Chapter 7 Exercises: Rate Statistics and Pace Adjustment

Section A: Per-Minute Calculations (Exercises 1-8)

Exercise 1: Basic Per-36 Calculation

A player has the following season statistics: - Minutes Played: 2,214 - Points: 1,476 - Rebounds: 615 - Assists: 344

Calculate the player's per-36-minute statistics for points, rebounds, and assists.

Exercise 2: Comparing Per-Minute Rates

Two players from the same team have the following statistics:

a) Calculate per-36 minute statistics for both players. b) Which player is the more efficient scorer per minute of playing time? c) Which player is the better rebounder per minute of playing time?

Exercise 3: Per-48 Minute Projection

A reserve player averages 18.4 minutes per game with the following per-game averages: - 8.2 points - 3.1 rebounds - 1.4 assists

a) Calculate their per-48 minute projections. b) Discuss two reasons why these projections might not accurately reflect what the player would produce if given starter minutes.

Exercise 4: College to NBA Translation

A college player has the following statistics over a 35-game season: - Minutes: 1,120 (32.0 per game) - Points: 595 (17.0 per game) - Rebounds: 280 (8.0 per game) - Assists: 105 (3.0 per game)

a) Calculate per-40 minute statistics (college game length). b) Calculate per-36 minute statistics (NBA convention). c) Discuss how these rate statistics might help project NBA performance.

Exercise 5: Minimum Minutes Threshold

You have data on the following players:

a) Calculate which players meet a 500-minute threshold for reliable per-36 statistics. b) Explain why Player D's per-36 scoring rate should not be compared directly to Player A's. c) What minimum games-played threshold would you recommend in addition to minutes?

Exercise 6: Per-Minute Efficiency Trade-offs

A team is considering extending a role player's minutes from 22 to 32 per game. The player's current statistics in 22 minutes: - 11.5 points on 48% FG - 4.2 rebounds - 1.8 assists - 1.2 turnovers

a) Calculate the player's per-36 projections. b) Research suggests efficiency typically drops 8-12% when minutes increase by 50% or more. Apply a 10% efficiency decline to project the player's actual production in 32 minutes. c) Compare the naive per-36 projection to the efficiency-adjusted projection.

Exercise 7: Multiple Per-Minute Baselines

Calculate per-36, per-40, and per-48 minute statistics for a player with: - Minutes: 2,952 - Points: 2,135 - True Shooting Attempts: 1,842

Create a table showing all three rate statistics and explain when each baseline is most appropriate.

Exercise 8: Rate Stat Anomalies

Identify and explain the flaw in each of the following statements:

a) "Player X has a 32.5 points per-36 average, making him the best scorer in the league." Additional context: Player X played only 180 minutes due to injury.

b) "The backup point guard has better per-36 assist numbers than the starter, so they should switch roles." Additional context: The backup primarily plays against opposing bench units.

c) "This rookie's per-48 numbers project him as an All-Star caliber player." Additional context: The rookie has played in garbage time in 80% of his minutes.

Section B: Possessions and Pace (Exercises 9-16)

Exercise 9: Basic Possession Estimation

Calculate the estimated possessions for a team with the following single-game statistics: - Field Goal Attempts: 88 - Free Throw Attempts: 24 - Offensive Rebounds: 11 - Turnovers: 14

Use the standard formula: Possessions = FGA + 0.44 x FTA - ORB + TOV

Exercise 10: Season Pace Calculation

A team has the following season statistics over 82 games: - Total Field Goal Attempts: 7,134 - Total Free Throw Attempts: 1,886 - Total Offensive Rebounds: 924 - Total Turnovers: 1,148 - Total Minutes Played: 19,880 (including overtime)

a) Calculate total estimated possessions. b) Calculate the team's pace (possessions per 48 minutes). c) How does this pace compare to a league average of 99.5?

Exercise 11: Comparing Possession Formulas

For the following team game statistics, calculate possessions using both formulas:

Team stats: FGA=92, FG=41, FTA=28, ORB=13, TOV=16 Opponent stats: FGA=86, FG=38, FTA=22, ORB=10, DRB=33, TOV=12

a) Standard formula: Poss = FGA + 0.44 x FTA - ORB + TOV b) Refined formula (team perspective only): Poss = FGA + 0.4 x FTA - 1.07 x (ORB/(ORB+Opp_DRB)) x (FGA-FG) + TOV c) Calculate the difference and explain why the refined formula might be more accurate.

Exercise 12: Free Throw Coefficient Sensitivity

Using the base statistics from Exercise 9, calculate possessions with: a) FT coefficient = 0.40 b) FT coefficient = 0.44 c) FT coefficient = 0.475

What is the range of possession estimates, and what does this tell you about the precision of possession estimation?

Exercise 13: Pace Trend Analysis

Given the following pace data for a franchise over five seasons:

a) Calculate the average pace over this period. b) Calculate the year-over-year pace change for each transition. c) A player who averaged 18.5 PPG in 2021-22 averaged 20.2 PPG in 2023-24. How much of this increase could be attributed to pace change alone?

