Boxplots
Pictorial method which is used to describe
o center of data
o amount of spread in data
o symmetry or lack of symmetry in data
o outliers in the data
Definition: lower fourth and upper fourth
o Sort n data points smallest to largest.
o Find the median of the data, call it
x
o Divide the data into a lower half and an upper half. Include the median in
each half
o The median of the lower half is called the lower fourth
o The median of the upper half is called the upper fourth
Example:
median
Data
1 4 6 18 40 41 43 44 45 46 48 49 50 58 67 101 256
lower fourth
upper fourth
Definition: fourth spread
The fourth spread of a data set, fs, is defined to be
fs = upper fourh – lower fourth
Example:
For the data above fs = 50 – 40 = 10
With the above definitions in mind, one formal way to define outliers is as follows:
o An observation farther than 1.5fs units from the closest fourth is an outlier
o An observation farther than 3fs units from the closest fourth is an extreme
outlier
Example:
Four the data above:
o fs = 10, 1.5fs = 15 and 3fs= 30
o lower fourth = 40, upper fourth = 50
o 1,4,6 are extreme outliers—greater than 30 units from 40 (lower fourth)
o 18 is an outlier it is more than 15 ( but not more than 30) units from 40
o 67 is an outlier – more than 15 units from 50 ( upper fourth)
o 101 and 256 are extreme outliers
Minitab’s Version using the data from the example:
o Minitab shows only 101 and 256 as extreme outliers.
o Minitab shows only 2 outliers. Notice, eliminating the outliers, the spread
below the median is grater than above the median.
0
100
200
C1
Text: Example 1.20
Radon Concentration –linked to childhood cancers
Data: Radon Concentration in households in which a child has been diagnosed
with cancer. (Measured in Bq/m3)
Data : (stem and leaf)
WITH
WITHOUT
Stem-and-leaf of With
Leaf Unit = 1.0
1
7
17
(10)
15
8
7
5
0
0
1
1
2
2
3
3
HI
N
= 42
Stem-and-leaf of Without
Leaf Unit = 1.0
3
567899
0001111233
5556667888
0112233
7
34
8
39,
45,
2
14
(9)
16
14
10
6
5
57, 210,
0
0
1
1
2
2
3
3
33
566777889999
111112234
77
1144
9999
3
89
HI
55, 55, 85,
N
= 39
Descriptive Statistics:
Descriptive Statistics: With, Without
Variable
With
Without
N
42
39
Mean
22.81
19.15
Median
16.00
12.00
TrMean
17.97
17.17
Variable
With
Without
Minimum
3.00
3.00
Maximum
210.00
85.00
Q1
10.75
8.00
Q3
22.25
29.00
StDev
31.66
16.99
SE Mean
4.88
2.72
With
200
100
0
With
Without
Note:
o Both the mean and medians indicate or suggest that the concentration of
radon in homes with diagnosed cancer is greater than those without
cancer
o The mean of the group with cancer is affected by the very extreme outlier
, 210. The trimmed means are very close.
o The standard deviations that there is more variation in the group with
cancer. However, the fourth spread of the group without is greater than
the fourth spread of the group with. Here the standard deviation was
affected by the extreme outlier( 210) in the group with cancer