Analytic Methods in Sports Using Mathematics and Statistics to Understand Data from Baseball, Football, Basketball, and Other…

1. 1Name the units and the attribute set before computing anything. Each team or player becomes one row, and each controllable input (passing yards, rushing yards, passing yards allowed, rushing yards allowed, or pitch velocity and movement) becomes one column. Keep the same attributes for every unit.
2. 2Check the units of measurement across columns. If every column is on the same scale, as with the four yardage columns, plain Euclidean distance is defensible. If columns are on different scales, set the weights to w_j = 1/S_j², where S_j is the sample standard deviation of variable j. This is the same as scaling each variable by its standard deviation (the R function scale does it).
3. 3Pick the distance function deliberately. Euclidean distance (the default in R's dist) squares each attribute gap. Manhattan distance sums absolute gaps. Minkowski distance generalizes both with a power p > 0 that you choose. Set it with the method argument of dist.
4. 4Compute the distance matrix, for example cfb4.dist <- dist(cfb4). Verify that d_ij ≥ 0, d_ii = 0 and d_ij = d_ji. The triangle inequality (d_ij ≤ d_ik + d_jk) is not strictly required, but MDS works better when it holds.
5. 5Read the raw distances first. Find the closest and farthest pairs and check them against the attribute rows. This catches data-entry or scaling problems before you plot anything.
6. 6Feed the distance matrix to cmdscale with eig=T, for example cmdscale(cfb4.dist, eig=T). Plot the points with type="n" and asp=1, then add labels with text(..., labels=labels(cfb4.dist)). The aspect ratio of 1 makes plotted distances agree with the cmdscale output.
7. 7Report the goodness-of-fit from the GOF component of the result. The Big Ten hockey analysis reports 0.814. Treat the map as a summary of the distances, not as the distances themselves, and go back to the matrix for any pair that matters.
8. 8If no attribute set exists, build the dissimilarity from the form of the data. For two teams, sum the absolute differences in points earned against each other team (the Big Ten hockey method). A small value means the two teams performed similarly against the common opponents.

The four 2023-season college football playoff teams have four yardage attributes (PY, RY, PYA, RYA). Alabama is 228, 173, 189, 124.5; Michigan is 219, 162, 153, 87.1; Texas is 287, 189, 241, 80.9; Washington is 344, 125, 263, 133.7. For Alabama versus Michigan the gaps are 9, 11, 36 and 37.4. Squared and summed they give about 2897, and the square root is about 54, matching the dist output. Alabama versus Texas gives about 91. Washington is far from everyone: 146 from Alabama, 177 from Michigan, 103 from Texas. Michigan and Alabama are the nearest pair, and Washington is the outlier. All four columns are yardage, so no rescaling is needed here. The distance matrix goes into cmdscale(cfb4.dist, eig=T), and the labelled plot with asp=1 shows the same pattern: Alabama and Michigan close together, Washington isolated.

• If the attributes are on different scales (for example velocity and weight), then weight each by 1/S_j² before computing distance. Otherwise the largest-unit variable dominates.
• If all attributes share a unit, as the four yardage columns do, then unscaled Euclidean distance is acceptable.
• If you want to emphasize certain attributes, then raise their weights w_j. Otherwise, standardize.
• If the data are results rather than attributes (games between teams), then define dissimilarity from performance against common opponents, as in the Big Ten hockey sum of absolute point differences (9 for Michigan versus Minnesota).
• If a pair looks wrong on the map, then check the GOF value and the raw distance matrix before drawing conclusions.

• Mixing attributes in different units without scaling. The tell: one column explains almost all of every distance.
• Treating the dist output as an ordinary matrix. The tell: row names are missing and the code errors. The object has labels, not rownames, so use labels(cfb4.dist).
• Plotting the MDS without asp=1. The tell: the on-screen distances disagree with the cmdscale output.
• Reading the map without checking goodness-of-fit. The tell: you interpret small gaps on the plot that the distance matrix does not support.
• Choosing the distance measure by habit. The tell: Manhattan, Euclidean and Minkowski choices give different neighbors and you never compared them, even though p is an analyst decision.

• ▸Units compared: list the teams or players, one per row.
• ▸Attributes used: list each controllable skill, physical or scheme measure, one per column.
• ▸Unit of each attribute: write the unit so mixed scales are visible.
• ▸Scaling decision: record 'none (same units)' or 'weights 1/S_j²'.
• ▸Distance function: Euclidean, Manhattan, or Minkowski with your chosen p.
• ▸Closest and farthest pairs: record from the distance matrix.
• ▸Symmetry and zero-diagonal check: confirm d_ij = d_ji and d_ii = 0.
• ▸Goodness-of-fit from cmdscale (eig=T): record the GOF value.
• ▸Interpretation: write one sentence on which units cluster, which is the outlier, and which attribute drives it.