Exercise 14: Offensive and Defensive Rating

A team has the following season totals: - Points Scored: 9,328 - Points Allowed: 9,156 - Estimated Possessions: 8,164

a) Calculate the team's Offensive Rating (points per 100 possessions). b) Calculate the team's Defensive Rating (points allowed per 100 possessions). c) If league average is 114.2 for both, evaluate this team's offensive and defensive performance relative to the league.

Exercise 15: Individual Per-100 Possessions

A player has the following statistics while on the court: - Points: 1,842 - Rebounds: 498 - Assists: 372 - Team Possessions (while player on court): 4,850

Calculate the player's per-100-possessions rates for points, rebounds, and assists.

Exercise 16: Pace and Game Outcomes

Two teams meet in a playoff game: - Team A (regular season pace: 104.2, ORtg: 118.5, DRtg: 112.3) - Team B (regular season pace: 94.8, ORtg: 115.2, DRtg: 108.6)

a) If the game is played at Team A's preferred pace (104.2), estimate each team's expected points using their regular season ratings. b) If the game is played at Team B's preferred pace (94.8), estimate each team's expected points. c) Which team benefits more from controlling pace? Explain.

Section C: Pace Adjustment (Exercises 17-24)

Exercise 17: Basic Pace Adjustment

Two players from different teams have the following statistics:

League average pace: 99.5

a) Calculate pace-adjusted statistics for both players. b) After adjustment, which player is the superior scorer? By how much? c) After adjustment, which player is the superior rebounder? By how much?

Exercise 18: Multiple Stat Pace Adjustment

Perform pace adjustment for the following player:

Raw stats: 19.8 PPG, 8.4 RPG, 2.1 APG, 1.3 SPG, 0.4 BPG, 2.8 TOV Team Pace: 101.8 League Pace: 99.5

Create a table showing raw and pace-adjusted statistics side by side.

Exercise 19: Pace Adjustment Direction

For each scenario, determine whether pace adjustment will increase or decrease the player's statistics, and by approximately what percentage:

a) Player on a team with pace 105.0, league average 99.5 b) Player on a team with pace 93.2, league average 99.5 c) Player on a team with pace 99.5, league average 99.5 d) Player on a team with pace 88.5, league average 99.5

Exercise 20: Era Comparison

Compare these two players from different eras:

a) Calculate each player's per-100-possessions scoring rate. b) Express each player's scoring relative to their league average pace. c) Which player was the more dominant scorer relative to their era?

Exercise 21: Complete Era Adjustment

Wilt Chamberlain in 1961-62: - PPG: 50.4 - Team Pace: ~130 - League Average PPG: ~118.8

Modern context (2022-23): - League Average PPG: ~114.7 - League Pace: ~100

a) Calculate Wilt's points per 100 possessions. b) Project Wilt's scoring at modern pace (100 possessions). c) Calculate what percentage above league average Wilt scored in his era. d) Apply that relative performance to modern league average scoring.

Exercise 22: Z-Score Comparison

Given the following league distributions for PPG:

Player from 1975-76: 28.5 PPG Player from 2022-23: 31.4 PPG

a) Calculate the z-score for each player's scoring. b) Which player was more dominant relative to their era? c) What are the limitations of this z-score comparison?

Exercise 23: Team Context Adjustment

A player is traded mid-season: - First half: 18.2 PPG on Team A (pace: 103.8) - Second half: 21.4 PPG on Team B (pace: 97.2)

League pace: 99.5

a) Pace-adjust both halves of the season to league average. b) Calculate the player's true improvement after accounting for pace. c) What other factors besides pace might explain the statistical change?

Exercise 24: Comprehensive Player Comparison

Compare these three players using multiple adjustment methods:

League pace: 99.5

a) Calculate pace-adjusted per-game statistics. b) Calculate per-36-minute statistics. c) Calculate pace-adjusted per-36-minute statistics. d) Rank the players by scoring using each method. Do rankings change?

Section D: Applied Problems (Exercises 25-32)

Exercise 25: Scouting Report Adjustment

You're scouting a player for your team (pace: 96.5) who currently plays for a team with pace 104.8. His current stats: - 16.4 PPG, 5.2 RPG, 3.8 APG

a) Project his expected statistics on your team. b) What other factors should you consider beyond pace when making this projection?

Exercise 26: Historical MVP Comparison

Three MVP seasons from different eras:

a) Calculate per-100-possession statistics for all three players. b) Rank them by pace-adjusted scoring, rebounding, and assists. c) Which era appears to have the most "inflated" raw statistics?

Exercise 27: Pace Impact on Team Statistics

A team considering a style change analyzes pace impact:

Current stats (pace 94.2): - PPG: 108.2 - Opp PPG: 104.8 - ORtg: 114.8 - DRtg: 111.2

If they increase pace to 102.0 while maintaining the same efficiency ratings:

a) Project their new PPG and Opp PPG. b) Will their point differential change? Explain. c) Under what circumstances would increasing pace benefit a team?

