Chapter 10 Exercises: Plus-Minus and On/Off Analysis

Section A: Conceptual Understanding (Questions 1-8)

Exercise 1: Defining Plus-Minus

Explain in your own words: a) What raw plus-minus measures b) Why it is called "plus-minus" (what do the plus and minus represent?) c) What data is required to calculate it d) One key limitation of the metric

Exercise 2: On-Court vs. Off-Court

A team has the following performance over a season: - With Player A on court: 8,500 points scored, 8,200 points allowed, 7,600 possessions - With Player A off court: 3,200 points scored, 3,400 points allowed, 3,000 possessions

a) Calculate the offensive rating with Player A on and off court b) Calculate the defensive rating with Player A on and off court c) Calculate the net rating with Player A on and off court d) Calculate Player A's on/off differential e) Interpret what these numbers suggest about Player A's impact

Exercise 3: Teammate Effects Thought Experiment

Consider two hypothetical players: - Player X plays all his minutes alongside LeBron James - Player Y plays all his minutes alongside replacement-level players

Both players have identical skills and contribute equally to their team's success.

a) Which player would likely have a higher raw plus-minus? Why? b) Which player would likely have a higher on/off differential? Why? c) How might this scenario mislead team decision-makers? d) What type of metric would better account for this problem?

Exercise 4: Sample Size Intuition

Rank the following scenarios from MOST reliable to LEAST reliable for drawing conclusions about a player's true impact:

A) +15 net rating over 2,500 minutes B) +25 net rating over 400 minutes C) +10 net rating over 4,000 minutes D) +8 net rating over 1,200 minutes (playoff minutes only) E) +30 net rating over 150 minutes

Explain your reasoning for the ranking.

Exercise 5: Lineup Analysis Fundamentals

A team uses the following three lineups extensively:

a) Calculate the net rating for each lineup b) Which lineup is most effective? Most ineffective? c) Why might the bench lineup have a worse net rating even if the bench players are solid? d) What does the mixed lineup's performance suggest about player combinations?

Exercise 6: The Replacement Level Problem

Player A and Player B are both starting point guards on different teams. Their on/off splits:

Player A: - Net Rating On: +6.5 - Net Rating Off: -8.0 - On/Off Differential: +14.5

Player B: - Net Rating On: +5.0 - Net Rating Off: +1.0 - On/Off Differential: +4.0

a) Based solely on on/off differential, which player appears more valuable? b) Based on net rating while on court, which player's team performs better? c) What might explain the large difference in their on/off differentials? d) If you could only sign one player, what additional information would you want?

Exercise 7: Decomposing On/Off

A player has the following on/off splits: - Offensive Rating On: 114.2, Off: 108.5 - Defensive Rating On: 109.8, Off: 106.2

a) Calculate the offensive on/off differential b) Calculate the defensive on/off differential (remember: lower is better for defense) c) Calculate the overall on/off differential d) Is this player's impact primarily offensive, defensive, or balanced? e) Would you characterize the defensive numbers as a concern? Why or why not?

Exercise 8: Garbage Time Considerations

Explain why the following statement is problematic:

"Player Z has the best plus-minus on our team at +320 for the season. He's clearly our most impactful player."

Consider: - When does Player Z typically play? - What types of game situations might inflate plus-minus? - What additional context would you need to evaluate this claim?

Section B: Calculation Practice (Questions 9-17)

Exercise 9: Basic Plus-Minus Calculation

Player M played the following minutes in a game:

a) Calculate Player M's plus-minus for the game b) Calculate how many minutes Player M played c) Calculate Player M's plus-minus per 36 minutes

Exercise 10: Possession Estimation

A team's box score shows: - Field Goal Attempts: 88 - Offensive Rebounds: 12 - Turnovers: 14 - Free Throw Attempts: 24

a) Estimate the number of possessions using the standard formula b) If the team scored 108 points, what is their offensive rating? c) If opponents scored 102 points, what is the net rating?

Exercise 11: On/Off Split Calculation

Player K's season data:

On Court: - Minutes: 2,400 - Team Points: 4,800 - Opponent Points: 4,560 - Estimated Possessions: 4,400

Off Court: - Minutes: 1,560 - Team Points: 2,800 - Opponent Points: 2,960 - Estimated Possessions: 2,800

Calculate: a) Offensive Rating (On and Off) b) Defensive Rating (On and Off) c) Net Rating (On and Off) d) On/Off Differential

Exercise 12: Two-Man Combination Analysis

Two players, J and K, have the following splits:

a) What is the team's net rating when J is on court (regardless of K)? b) What is the team's net rating when K is on court (regardless of J)? c) What is the synergy between J and K? (Hint: Compare actual "both on" to expected) d) Which player appears more valuable as an individual? Explain.

