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.