1.1 Boxplots 15
Figure 1.1. An old Swiss 1000-franc bank note.
The boxplot is a graphical technique that displays the distribution of variables. It helps us
see the location, skewness, spread, tail length and outlying points.
It is particularly useful in comparing different batches. The boxplot is a graphical repre-
sentation of the Five Number Summary. To introduce the Five Number Summary, let us
consider for a moment a smaller, one-dimensional data set: the population of the 15 largest
U.S. cities in 1960 (Table 1.1).
In the Five Number Summary, we calculate the upper quartile F
U
, the lower quartile F
L
,
the median and the extremes. Recall that order statistics {x
(1)
, x
(2)
, . . . , x
(n)
} are a set of
ordered values x
1
, x
2
, . . . , x
n
where x
(1)
denotes the minimum and x
(n)
the maximum. The
median M typically cuts the set of observations in two equal parts, and is defined as
M =
(
x
(
n+1
2
)
n odd
1
2
n
x
(
n
2
)
+ x
(
n
2
+1)
o
n even
. (1.1)
The quartiles cut the set into four equal parts, which are often called fourths (that is why we
use the letter F ). Using a definition that goes back to Hoaglin, Mosteller and Tukey (1983)
the definition of a median can be generalized to fourths, eights, etc. Considering the order
statistics we can define the depth of a data value x
(i)
as min{i, n − i + 1}. If n is odd, the
depth of the median is
n+1
2
. If n is even,
n+1
2
is a fraction. Thus, the median is determined
to be the average between the two data values belonging to the next larger and smaller order
statistics, i.e., M =
1
2
n
x
(
n
2
)
+ x
(
n
2
+1)
o
. In our example, we have n = 15 hence the median
M = x
(8)
= 88.
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