In-class activity 10- Rank Correlations

Overview & Background In 2012 and 2013, 10 teams made the Major League Baseball (MLB) playoffs: the six division winners and four wild card teams. To evaluate whether regular season success predicts postseason performance, we assign ordinal ranks to teams based on their playoff finishes:

Rank 1: World Series Winner (Champion)

Rank 2: World Series Runner-Up

Rank 3: League Championship Series Losers (2 teams)

Rank 4: Division Series Losers (4 teams)

Rank 5: Wild Card Game Losers (2 teams)

R Code Implementation Chunk 1: Vector Definitions We first construct the rank vector and win vectors for both seasons ordered by playoff finish:

# Define team ranks vector (1 = Champion, 5 = Early Elimination)
teamRank <- c(1, 2, 3, 3, 4, 4, 4, 4, 5, 5)

# 2012 Regular Season Wins (ordered by teamRank)
# Rank 1: SF (94) | Rank 2: DET (88)
# Rank 3: NYY (95), STL (88)
# Rank 4: BAL (93), OAK (94), WSH (98), CIN (97)
# Rank 5: TEX (93), ATL (94)
wins2012 <- c(94, 88, 95, 88, 93, 94, 98, 97, 93, 94)

# 2013 Regular Season Wins (ordered by teamRank)
# Rank 1: BOS (97) | Rank 2: STL (97)
# Rank 3: LAD (92), DET (93)
# Rank 4: TB (92), OAK (96), PIT (94), ATL (96)
# Rank 5: CLE (92), CIN (90)
wins2013 <- c(97, 97, 92, 93, 92, 96, 94, 96, 92, 90)

Chunk 2: Exercises & Correlation AnalysisWe calculate Pearson’s correlation coefficient \(r\) for both seasons using R’s cor() function:

# Exercise 1: Correlation for 2012 Season
cor_2012 <- cor(teamRank, wins2012)
cor_2012

[1] 0.3477129

# Exercise 2: Correlation for 2013 Season
cor_2013 <- cor(teamRank, wins2013)
cor_2013

[1] -0.6556945

Mathematical MethodologyThe Pearson correlation coefficient \(r\) measuring the linear association between playoff rank (\(x\)) and regular season wins (\(y\)) is computed as:\[r = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n} (x_i - \bar{x})^2 \cdot \sum_{i=1}^{n} (y_i - \bar{y})^2}}\]For both datasets, \(n = 10\), \(\bar{x} = 3.5\), and \(\sum (x_i - \bar{x})^2 = 14.5\).2012 Detailed ComputationMean wins (\(\bar{y}_{2012}\)): \(93.4\)Sum of cross-products (\(\text{SP}_{xy}\)): \(13.0\)Sum of squared deviations (\(\text{SS}_y\)): \(96.4\)\[r_{2012} = \frac{13.0}{\sqrt{14.5 \times 96.4}} = \frac{13.0}{\sqrt{1397.8}} \approx \mathbf{0.3477129}\]2013 Detailed ComputationMean wins (\(\bar{y}_{2013}\)): \(93.9\)Sum of cross-products (\(\text{SP}_{xy}\)): \(-18.5\)Sum of squared deviations (\(\text{SS}_y\)): \(54.9\)\[r_{2013} = \frac{-18.5}{\sqrt{14.5 \times 54.9}} = \frac{-18.5}{\sqrt{796.05}} \approx \mathbf{-0.6556945}\]Statistical InterpretationDirection of Ranks:Because Rank 1 represents the best outcome and Rank 5 represents early elimination:A negative correlation (\(r < 0\)) indicates that higher regular season win totals correspond with lower rank numbers (better playoff performance). In 2013 (\(r = -0.656\)), regular season wins predicted postseason success well.A positive correlation (\(r > 0\)) indicates that higher regular season win totals correspond with higher rank numbers (worse playoff performance). In 2012 (\(r = +0.348\)), regular season wins were inversely related to playoff outcomes.Postseason Volatility:The sign flip from \(+0.348\) in 2012 to \(-0.656\) in 2013 highlights how small sample sizes (short playoff series) and inherent randomness make regular season win totals an unreliable single predictor for playoff champion outcomes.