EXAMPLE 2.9.8                    

Members of the preview audience for the new Arnie Schvartzengagger film,

The Predagagger XII: Things Blow Up, were asked to rank the film from 0 ("stinks like a dead cyborg") to 6 ("XII thumbs up!"). The results are summarized in the table below.

 

value 0 frequency 96

value 1 frequency 32

value 2 frequency 28

value 3 frequency 20

value 4 frequency 12

value 5 frequency 10

value 6 frequency 2

 

1. True or false: The median is 3.

2. True or false: The mean is 3.

3. True or false: Twenty-eight people gave the film a rank of 2.

4. True or false: 128 people gave the film a rank of less than 28.

 

SOLUTION

1.  First we add frequencies to find n, the number of data points in this distribution.

n = 96 + 32 + 28 + 20 + 12 + 10 + 2 = 200

The table represents an ordered list of 200 numbers.  To find the median, we use the fact that 201 divided by2 indicates the Òmiddle positionÓ in this list.  That is, the Òmiddle of the listÓ is 100.5, or between the 100th and 101st numbers.  To find the median we need to count frequencies until we find the 100th and 101st numbers.

The first 96 numbers are all Ò0sÓ and the next 32 numbers are all Ò1s,Õ so the 100th and 101st numbers are both Ò1.Ó  The median is 1, not 3.

The answer to question #1 is ÒFalse.Ó

 

 

2.  Now we find the mean:

 

Mean = [(96 times 0) plus (32 times 1) plus (28 times 2) plus (20 times 3) plus (12 times 4) plus (10 times 5) plus (2 times 6)] divided by 200

 

= 258 divided by 200

 

= 1.29

 

The answer to question #2 is ÒFalse.Ó

 

3. True or false: Twenty-eight people gave the film a rank of 2.

Since the value Ò2Ó has a frequency of 28, this statement is true.

 

 

4. True or false: 128 people gave the film a rank of less than 28.

False; 128 people gave the film a ranking of less than 2, not less than 28.