Exercise 13: Confidence Interval Calculation

A lineup has played 300 minutes together with a +12.0 net rating. Using the approximation that standard deviation of net rating is 37 per 100 possessions:

a) Estimate the number of possessions (use 2.0 possessions per minute) b) Calculate the standard error of the net rating estimate c) Calculate a 95% confidence interval for the true net rating d) Can we confidently say this lineup is above average (+0)? Explain.

Exercise 14: Bayesian Shrinkage

A rookie plays 800 possessions with a +10.0 observed net rating. Using a prior mean of 0 (league average) and prior strength of 500:

a) Calculate the Bayesian posterior estimate of true net rating b) How much did the estimate shrink from the observed value? c) If the same player had 2,000 possessions with the same +10.0 observed rating, what would the posterior be? d) Explain why the posterior is different in part (c) even though the observed rating is the same.

Exercise 15: Multi-Year Aggregation

A player's season-by-season data:

a) Calculate the total plus-minus across all three seasons b) Calculate the plus-minus per 1,000 minutes for each season c) Calculate the weighted average plus-minus per 1,000 minutes (weighted by minutes) d) Which season was the player most impactful per minute played?

Exercise 16: Lineup Efficiency Comparison

Three lineups with different sample sizes:

a) Calculate approximate 95% confidence intervals for each lineup b) Which lineup would you trust most to perform well going forward? Why? c) Is Lineup B statistically significantly better than Lineup A? Explain.

Exercise 17: Opponent Adjustment (Simplified)

Player P has a +5.0 raw plus-minus per 100 possessions. Analysis reveals: - Average opponent quality faced: +2.0 (above average opponents) - Average teammate quality: +1.5 (above average teammates)

Using a simplified adjustment formula: Adjusted PM = Raw PM - (0.5 * Teammate Boost) + (0.5 * Opponent Difficulty)

a) Calculate the teammate adjustment b) Calculate the opponent adjustment c) Calculate the adjusted plus-minus d) Interpret what this adjustment tells us about the player's value

Section C: Application and Analysis (Questions 18-25)

Exercise 18: Case Analysis - The Backup Effect

A team's starting center has these splits:

Starting Center (Player C): - On Court Net Rating: +4.5 - Off Court Net Rating: -8.2 - On/Off: +12.7

The backup center (Player D) has: - On Court Net Rating: -6.5 (when C is off) - Total Minutes: 1,200

a) How much of Player C's on/off differential might be attributed to the backup being poor? b) If the backup were average (0.0 net rating), what would Player C's expected on/off be? c) What roster move might help the team, and how would it affect Player C's on/off stats? d) Write a brief paragraph advising the GM on interpreting Player C's value.

Exercise 19: Lineup Construction

You are a coach with the following five-man lineup data:

Two-man synergies (bonus net rating when both play together): - A + E: +2.5 - B + D: +1.8 - F + J: -1.0 - C + H: +0.5

a) What is the expected net rating of your starting lineup (A, B, C, D, E) accounting for synergies? b) What bench lineup would you recommend? c) Design a mixed lineup that maximizes synergy effects d) What cautions would you have about relying on this analysis?

Exercise 20: Season Evaluation

Evaluate the following player's season using plus-minus data:

Player Stats: - Minutes: 2,850 (34.8 mpg) - Raw Plus-Minus: +412 (season total) - On/Off Differential: +9.8 - Net Rating On: +7.2 - Net Rating Off: -2.6

Context: - Team Record: 52-30 - Team Net Rating: +4.1 - Player's backup is a second-round rookie

a) Calculate plus-minus per 36 minutes b) How does the player's on-court net rating compare to team average? c) What concerns might you have about the on/off differential? d) Write a two-paragraph evaluation of this player's impact based on plus-minus metrics alone, then a third paragraph on what additional information you'd want.

Exercise 21: Clutch Time Analysis

A player's clutch statistics (final 5 minutes, within 5 points):

a) Calculate plus-minus per 36 minutes in clutch vs. non-clutch situations b) Calculate the approximate possessions in clutch time c) Calculate a 95% confidence interval for clutch net rating d) Can we conclude this player is a "clutch performer"? Explain your statistical reasoning. e) What other factors might explain the clutch vs. non-clutch difference?