1. 1List the circumstances around the event before you model it. The passages name pitcher faced, runners on base, ballpark, and home or away. Weather, opposition quality, and game leverage belong on the same list. Write each factor down before you compute anything.
2. 2Test the constant-probability assumption. The Pujols example assumes the same hit probability in every at bat. Ask whether the circumstances you listed would plausibly move that probability. If they would, a single pooled number is an idealization, not a fact.
3. 3Match the level of detail to the purpose. A yearly batting average can tolerate the pooled assumption. A prediction for one specific matchup, such as Pujols facing Sabathia in Yankee Stadium, needs a more detailed analysis. State the purpose first, then decide how many context factors to include.
4. 4Convert the context into a conditional probability. Define the event of interest (B, the Bears win) and the circumstance (C, Cutler throws 4 interceptions). Estimate P(B given C) from only the games that satisfy C. Do not read it off the overall P(B).
5. 5Anchor the baseline to the setting. In the book's Bears-Packers experiment the Bears play at home and P(B) = 0.25. That 0.25 is a long-run frequency for that setup. Reuse it only when the next game matches the setup.
6. 6Adjust by combining the conditional pieces. Section 3.8 (law of total probability) and Section 3.11 (adjusting sports statistics) apply here. Split the sample by context, estimate within each context, then reweight to a common standard. This makes players or teams comparable across different circumstances.
7. 7Check sample size per context. The law of large numbers means relative frequencies approach true values only as the sample grows. Slicing by context shrinks each cell, so confirm every conditional estimate rests on enough games or at bats.
8. 8Report the adjusted number next to the raw number, and name the factors you adjusted for.

The source passages give the Bears-Packers experiment: the Bears play the Packers at home, and P(B) = 0.25, where B is the event that the Bears win. Read literally, that means the Bears win about 25% of a long run of such games. Now add a circumstance: C is the event that Jay Cutler throws 4 interceptions. You no longer care about all games, only those satisfying C. Restrict the sequence to games where C occurs and ask what fraction of those the Bears win. That is the conditional probability of B given C. It is a different number from 0.25, because the context changed. The same logic applies to Pujols. His season-long hit probability treats every at bat alike, but the probability depends on pitcher, runners on base, and park. For a yearly batting average the pooled number is fine. For Pujols against Sabathia in Yankee Stadium, you need the conditional version.

• If the question concerns a season-long total or average, accept a pooled probability. If it concerns one specific matchup or situation, condition on the circumstances.
• If a factor plausibly changes the event's probability (opposing pitcher, runners on base, park, home or away), then include it. Otherwise record it as an assumption you made.
• If two players or teams faced different circumstances, then adjust before comparing them (Section 3.11). Do not compare raw numbers.
• If conditioning leaves you with very few observations, then stop slicing. The law of large numbers needs sample size, so a small conditional cell is unreliable.
• If a stated probability comes from one setup (Bears at home against the Packers), then do not apply it to a different setup without re-estimating.

• Treating a pooled probability as universal. The tell: the same hit or win probability is applied to every at bat or game regardless of pitcher, park, or runners.
• Using the overall P(B) when a known circumstance applies. The tell: the 0.25 Bears figure is quoted for a game in which Cutler threw 4 interceptions.
• Comparing raw statistics earned under different circumstances. The tell: two players are ranked without any mention of opposition, park, or situation.
• Over-slicing the data. The tell: a conditional estimate rests on a handful of games and swings widely, which the law of large numbers warns against.
• Adjusting for factors the team controls and calling them context. The tell: the adjustment list includes things like lineup choices rather than home field, opponent, or weather.

• ▸Event of interest: define the outcome (e.g., Bears win, Pujols gets a hit).
• ▸Baseline probability and its setup: record the pooled figure and the conditions it came from (e.g., 0.25, Bears at home vs. Packers).
• ▸Context factors present: list pitcher, runners on base, park, home or away, opposition, weather, game leverage.
• ▸Purpose of the analysis: season-long summary or specific matchup prediction.
• ▸Is constant probability reasonable?: yes or no, with one line of justification.
• ▸Conditioning event: the specific circumstance (e.g., Cutler throws 4 interceptions).
• ▸Games or at bats meeting the condition: count the observations behind the conditional estimate.
• ▸Conditional probability estimate: the fraction of qualifying observations where the event occurs.
• ▸Adjusted vs. raw comparison: show both numbers and name the factors adjusted for.

