Question

One important statistic in baseball is a pitcher’s earned run average, or ERA. This number represents the average number of earned runs given up by the pitcher per nine innings. The following table lists the portion of the ERAs for pitchers playing for the New York Yankees and the Baltimore Orioles as of July 22, 2010; the complete data, labeled ERA , are available on the text website.

New York ERA Baltimore ERA
Sabathia 3.13 Guthrie 4.58
Pettitte 2.88 Millwood 5.77
Burnett 4.99 Matusz 5.21
Hughes 3.99 Bergeson 6.51
Vazquez 4.68 Hernandez 4.29
Chamberlain 5.80 Berken 2.50
Gaudin 6.81 Hendrickson 5.23
Rivera 0.98 Albers 4.31
Robertson 4.86 Arrieta 4.87
Park 5.93 Simon 3.14
Mitre 2.88 Ohman 2.57
Logan 3.92 Tillman 7.92
Marte 4.08 Mata 7.79
Aceves 3.00 Meredith 5.40
Moseley 7.50 Uehara 2.92
Melancon 9.00 Castillo 10.13
Albaladejo 5.40 Johnson 6.52
Mickolio 7.36
Gonzalez 18.00

Required:
a. Calculate the mean and the median ERAs for the New York Yankees.
b. Calculate the mean and the median ERAs for the Baltimore Orioles.
c. Based solely on your calculations above, which team is likely to have the better winning record?

1. New York Yankees
2. Baltimore Orioles

Answer 1

(a):
`\bar x = 5.54` and
`Median = 4.86`

(b):
`\bar x = 5.27` and
`Median = 5.21`

(c): New York Yankees

Explanation:

Given

Data of New York ERA and Baltimore ERA

Solving (a): Mean and Median of New York ERA

Mean is calculated as:

`\bar x = (\sum x)/(n)`

For New York ERA, n = 19. So, we have:

`\bar x = (3.13 + 2.88 + 4.99 + 3.99 + 4.68 + 5.80 + 6.81 + 0.98 + 4.86 + 5.93 + 2.88 + 3.92 + 4.08 + 3.00 + 7.50 + 9.00 + 5.40 + 7.36 + 18.00)/(19)`

`\bar x = (105.19)/(19)`

`\bar x = 5.53631578947`

`\bar x = 5.54`

To calculate the median value, we first arrange the data (in ascending order):

So, we have:

0.98, 2.88, 2.88, 3.00, 3.13, 3.92, 3.99, 4.08, 4.68, 4.86, 4.99, 5.40, 5.80, 5.93, 6.81, 7.50, 7.36, 9.00, 18.00

The median value of odd number of data is:

`Median = (1)/(2)(n+1)\ th`

Substitute 19 for n

`Median = (1)/(2)(19+1)\ th`

`Median = (1)/(2)(20)\ th`

`Median = 10th`

So, the median is the 10th item.

`Median = 4.86`

(b): Mean and Median of Baltimore ERA

Mean is calculated as:

`\bar x = (\sum x)/(n)`

For Baltimore ERA, n = 17. So, we have:

`\bar x = (4.58+ 5.77+ 5.21+ 6.51+ 4.29+ 2.50+ 5.23+ 4.31+ 4.87+ 3.14+ 2.57+ 7.92+ 7.79+ 5.40+ 2.92+ 10.13+ 6.52)/(17)`

`\bar x = (89.66)/(17)`

`\bar x = 5.27411764706`

`\bar x = 5.27` --- approximated

Arrange in ascending order:

2.50 , 2.57 , 2.92 , 3.14 , 4.29 , 4.31 , 4.58 , 4.87 , 5.21, 5.23 , 5.40 ,5.77 , 6.51 , 6.52, 7.79 , 7.92 , 10.13

`Median = (1)/(2)(n+1)\ th`

`Median = (1)/(2)(17+1)th`

`Median = (1)/(2)(18)th`

`Median = 9th`

So, the median is the 9th item.

`Median = 5.21`

(c): Who has a better record

Statistically, the average ERA of New York Yankees is better than the average ERA of Baltimore Orioles.

Hence, New York Yankees hold the better record