Exercise 28: Rate Stat Report Card

Create a comprehensive rate statistics report for this player:

Season Totals (82 games): - Minutes: 2,788 - Points: 2,156 - FGA: 1,642 - FGM: 782 - FTA: 524 - FTM: 445 - Rebounds: 548 - Assists: 486 - Steals: 98 - Blocks: 42 - Turnovers: 234 - Team Pace: 101.2 - League Pace: 99.5

Calculate: a) Per-game averages b) Per-36-minute statistics c) Per-48-minute statistics d) Estimated possessions and per-100-possession statistics e) Pace-adjusted per-game statistics

Exercise 29: Rotation Analysis

A coach wants to analyze the efficiency of different rotation patterns:

a) Calculate points per 100 possessions for each lineup. b) Calculate points per 36 minutes for each lineup. c) Which metric better captures lineup efficiency? Why? d) What additional context would help interpret these numbers?

Exercise 30: League-Wide Pace Analysis

Given pace data for all 30 NBA teams (hypothetical):

Fast (Pace > 101): 8 teams, average ORtg 113.2, average DRtg 114.8 Medium (Pace 97-101): 14 teams, average ORtg 114.0, average DRtg 113.6 Slow (Pace < 97): 8 teams, average ORtg 112.8, average DRtg 111.4

a) Calculate net rating (ORtg - DRtg) for each group. b) Which pace category has the best average net rating? c) Is there a correlation between pace and team success in this data? d) What confounding factors might affect this analysis?

Exercise 31: Historical Pace Adjustment Project

Adjust these historical scoring seasons to a common baseline of 100 possessions per game:

a) Calculate pace-adjusted PPG for each season. b) Rank the seasons by pace-adjusted scoring. c) Which adjustment changes the ranking most significantly from raw PPG?

Exercise 32: Pace Manipulation Detection

A team's statistics raise questions about potential stat manipulation:

a) Calculate what the team's overall PPG would be using just close-game pace and efficiency. b) How much do blowout game statistics inflate the team's overall numbers? c) Propose a "competition-adjusted" rating that weights close games more heavily.

Section E: Programming Exercises (Exercises 33-40)

Exercise 33: Possession Calculator Function

Write a Python function that: - Takes team box score statistics as input - Calculates possessions using multiple formulas (standard, refined) - Returns a dictionary with all estimates and their differences

Test your function with sample data and compare results.

Exercise 34: Pace Adjustment Class

Implement a Python class called PaceAdjuster that: - Stores league average pace as an instance variable - Has methods for adjusting single stats and dictionaries of stats - Includes validation to prevent negative adjustments - Has a method to reverse pace adjustment (from adjusted back to raw)

Exercise 35: Era Comparison Tool

Create a Python program that: - Stores historical league data (pace, average stats) for multiple seasons - Allows comparison of players from different eras - Outputs a formatted comparison table with both raw and adjusted statistics

Exercise 36: Per-Minute Rate Calculator

Write a function that: - Accepts player statistics and minutes played - Calculates per-36, per-40, and per-48 minute rates - Includes a minimum minutes threshold warning - Returns confidence intervals for small samples

Exercise 37: Pace Trend Visualization

Create a Python script using matplotlib that: - Plots historical NBA pace from 1974-2024 - Highlights different eras with different colors - Adds annotations for significant pace changes - Includes trend lines for each era

Exercise 38: Comprehensive Rate Stats Report

Write a Python program that: - Takes a player's raw statistics as input - Outputs a comprehensive report including: - Raw per-game statistics - Per-36, per-40, per-48 minute rates - Per-100 possession rates - Pace-adjusted statistics - Comparison to league average

Exercise 39: Data Validation Module

Create a Python module that validates basketball statistics for rate calculations: - Checks for valid ranges (no negative stats, minutes <= 48*games, etc.) - Warns when samples are too small for reliable rate stats - Identifies potential data entry errors (e.g., more assists than teammate FGM)

Exercise 40: API Integration Exercise

Using the basketball-reference or NBA stats API: - Fetch current season data for a team - Calculate team pace using the refined formula - Compare calculated pace to official figures - Document any discrepancies and hypothesize causes

Answer Key Summary

Full solutions for exercises 1-32 are available in the instructor's materials. For programming exercises (33-40), sample solutions are provided in the companion code repository.

Key formulas for quick reference:

Per-Minute: $$\text{Stat per X} = \frac{\text{Raw Stat}}{\text{Minutes}} \times X$$

Possessions (Standard): $$\text{Poss} = \text{FGA} + 0.44 \times \text{FTA} - \text{ORB} + \text{TOV}$$

Pace: $$\text{Pace} = \frac{\text{Poss}}{\text{Minutes}} \times 48$$

Per-100 Possessions: $$\text{Stat per 100} = \frac{\text{Raw Stat}}{\text{Poss}} \times 100$$

Pace Adjustment: $$\text{Adjusted} = \text{Raw} \times \frac{\text{League Pace}}{\text{Team Pace}}$$

Z-Score: $$z = \frac{\text{Player Stat} - \mu}{\sigma}$$