1.2: Conditional Probability

Level 2 | Conditional Probability

Using play-by-play data, calculate: 1. P(First Down | Third Down) 2. P(First Down | Third Down AND Yards to Go ≤ 3) 3. P(First Down | Third Down AND Yards to Go > 7) 4. P(Pass | Third Down AND Yards to Go > 7)

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Exercise 1.3: Joint Probability

Level 2 | Joint Probability

Calculate: 1. P(Pass AND Touchdown) 2. P(Run AND Touchdown) 3. P(Third Down AND Pass AND First Down) 4. Are "Pass" and "Touchdown" independent events? How can you test this?

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Exercise 1.4: Expected Value

Level 2 | Expected Value

For a fourth-and-1 situation at the opponent's 35-yard line, calculate expected points for: 1. Go for it (assume 70% conversion rate) 2. Punt (assume touchback) 3. Field goal attempt

Which option maximizes expected points?

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Exercise 1.5: Bayes' Theorem Application

Level 3 | Bayesian Inference

A team wins 60% of games when they score first. They score first 52% of the time. 1. What's P(Win)? 2. If a team wins, what's the probability they scored first? 3. Use Bayes' theorem to calculate P(Score First | Win)

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Section 2: EPA Analysis (Level 2-3)

Exercise 2.1: EPA Interpretation

Level 2 | EPA Concepts

Using the play-by-play data: 1. What is the league-wide average EPA on passes? 2. What is the league-wide average EPA on runs? 3. What is the standard deviation of EPA for each play type? 4. Why is the pass EPA higher than run EPA?

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Exercise 2.2: QB EPA Rankings

Level 2 | Player Evaluation

Calculate EPA per dropback for all QBs with 200+ attempts: 1. Rank QBs by EPA per dropback 2. Calculate the 95% confidence interval for the top 5 QBs 3. Identify which QBs' confidence intervals overlap with league average

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Exercise 2.3: Situational EPA

Level 3 | Context Analysis

Calculate average EPA for passes in different situations: 1. First down vs third down 2. Leading by 7+ vs trailing by 7+ 3. First half vs fourth quarter 4. Red zone vs rest of field

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Exercise 2.4: Team EPA Decomposition

Level 3 | Aggregation

For each team, calculate: 1. Total offensive EPA 2. Pass EPA contribution (total pass EPA) 3. Rush EPA contribution (total rush EPA) 4. What percentage of each team's EPA comes from passing?

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Exercise 2.5: EPA Model Building

Level 4 | Model Construction

Build a simple Expected Points model: 1. Calculate historical average points scored from each yardline (1-99) 2. Create a lookup table of EP by yardline 3. Use this to calculate "homemade" EPA for plays 4. Compare your EPA values to the provided EPA column

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Section 3: Win Probability (Level 2-4)

Exercise 3.1: WP Interpretation

Level 2 | WP Concepts

Using play-by-play data: 1. Find the play with the highest WPA in a given season 2. Find the play with the lowest WPA 3. What types of plays generate the most extreme WPA values?

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Exercise 3.2: Clutch Performance

Level 3 | Player Evaluation

Calculate total WPA for QBs: 1. Rank QBs by total WPA 2. Compare rankings by total WPA vs EPA per play 3. Which QBs are "clutch" (high WPA rank, lower EPA rank)?

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Exercise 3.3: Game Flow Analysis

Level 3 | Time Series

For a specific game: 1. Plot win probability over time for both teams 2. Identify the "turning point" (largest single-play WPA swing) 3. Calculate each team's cumulative WPA through the game

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Exercise 3.4: Comeback Probability

Level 4 | Model Analysis

Analyze comeback likelihood: 1. What is the historical win rate when trailing by 14+ in the 4th quarter? 2. What is the win rate when trailing by 7 with 2 minutes left? 3. Build a simple model predicting win probability from score differential and time remaining

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Exercise 3.5: WPA Variance Analysis

Level 4 | Statistical Analysis

Compare WPA distributions: 1. Calculate the standard deviation of WPA for passes vs runs 2. Test whether the difference is statistically significant 3. What does this tell us about which play types are "higher leverage"?

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Section 4: Hypothesis Testing (Level 3-4)

Exercise 4.1: Home Field Advantage

Level 3 | Two-Sample Test

Test whether home field advantage exists: 1. Calculate mean EPA for home vs away offenses 2. Perform a two-sample t-test 3. Calculate the effect size (Cohen's d) 4. Is the effect practically significant?

