Question

48. Recall that the file Baseball Salaries.xlsx contains data on 868 MLB players in the 2015 season. Using this data set, answer the following questions:
a) Find the mean and median of the salaries of all shortstops. Are any of these measures influenced significantly by one or more unusual observations?
b) Find the standard deviation, first and third quartiles, and 5th and 95th percentiles for the salaries of all shortstops. Are any of these measures influenced significantly by one or more unusual observations? Create a histogram of the salaries of all shortstops.
c) Are any of these measures influenced significantly by one or more unusual observations?

Figure 2.9 3 1 Player Salary Team Position Baseball Salaries 2 A.J. Burnett 3 A.J. Ellis Pittsburgh Pirates Pitcher $8,500,000 $4,250,000 Catcher 4 A.J. Griffin Oakland Athletics Pitcher $517,500 AJ.Pierzynski 6 A.J. Pollock Catcher $2,000,000 $519,500 Atlanta Braves Arizona Diamondbacks Center Fielder Pitcher 530,000 $514,200 7 A.J. Ramos Miami Marlins Pitcher 8 Aaron Barrett Washington Nationals Miami Marlins Pitcher $1,975,000 9 Aaron Crow Philadelphia Phillies $5,000,000 Second Baseman $12,000,000 Pitcher 10 Aaron Harang Arizona Diamondbacks 11 Aaron Hill $527,000 $510,800 Pitcher 12 Aaron Loup 13 Aaron Sanchez Pitcher Toronto Blue Jays 14 Aaron Thom Minnesota Twins Pitcher 520,000

Answer 1

We can find the mean ,median, standard deviation,1^st and 3^rd quartiles , 5^th and 95^th percentile using the excel as follows :

1 8500000 2 4250000 3 517500 4 2000000 5 519500 6 530000 7 514200 8 1975000 9 5000000 10 12000000 11 527000 12 510800 13 520000 15 FAVERAGE(A1:A13) 16MEDIAN(A1:A13 2874154 530000 17 STDEV.S(A1:A13) 3671004 18 QUARTILE(A1:A13,1) 519500 19 EQUARTILE(A1 A13,3) 20 PERCENTILE(A1:A13,0.05) 512840 21 PERCENTILE(A1:A13,0.95 4250000 9900000

a) Mean = 2874154  

Median = 530000

Mean is influenced significantly by one or more unusual observations called as an outlier.

b)  

Standard deviation = 3671004

1^st Quartile = 519500

3^rd Quartile = 4250000

5^th Percentile = 512840

95^th percentile = 9900000

Standard deviation is influenced significantly by one or more unusual observations called as an outlier.

Histogram :

Let us assume that we have to construct 6 classes

First we need to find the class intervals ( lower and upper ) for that we need to find the class width.

Class width =
(Marimum- minimum) classes
=
(12000000-510800) 6
  

Class width = 1914867

Lower limit of 1st class would be the minimum number = 510800

Lower limit of 1st class would be 510800 + class width = 510800 + 1914867 = 2425667

Similarly we can find the Lower limits of remaining classes

Upper limit of 1st class = lower limit of 1st class - 1 = 2425667 - 1 = 2425666

Upper limit of 2nd class = 2425666 + class width = 2425666 + 1914867 = 4723506

Similarly we can find the upper limits of remaining classes

Then we find the frequency of each class intervals.

Lower Upper Frequency 510800 2425666 2425667 4723506 4340534 7021346 6255401 9319186 8170268 11617026 10085135 139148661

Then to draw histogram only select frequencies go to Insert menu ---> Charts ---- > Select Clustered column

it will give the histogram :

Histogranm 10 ■ Frequency 0 Salary

C) The data set contain one unusual observation 12000000 and it affects mean and standard deviation of the data set.