Activity 10

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:

In your R console, create a corresponding rank vector by typing teamRank = c(1,2,3,3,4,4,4,4,5,5) In this quick question, we’ll see how well these rankings correlate with the regular season wins of the teams. In 2012, the ranking of the teams and their regular season wins were as follows: Rank 1: San Francisco Giants (Wins = 94) Rank 2: Detroit Tigers (Wins = 88) Rank 3: New York Yankees (Wins = 95), and St. Louis Cardinals (Wins = 88) Rank 4: Baltimore Orioles (Wins = 93), Oakland A’s (Wins = 94), Washington Nationals (Wins = 98), Cincinnati Reds (Wins = 97) Rank 5: Texas Rangers (Wins = 93), and Atlanta Braves (Wins = 94) Create a vector in R called wins2012, that has the wins of each team in 2012, in order of rank (the vector should have 10 numbers). In 2013, the ranking of the teams and their regular season wins were as follows: Rank 1: Boston Red Sox (Wins = 97) Rank 2: St. Louis Cardinals (Wins = 97) Rank 3: Los Angeles Dodgers (Wins = 92), and Detroit Tigers (Wins = 93) Rank 4: Tampa Bay Rays (Wins = 92), Oakland A’s (Wins = 96), Pittsburgh Pirates (Wins = 94), and Atlanta Braves (Wins = 96) Rank 5: Cleveland Indians (Wins = 92), and Cincinnati Reds (Wins = 90) Create another vector in R called wins2013, that has the wins of each team in 2013, in order of rank (the vector should have 10 numbers).

# 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=∑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 ComputationMean 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 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.