1. 1Define the metric as a numeric coding of an event. Decide what counts as the event and what is excluded. Batting average codes any hit as 1 and any non-hit as 0, so it is the mean of a 0–1 variable. Any proportion works the same way. Corsi-percentage and Fenwick-percentage are shot shares, computed only in 5-on-5 play with shorthanded shots excluded. Write the denominator and the exclusions down before you calculate.
2. 2If the outcome is ordered, choose the weights explicitly. Slugging average assigns no hit 0, single 1, double 2, triple 3, home run 4, and averages those values over at bats. Treat the weights as a modeling choice, not a fact. They imply that a single-to-double gap equals a triple-to-home-run gap, and the resulting mean reflects that assumption.
3. 3Test the weighting by finding outcome sets that tie. Build two plausible player lines with the same total, then ask whether you would call them equivalent. If you would not, change the weights or use a different metric.
4. 4Standardize rate statistics to a common base. Compute the per-unit rate first, then scale it. Runs allowed per 9 innings is runs allowed divided by innings pitched, multiplied by 9. Choose the direction of a ratio deliberately: the book treats home runs per at bat versus at bats per home run as a real choice.
5. 5Choose the summary statistic for the data type. For at-bat outcomes the median is usually no hit for any player with a batting average under .500, so it carries no information. Use the mean of the coded variable, or report the outcome frequencies directly.
6. 6Attach a margin of error before you rank or compare. Game results depend on skill and luck, so a metric computed from a limited sample carries noise. Compute the margin of error (or simulate it for complicated statistics) and treat gaps smaller than it as unresolved.
7. 7Check relationships between metrics with correlation, then test practical size. When comparing Corsi-percentage, Fenwick-percentage and shot-percentage against point-percentage (points ÷ 2 × games), correlate each with it across team-seasons. Then ask whether the effect is large enough to matter, not only whether it is statistically significant.
8. 8Prefer fielding-independent or context-adjusted versions when the raw stat mixes in other players' work. For pitchers, WAR often uses fielding-independent measures that adjust for fielding's contribution to pitching statistics. State the version used, since several definitions of WAR exist.

Take Joey Votto's 2011 season: 599 at bats, 113 singles, 40 doubles, 3 triples, 29 home runs, 414 no hits. Hits are 113+40+3+29 = 185, so batting average (the mean of the 0–1 hit variable) is 185/599 ≈ .309. Total bases are 113 + 2(40) + 3(3) + 4(29) = 318, so slugging average is 318/599 ≈ .531. The median at-bat outcome is 'no hit,' which tells you nothing, as it does for any player under .500. Now test the weighting. Batter A has 500 at bats with 150 singles, 10 doubles and 10 triples; Batter B has 500 at bats with 50 singles, 15 doubles and 30 home runs. Both have 200 total bases, so both slug .400. Batter A has 170 hits and Batter B has 95, so batting average separates them where slugging does not. Whether slugging's tie is acceptable depends on your goal; if it is not, change the weights.

• If the variable is categorical and ordered, then convert it to numbers only after stating the weights and testing them with tied outcome sets.
• If more than 50% of events are the 'zero' outcome (e.g., batting average below .500), then do not use the median; use the mean of the coded variable or the frequency table.
• If two players or teams differ by less than the margin of error, then do not rank them on that metric.
• If a relationship is statistically significant, then check its size in the metric's own units before acting. Age and WHIP is significant (slope −0.00543, margin 0.00526), yet the 23-versus-36 gap is only about 0.07 WHIP.
• If situations differ, as with shorthanded play, then restrict the metric to comparable situations (5-on-5) before comparing teams.

• Treating arbitrary weights as truth. Tell: you defend slugging average as measuring value, but two very different hitters tie at 200 total bases and you cannot explain why that is fine.
• Reporting the median for a low-frequency success. Tell: every player's median at-bat outcome reads 'no hit,' so the summary cannot separate hitters.
• Confusing statistical significance with importance. Tell: you call age a driver of WHIP because the slope clears its margin of error, even though it moves WHIP by only about 0.07 across nearly the whole age range.
• Mixing situations in a rate. Tell: shot-share numbers include shorthanded shots, so teams that take more penalties look different for reasons unrelated to 5-on-5 play.
• Reading small-sample differences as skill. Tell: you rank players on a metric without any margin of error, and the order reshuffles the next time the data is sampled, because luck and randomness are part of every game result.

• ▸Metric name and sport: name the statistic and the sport/league it comes from.
• ▸Event definition: state exactly what counts as the event (e.g., hit = 1, non-hit = 0) and what is excluded.
• ▸Denominator and situation filter: state the base (at bats, innings, shots) and any restriction, such as 5-on-5 only.
• ▸Outcome coding/weights: list each outcome and its numeric value; mark whether the values are arbitrary.
• ▸Tie test: write two player lines that produce the same value and note whether you accept them as equivalent.
• ▸Rate direction/scale: state the unit (per 9 innings, per at bat or at bats per event) and why you chose it.
• ▸Sample size and margin of error: record the number of observations and the margin of error (or simulation result).
• ▸Relationship check: record the correlation with the target outcome (e.g., point-percentage) and whether it is significant.
• ▸Practical-size check: translate the effect into the metric's own units across the realistic range and judge whether it matters.