Exercise 22: Defensive Specialist Evaluation

A defensive specialist has these on/off splits:

On Court: - Offensive Rating: 106.5 - Defensive Rating: 101.2

Off Court: - Offensive Rating: 112.8 - Defensive Rating: 110.5

a) Calculate the offensive on/off differential b) Calculate the defensive on/off differential c) Calculate the overall on/off differential d) Is this player a net positive? Show the math. e) Argue both sides: Why might a team want this player? Why might they not?

Exercise 23: Lineup Discovery

You have access to a database with all five-man lineup combinations for a team. The top 5 lineups by net rating (minimum 100 minutes) are:

a) Which player appears in the most top-5 lineups? b) Which two-player combination appears most frequently? c) Player A and B appear together in lineups 1, 2, and 3. If you could only have one, which would you keep based on this data? d) Why should you be cautious about drawing strong conclusions from this analysis?

Exercise 24: Regression Setup

You are preparing data for an adjusted plus-minus regression. You have the following possession:

• Home team players: [Player 1, Player 2, Player 3, Player 4, Player 5]
• Away team players: [Player 6, Player 7, Player 8, Player 9, Player 10]
• Point margin: +3 (home perspective)
• Duration: 15 possessions

a) Create the design matrix row for this observation (use +1 for home, -1 for away, 0 for not on court) b) What is the target variable (y) for this observation (in per-100-possession terms)? c) What would the weight be for this observation? d) If Player 1 and Player 2 always play together, what statistical problem would this create?

Exercise 25: Comprehensive Team Analysis

A team has these splits for all rotation players:

a) Who is the team's most impactful player by on/off differential? b) Why do bench players have negative on/off differentials despite Bench H having positive net rating on court? c) Calculate the approximate team net rating (hint: weight by minutes) d) The team is considering trading Starter E (on/off +0.3) for an upgrade. What on/off differential would a replacement need to significantly improve the team? e) Write a brief analytical report (one page) summarizing this team's plus-minus profile, identifying strengths, weaknesses, and recommendations.

Section D: Advanced Problems (Questions 26-32)

Exercise 26: Multi-Year RAPM Estimation

A player has the following single-season RAPM estimates with standard errors:

a) Calculate a precision-weighted average RAPM (weight by 1/SE^2) b) What is the standard error of the combined estimate? c) Is the player statistically significantly above average (0)? Show work. d) Why might we still want to weight recent seasons more heavily despite precision weighting?

Exercise 27: Lineup Synergy Detection

You have 200 lineups with at least 50 minutes. You want to test whether certain player pairs have significant synergy (perform better together than expected).

Expected two-man net rating (additive model): (Player A NetRtg + Player B NetRtg) / 2 + Team Constant

For the pair of Player X (+4.0) and Player Y (+2.0) with team constant -1.0: - Expected when together: (4.0 + 2.0) / 2 - 1.0 = +2.0 - Observed when together: +6.5 (over 400 minutes)

a) Calculate the synergy (observed - expected) b) Estimate the standard error (use 37/sqrt(possessions/100), assuming 800 possessions) c) Calculate a z-score for the synergy d) Is this synergy statistically significant at p < 0.05? e) List three basketball reasons why this synergy might exist

Exercise 28: Opponent Strength Adjustment

A player's schedule analysis reveals:

a) Calculate the weighted average raw net rating b) Calculate the weighted average opponent strength c) Adjust the player's net rating for opponent strength (subtract 0.5 * opponent adjustment from 0) d) How much did the adjustment change the player's rating? e) Is this player better against good teams or weak teams? What might this indicate?

Exercise 29: Possession-Level Variance

You observe 1,000 possessions for a player with the following outcomes:

a) Calculate total points for and against b) Calculate the observed plus-minus per 100 possessions c) Calculate the variance of single-possession point differential d) Calculate the standard error of the plus-minus estimate e) How many possessions would you need to cut the standard error in half?

