The general rule for determining the position of the median is that the median will always be in position (n + 1) / 2 in an ordered list. The positions can be indicated from smallest to largest (ascending order) or from largest to smallest (descending order). The median is in the same position and is the same value, regardless of the ordering method you use.
It is important to remember that this rule indicates the position of the median, and not the value of the median. The value of the median is the value at that position. In our 13-noodle example, n = 13, and the position of the median is determined by (13 + 1) / 2 = 14 / 2 = 7. So the median is in position (7), and the value of Med is the length of the seventh noodle.
Note that there are six noodles to the left (1L to 6L) and six noodles to the right (6R to 1R) of the median. To find the positions of the remaining quantities for the Five-Number Summary, it's convenient to label the noodles to the left of the median in ascending order and the noodles to the right of the median in descending order:
Again, notice that Med is in position (7) from each end of the ordered list, and notice that Min (the shortest noodle) and Max (the longest noodle) are each in position (1) on their respective ends of the ordered data.
Problem E2
What is the position of Q1 in this ordered list?
Remember that you should only consider noodles to the left of the median. Do not include the median itself in this count. Close Tip
Problem E3
What is the position of Q3 in this ordered list?
Problem E4
Here is the ordered list of the numerical values for the 13 noodles and the corresponding position of each measurement from its respective end of the data:
13 Noodles (in millimeters)
13
23
28
33
41
56
74
81
91
102
118
122
127
(1L)
(2L)
(3L)
(4L)
(5L)
(6L)
(7)
(6R)
(5R)
(4R)
(3R)
(2R)
(1R)
Use the information from the position of the data in this ordered list and the results from Problems E2 and E3 to build the Five-Number Summary for this data.
Remember that a summary value that lies in a position halfway between two items in an ordered list is the average of the adjacent pair of values. Close Tip
Problem E5
Use the techniques you've learned in Part E to build the Five-Number Summary for the following set of measurements, where n = 15:
n = 15
10
12
15
18
22
24
25
26
34
39
45
51
62
75
89
Problem E6
Build the Five-Number Summary for the following set of measurements, where n = 20:
n = 20
1
1
1
3
4
4
5
10
10
17
20
24
26
33
34
38
39
50
53
53
Ignore the values of the data when finding the positions of the median and quartiles. It is possible for the values surrounding the median and quartiles to be identical. Close Tip
Problem E7
Here are the lengths of 20 pine needles, to the nearest millimeter, from Session 1, Problem H1:
Length of Needles (in millimeters)
117
56
48
69
71
120
111
49
68
110
109
64
93
43
109
37
93
40
86
47
a.
Determine the Five-Number Summary for these 20 measurements.
b.
Draw a box plot for these 20 measurements.
c.
Give a brief interpretation of this summary. What does it tell you about the lengths of the pine needles?
Don't forget that in order to build a Five-Number Summary or a box plot, you will need to order the list first! Close Tip
