Activity 10

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 Analysis We 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=∑ni=1(xi−x¯)(yi−y¯) ∑ni=1(xi−x¯)2⋅∑ni=1(yi−y¯)2

For both datasets, n=10, x¯=3.5, and ∑(xi−x¯)2=14.5.2012 Detailed ComputationMean wins (y¯2012): 93.4 Sum of cross-products (SPxy): 13.0 Sum of squared deviations (SSy): 96.4

r2012=13.014.5×96.4−−−−−−−−−√=13.01397.8−−−−−√≈0.3477129

2013 Detailed Computation Mean wins (y¯2013): 93.9 Sum of cross-products (SPxy): −18.5 Sum of squared deviations (SSy): 54.9

r2013=−18.514.5×54.9−−−−−−−−−√=−18.5796.05−−−−−√≈−0.6556945

Statistical Interpretation Direction 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.