Exercise 30: Role Change Analysis

A player changed roles mid-season:

First Half (Games 1-41): - Role: Sixth man - Minutes per game: 24 - Net Rating On: +6.5 - On/Off: +8.2

Second Half (Games 42-82): - Role: Starter - Minutes per game: 34 - Net Rating On: +3.2 - On/Off: +4.8

a) Calculate total minutes and weighted average net rating on court b) Why might the player's numbers be worse as a starter? c) List three basketball factors that changed with the role change d) How should we interpret this player's "true" value? e) If you were projecting next season where he'll start, which half is more predictive?

Exercise 31: Trade Analysis Using Plus-Minus

Team A is considering trading Player X for Player Y:

Player X (on Team A currently): - Net Rating On: +5.5 - On/Off: +7.2 - Team Net Rating: +3.0

Player Y (on Team B currently): - Net Rating On: +4.0 - On/Off: +9.5 - Team Net Rating: -1.0

a) Which player has better on-court impact in absolute terms? b) Which player appears more valuable relative to their backup? c) Estimate Team A's projected net rating if they swap (assume X's value transfers to Team B and vice versa) d) What are three major uncertainties in this projection? e) What additional data would make this analysis more reliable?

Exercise 32: Building an Adjusted Plus-Minus Model

You have play-by-play data for 100 games with 450 unique players. Each game has approximately 200 possessions.

a) How many observations (rows) would your regression have? b) How many predictors (columns) would your design matrix have? c) Why would ordinary least squares regression fail with this setup? d) Describe how ridge regression would help e) If you added a prior that all players are expected to be average (0), how would this change extreme estimates?

Section E: Critical Thinking and Essay Questions (Questions 33-40)

Exercise 33: The Attribution Problem

Write a 500-word essay addressing the following question:

"Raw plus-minus tells us what happened when a player was on the court, but not why it happened. Why is this distinction important for player evaluation, and what approaches help address this limitation?"

Exercise 34: Historical Context

Research and write about:

a) When did plus-minus first gain prominence in NBA analysis? b) Which teams were early adopters of plus-minus analysis? c) How has the availability of play-by-play data changed plus-minus research? d) What role did the "Moneyball" era play in basketball analytics adoption?

Exercise 35: Comparing Metrics

Compare and contrast: - Raw Plus-Minus - On/Off Differential - Box Plus-Minus (BPM) - Regularized Adjusted Plus-Minus (RAPM)

For each, describe: (1) what it measures, (2) data requirements, (3) key strengths, (4) key weaknesses.

Exercise 36: Ethical Considerations

Discuss the ethical implications of using plus-minus metrics for: a) Contract negotiations b) Trade decisions c) Playing time allocation d) Public perception of players

Consider: What happens when a player's plus-minus is heavily influenced by factors outside their control?

Exercise 37: Designing a Study

Design a research study to answer the question: "Do players with high on/off differentials have more impact in the playoffs than the regular season?"

Include: - Hypothesis - Data requirements - Methodology - Statistical tests - Potential confounds - Expected limitations

Exercise 38: Communication Challenge

You need to explain plus-minus analysis to: a) A general manager with no statistics background b) A player who has poor plus-minus despite good box score stats c) A fan on social media who claims plus-minus is "the only stat that matters"

Write a brief (200-word) explanation appropriate for each audience.

Exercise 39: Future Directions

What technological or data advances might improve plus-minus analysis in the next decade? Consider: - Tracking data - Machine learning - Real-time computation - New data sources

Write a 400-word forward-looking analysis.

Exercise 40: Integration Exercise

A team asks you to evaluate a free agent using plus-minus analysis. The player: - Has a +6.2 on/off differential - Played on a team with net rating +7.5 - Had teammates averaging +3.0 RAPM - Faced opponents averaging +1.5 RAPM - Played 2,400 minutes - Is 28 years old

Write a comprehensive evaluation (800-1000 words) that: 1. Interprets the raw numbers 2. Adjusts for context 3. Acknowledges uncertainties 4. Makes a recommendation with confidence level 5. Identifies what additional analysis would help

Answer Key Notes

Detailed solutions for calculation problems (Exercises 9-17) and selected application problems are available in the instructor's manual. For exercises requiring essay responses or analysis, evaluation rubrics are provided separately.

Key formulas for reference: - Offensive/Defensive Rating = (Points / Possessions) * 100 - Possessions = FGA - OREB + TOV + 0.44 * FTA - On/Off Differential = Net Rating On - Net Rating Off - Standard Error = SD / sqrt(n) - Bayesian Posterior = (Prior * Prior_Strength + Observed * Sample_Size) / (Prior_Strength + Sample_Size)