In: Statistics and Probability
3. Use "MLB_Salaries" data in Chapter3.xlsx to answer the following questions. For questions that require Excel, include the appropriate output (copy + paste) along with an explanation. Data description: An article in The Wall Street Journal (July 11, 2008) outlined a number of reasons as to why the 16 teams in Major League Baseball’s National League (NL) are inferior to the 14 teams in the American League (AL). One reason for the imbalance pointed to the disparity in opening-day payrolls: the average AL payroll is greater than the NL average. A portion of the data showing opening-day payroll (in $) for each team is shown in the accompanying table.
Questions:
a. Discuss the mean and the median of AL and NL opening-day salaries and comment on skewness.
b. Compare the range and the standard deviation of AL and NL opening-day salaries.
c. Use these summary measures to comment on the findings in The Wall Street Journal.
American League | Payroll | National League | Payroll |
New York Yankees | 206333389 | Chicago Cubs | 146609000 |
Boston Red Sox | 162447333 | Philadelphia Phillies | 141928379 |
Detroit Tigers | 122864928 | New York Mets | 134422942 |
Chicago White Sox | 105530000 | San Francisco Giants | 98641333 |
Los Angeles Angels | 104963866 | Los Angeles Dodgers | 95358016 |
Minnesota Twins | 97559166 | St. Louis Cardinals | 93540751 |
Seattle Mariners | 86510000 | Houston Astros | 92355500 |
Baltimore Orioles | 81612500 | Atlanta Braves | 84423666 |
Tampa Bay Rays | 71923471 | Colorado Rockies | 84227000 |
Kansas City Royals | 71405210 | Milwaukee Brewers | 81108278 |
Toronto Blue Jays | 62234000 | Cincinnati Reds | 71761542 |
Cleveland Indians | 61203966 | Washington Nationals | 61400000 |
Texas Rangers | 55250544 | Arizona Diamondbacks | 60718166 |
Oakland Athletics | 51654900 | Florida Marlins | 57034719 |
San Diego Padres | 37799300 | ||
Pittsburgh Pirates | 34943000 |
American League | Payroll | National League | Payroll |
New York Yankees | 206333389 | Chicago Cubs | 146609000 |
Boston Red Sox | 162447333 | Philadelphia Phillies | 141928379 |
Detroit Tigers | 122864928 | New York Mets | 134422942 |
Chicago White Sox | 105530000 | San Francisco Giants | 98641333 |
Los Angeles Angels | 104963866 | Los Angeles Dodgers | 95358016 |
Minnesota Twins | 97559166 | St. Louis Cardinals | 93540751 |
Seattle Mariners | 86510000 | Houston Astros | 92355500 |
Baltimore Orioles | 81612500 | Atlanta Braves | 84423666 |
Tampa Bay Rays | 71923471 | Colorado Rockies | 84227000 |
Kansas City Royals | 71405210 | Milwaukee Brewers | 81108278 |
Toronto Blue Jays | 62234000 | Cincinnati Reds | 71761542 |
Cleveland Indians | 61203966 | Washington Nationals | 61400000 |
Texas Rangers | 55250544 | Arizona Diamondbacks | 60718166 |
Oakland Athletics | 51654900 | Florida Marlins | 57034719 |
San Diego Padres | 37799300 | ||
Pittsburgh Pirates | 34943000 | ||
Total | 1341493273 | 1376271592 |
a.
Mean of American League Payroll = 1341493273 / 14 = 95820948.07
Mean of National League Payroll = 1376271592 / 16 = 86016974.5
Now, we arrange the data in ascending order as :
S.No. | American League | Payroll | S.No. | National League | Payroll |
1 | 51654900 | Oakland Athletics | 1 | Pittsburgh Pirates | 34943000 |
2 | 55250544 | Texas Rangers | 2 | San Diego Padres | 37799300 |
3 | 61203966 | Cleveland Indians | 3 | Florida Marlins | 57034719 |
4 | 62234000 | Toronto Blue Jays | 4 | Arizona Diamondbacks | 60718166 |
5 | 71405210 | Kansas City Royals | 5 | Washington Nationals | 61400000 |
6 | 71923471 | Tampa Bay Rays | 6 | Cincinnati Reds | 71761542 |
7 | 81612500 | Baltimore Orioles | 7 | Milwaukee Brewers | 81108278 |
8 | 86510000 | Seattle Mariners | 8 | Colorado Rockies | 84227000 |
9 | 97559166 | Minnesota Twins | 9 | Atlanta Braves | 84423666 |
10 | 104963866 | Los Angeles Angels | 10 | Houston Astros | 92355500 |
11 | 105530000 | Chicago White Sox | 11 | St. Louis Cardinals | 93540751 |
12 | 122864928 | Detroit Tigers | 12 | Los Angeles Dodgers | 95358016 |
13 | 162447333 | Boston Red Sox | 13 | San Francisco Giants | 98641333 |
14 | 206333389 | New York Yankees | 14 | New York Mets | 134422942 |
15 | Philadelphia Phillies | 141928379 | |||
16 | Chicago Cubs | 146609000 |
Median = [ (n+1)/2 ]th value
Median for American League Payroll = [ (14+1)/2 ]th value = 7.5th value = 7th value + ( 8th value - 7th value)*0.5
= 81612500 + ( 86510000 - 81612500)*0.5 = 84061250
Median for National League Payroll = [ (16+1)/2 ]th value = 8.5th value = 8th value + ( 9th value - 8th value)*0.5
= 84227000+ ( 84423666- 84227000)*0.5 = 84325333
b.
Range for American League Payroll = 206333389 - 51654900 = 154678489
Range for National League Payroll = 146609000 - 34943000 = 111666000
Sample Standard deviation is obtained using the formula :
We get :
Sample Standard deviation for American League Payroll = 43803002.27
Sample Standard deviation for National League Payroll = 33403170.09
c.
We can see that Mean of American League Payroll is greater than Mean of National League Payroll.
Median of American League Payroll is approximately equal to Median of National League Payroll.
Sample Standard deviation for American League Payroll is greater than Sample Standard deviation for National League Payroll.
Hence, one of the findings in The Wall Street Journal is that the average AL payroll is greater than the NL average which is observed from the given data.