EXAMPLE 3.2.8

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	Frequency
0	96
1	32
2	28
3	20
4	12
5	10
6	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 201/2 indicates the “middle position” in this list. That is, the “middle of the list” is 100.5, or between the 100^th and 101^st numbers. To find the median we need to count frequencies until we find the 100^th and 101^st numbers.

The first 96 numbers are all “0s” and the next 32 numbers are all “1s,’ so the 100^th and 101^st numbers are both “1.” The median is 1, not 3.

The answer to question #1 is “False.”

2. Now we find the mean:

The answer to question #2 is “False.”

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

Since the value “2” has a frequency of 28, this statement is true.

4. True or false: 128 people gave the film a rank of less than 2.

True; 96 people gave the film people gave the film a ranking of 1, and another 32 people gave the film a rank of 1, so 128 people gave the the film a rank of less than 2.