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Exercise 4.2: QB Performance by Weather

Level 3 | Comparative Analysis

Test whether a specific QB performs differently in cold weather: 1. Define "cold" as temperature < 40°F 2. Compare EPA in cold vs normal conditions 3. Perform appropriate hypothesis test 4. Account for sample size in your conclusions

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Exercise 4.3: Receiver Comparison

Level 3 | Pairwise Comparison

Compare two receivers' catch rates: 1. Calculate catch rate and targets for each 2. Test whether the difference is significant (z-test for proportions) 3. Calculate the confidence interval for the difference

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Exercise 4.4: Multiple Comparisons

Level 4 | Multiple Testing

Compare all 32 teams' offensive EPA to league average: 1. Perform 32 one-sample t-tests 2. Apply Bonferroni correction 3. Apply Benjamini-Hochberg FDR correction 4. How many teams are significantly different with each method?

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Exercise 4.5: A/B Testing Simulation

Level 4 | Power Analysis

Design an experiment to detect play-calling differences: 1. Calculate the effect size you want to detect 2. Determine required sample size for 80% power 3. How many games would this require? 4. Simulate data to validate your power calculation

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Section 5: Regression (Level 3-5)

Exercise 5.1: Simple Linear Regression

Level 3 | Linear Regression

Build a regression model predicting EPA from air yards: 1. Fit a simple linear regression 2. Interpret the slope coefficient 3. Calculate R-squared 4. Plot the regression line with the data

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Exercise 5.2: Multiple Regression

Level 3 | Multiple Features

Predict EPA using multiple features: 1. Include: down, ydstogo, yardline_100, score_differential, pass 2. Fit the model and interpret coefficients 3. Which features are most important? 4. Check for multicollinearity

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Exercise 5.3: Logistic Regression

Level 4 | Classification

Build a model to predict play success: 1. Fit logistic regression with game state features 2. Interpret coefficients as odds ratios 3. Calculate accuracy, precision, and recall 4. Plot the ROC curve

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Exercise 5.4: Regularized Regression

Level 4 | Regularization

Compare Ridge and Lasso regression: 1. Fit both models with alpha = 1.0 2. Compare coefficient sizes 3. Which features are "zeroed out" by Lasso? 4. Cross-validate to find optimal alpha

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Exercise 5.5: Regression Diagnostics

Level 5 | Model Validation

Perform comprehensive regression diagnostics: 1. Plot residuals vs fitted values 2. Check for heteroscedasticity 3. Test for normality of residuals 4. Identify influential observations (Cook's distance)

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Section 6: Advanced Topics (Level 4-5)

Exercise 6.1: Bootstrap Confidence Intervals

Level 4 | Resampling

Calculate bootstrap confidence intervals: 1. For a QB's EPA per play, bootstrap 1000 samples 2. Calculate 95% CI using percentile method 3. Compare to the t-interval 4. When would bootstrap be preferred?

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Exercise 6.2: Permutation Testing

Level 4 | Non-Parametric Test

Use permutation testing to compare two QBs: 1. Create null distribution by permuting labels 2. Calculate p-value from permutation distribution 3. Compare to t-test result 4. When is permutation testing more appropriate?

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Exercise 6.3: Sample Size Calculation

Level 4 | Power Analysis

For a QB comparison study: 1. Assume you want to detect 0.1 EPA/play difference 2. Calculate required sample size for 80% power 3. How many games does this represent? 4. What if you only have 100 plays per QB?

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Exercise 6.4: Causal Inference Challenge

Level 5 | Causal Analysis

Address the causal question: "Does passing more lead to winning?" 1. Calculate correlation between pass rate and wins 2. Discuss why correlation ≠ causation here 3. What confounders might exist? 4. Propose a quasi-experimental approach

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Exercise 6.5: Bayesian Analysis

Level 5 | Bayesian Statistics

Perform Bayesian estimation of a kicker's true FG percentage: 1. Use Beta prior (based on historical data) 2. Update with current season makes/attempts 3. Calculate posterior mean and 95% credible interval 4. Compare to frequentist confidence interval

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Submission Guidelines

For each exercise: 1. Include all code with comments 2. Show all statistical output 3. Provide written interpretation 4. Discuss practical significance, not just statistical significance

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