1. 1Fix the subject before you collect anything. The subject can be a game, a team-season, or a team's 2011 and 2012 versions treated as separate subjects. The source says the choice depends on the goals of the analysis. For season outcomes, use one row per team-season. For game outcomes, use one row per game.
2. 2List the outcome variables and label each by type. Points scored and points allowed are quantitative. Point differential is scored minus allowed, also quantitative. Win or loss is categorical, since 'average result' of W and L makes no sense without conversion. Keep these separate in your sheet.
3. 3Keep the raw game log in the order played (W W L W L ...) and then summarize it. The log has too much detail for most purposes. Count the wins and losses to get the frequency table. Divide each count by total games to get relative frequencies, reported as a proportion or percentage.
4. 4Convert W/L to a 0-1 variable (win = 1, loss = 0). The winning percentage is then the average of that variable, the same way batting average averages hit/non-hit. Any proportion is an average of 0-1 variables, so the same machinery applies.
5. 5Summarize each quantitative outcome with a mean and a spread. For spread, take each game's distance from the mean, square it, and average the squares to get the variance. Then take the square root to get the standard deviation, which is back in the original units (points, not points squared). Do this for points scored, points allowed, and differential.
6. 6Put rates on a common scale when exposure differs. The source's example is runs allowed per 9 innings: runs allowed divided by innings pitched, times 9. Apply the same logic when comparing outcomes across different numbers of games or opportunities.
7. 7State what experiment and event you are describing. 'The Packers playing the Bears' is an experiment and 'Packers win' is an event of it. 'Yankees win' is not an event of a single Jeter-Beckett at bat, because that at bat does not tell you whether it occurred. Match the outcome to a unit that resolves it.
8. 8Treat any single-number summary as one choice among weightings. State what it ignores. Two teams can have identical totals built from very different distributions, and the summary hides that.

Take the 2011 Yankees. The ordered game log begins W W L W L W L W L W W L ..., meaning the first two games were wins and the third a loss. That list is too detailed for most purposes, so you summarize it as a frequency table: 97 wins and 65 losses. Together that is 162 games. The relative frequency of wins is 97/162, about 59.9%. Coding each win as 1 and each loss as 0 makes 0.599 the average of the 162 values. That matches how batting average averages 0-1 outcomes. To see what a summary hides, use the source's slugging example. One batter has 150 singles, 10 doubles, and 10 triples in 500 at bats, and another has 50 singles, 15 doubles, and 30 home runs. Both total 200 bases, so they have the same slugging average. Season-level summaries work the same way: two teams can share a record or a differential and still have very different game-by-game distributions. That is why you also report the standard deviation, not just the mean.

• If your data are one row per game, then treat each game as the subject. If you want one number per season, then aggregate to team-season and say so.
• If a variable is W/L, then convert it to 0-1 before averaging. Never average a categorical variable directly.
• If two teams look equal on a single summary (same record, same total), then compare spread and the game-by-game distribution before calling them equivalent.
• If teams played different numbers of games or opportunities, then convert to a rate per fixed unit before comparing.
• If you want a typical value for a skewed or mostly-one-outcome variable, then check the median against the mean. The source's example is Votto's median at-bat result of 'no hit', which tells you almost nothing.

• Treating W/L as numeric without coding it. The tell: an 'average result' that has no interpretation.
• Assuming a summary number proves equivalence. The tell: two teams with the same record or total are called identical without anyone checking how the results were distributed.
• Defining the outcome at the wrong unit. The tell: you try to call 'team wins' an event of a single plate appearance or play, when it cannot be determined from that unit.
• Reporting only the mean and ignoring variation. The tell: no standard deviation, and surprise when outcomes vary game to game, though the source says outcomes vary even when the same two teams play repeatedly.
• Treating a weighting scheme as natural. The tell: a composite metric is defended as 'obviously right' when the source stresses that such assignments are somewhat arbitrary.

• ▸Subject definition: game, team-season, or team-version-by-year, plus one line on why it fits the goal.
• ▸Number of games: total games in the sample.
• ▸Wins and losses: count of each from the ordered game log.
• ▸Win proportion: wins divided by games, as a proportion or percentage.
• ▸Mean points scored and mean points allowed per game.
• ▸Mean point differential per game: mean scored minus mean allowed.
• ▸Standard deviation of point differential: square root of the average squared distance from the mean.
• ▸Rate conversions: any per-unit rate used (for example, per 9 innings) and its formula.
• ▸Weighting caveat: one line on what the chosen summary ignores or treats as equivalent.