<s>
In	O
descriptive	B-General_Concept
statistics	I-General_Concept
,	O
a	O
box	B-Application
plot	I-Application
or	O
boxplot	B-Application
is	O
a	O
method	O
for	O
graphically	O
demonstrating	O
the	O
locality	O
,	O
spread	O
and	O
skewness	B-General_Concept
groups	O
of	O
numerical	O
data	O
through	O
their	O
quartiles	O
.	O
</s>
<s>
In	O
addition	O
to	O
the	O
box	O
on	O
a	O
box	B-Application
plot	I-Application
,	O
there	O
can	O
be	O
lines	O
(	O
which	O
are	O
called	O
whiskers	O
)	O
extending	O
from	O
the	O
box	O
indicating	O
variability	O
outside	O
the	O
upper	O
and	O
lower	O
quartiles	O
,	O
thus	O
,	O
the	O
plot	O
is	O
also	O
termed	O
as	O
the	O
box-and-whisker	B-Application
plot	I-Application
and	O
the	O
box-and-whisker	B-Application
diagram	I-Application
.	O
</s>
<s>
Box	B-Application
plots	I-Application
are	O
non-parametric	B-General_Concept
:	O
they	O
display	O
variation	O
in	O
samples	O
of	O
a	O
statistical	O
population	O
without	O
making	O
any	O
assumptions	O
of	O
the	O
underlying	O
statistical	O
distribution	O
(	O
though	O
Tukey	O
's	O
boxplot	B-Application
assumes	O
symmetry	O
for	O
the	O
whiskers	O
and	O
normality	O
for	O
their	O
length	O
)	O
.	O
</s>
<s>
The	O
spacings	O
in	O
each	O
subsection	O
of	O
the	O
box-plot	O
indicate	O
the	O
degree	O
of	O
dispersion	O
(	O
spread	O
)	O
and	O
skewness	B-General_Concept
of	O
the	O
data	O
,	O
which	O
are	O
usually	O
described	O
using	O
the	O
five-number	B-Language
summary	I-Language
.	O
</s>
<s>
In	O
addition	O
,	O
the	O
box-plot	O
allows	O
one	O
to	O
visually	O
estimate	O
various	O
L-estimators	O
,	O
notably	O
the	O
interquartile	B-General_Concept
range	I-General_Concept
,	O
midhinge	B-General_Concept
,	O
range	B-General_Concept
,	O
mid-range	O
,	O
and	O
trimean	O
.	O
</s>
<s>
Box	B-Application
plots	I-Application
can	O
be	O
drawn	O
either	O
horizontally	O
or	O
vertically	O
.	O
</s>
<s>
The	O
range-bar	O
method	O
was	O
first	O
introduced	O
by	O
Mary	O
Eleanor	O
Spear	O
in	O
her	O
book	O
"	O
Charting	O
Statistics	O
"	O
in	O
1952	O
and	O
again	O
in	O
her	O
book	O
"	O
Practical	O
Charting	O
Techniques	O
"	O
in	O
1969	O
.	O
</s>
<s>
The	O
box-and-whisker	B-Application
plot	I-Application
was	O
first	O
introduced	O
in	O
1970	O
by	O
John	O
Tukey	O
,	O
who	O
later	O
published	O
on	O
the	O
subject	O
in	O
his	O
book	O
"	O
Exploratory	B-General_Concept
Data	I-General_Concept
Analysis	I-General_Concept
"	O
in	O
1977	O
.	O
</s>
<s>
A	O
boxplot	B-Application
is	O
a	O
standardized	O
way	O
of	O
displaying	O
the	O
dataset	O
based	O
on	O
the	O
five-number	B-Language
summary	I-Language
:	O
the	O
minimum	O
,	O
the	O
maximum	O
,	O
the	O
sample	O
median	O
,	O
and	O
the	O
first	O
and	O
third	O
quartiles	O
.	O
</s>
<s>
In	O
addition	O
to	O
the	O
minimum	O
and	O
maximum	O
values	O
used	O
to	O
construct	O
a	O
box-plot	O
,	O
another	O
important	O
element	O
that	O
can	O
also	O
be	O
employed	O
to	O
obtain	O
a	O
box-plot	O
is	O
the	O
interquartile	B-General_Concept
range	I-General_Concept
(	O
IQR	B-General_Concept
)	O
,	O
as	O
denoted	O
below	O
:	O
</s>
<s>
Another	O
popular	O
choice	O
for	O
the	O
boundaries	O
of	O
the	O
whiskers	O
is	O
based	O
on	O
the	O
1.5	O
IQR	B-General_Concept
value	O
.	O
</s>
<s>
From	O
above	O
the	O
upper	O
quartile	O
(	O
Q3	O
)	O
,	O
a	O
distance	O
of	O
1.5	O
times	O
the	O
IQR	B-General_Concept
is	O
measured	O
out	O
and	O
a	O
whisker	O
is	O
drawn	O
up	O
to	O
the	O
largest	O
observed	O
data	O
point	O
from	O
the	O
dataset	O
that	O
falls	O
within	O
this	O
distance	O
.	O
</s>
<s>
Similarly	O
,	O
a	O
distance	O
of	O
1.5	O
times	O
the	O
IQR	B-General_Concept
is	O
measured	O
out	O
below	O
the	O
lower	O
quartile	O
(	O
Q1	O
)	O
and	O
a	O
whisker	O
is	O
drawn	O
down	O
to	O
the	O
lowest	O
observed	O
data	O
point	O
from	O
the	O
dataset	O
that	O
falls	O
within	O
this	O
distance	O
.	O
</s>
<s>
Because	O
the	O
whiskers	O
must	O
end	O
at	O
an	O
observed	O
data	O
point	O
,	O
the	O
whisker	O
lengths	O
can	O
look	O
unequal	O
,	O
even	O
though	O
1.5	O
IQR	B-General_Concept
is	O
the	O
same	O
for	O
both	O
sides	O
.	O
</s>
<s>
Some	O
box	B-Application
plots	I-Application
include	O
an	O
additional	O
character	O
to	O
represent	O
the	O
mean	O
of	O
the	O
data	O
.	O
</s>
<s>
If	O
the	O
data	O
are	O
normally	O
distributed	O
,	O
the	O
locations	O
of	O
the	O
seven	O
marks	O
on	O
the	O
box	B-Application
plot	I-Application
will	O
be	O
equally	O
spaced	O
.	O
</s>
<s>
On	O
some	O
box	B-Application
plots	I-Application
,	O
a	O
cross-hatch	O
is	O
placed	O
before	O
the	O
end	O
of	O
each	O
whisker	O
.	O
</s>
<s>
Since	O
the	O
mathematician	O
John	O
W	O
.	O
Tukey	O
first	O
popularized	O
this	O
type	O
of	O
visual	O
data	O
display	O
in	O
1969	O
,	O
several	O
variations	O
on	O
the	O
classical	O
box	B-Application
plot	I-Application
have	O
been	O
developed	O
,	O
and	O
the	O
two	O
most	O
commonly	O
found	O
variations	O
are	O
the	O
variable	O
width	O
box	B-Application
plots	I-Application
and	O
the	O
notched	O
box	B-Application
plots	I-Application
shown	O
in	O
Figure	O
4	O
.	O
</s>
<s>
Variable	O
width	O
box	B-Application
plots	I-Application
illustrate	O
the	O
size	O
of	O
each	O
group	O
whose	O
data	O
is	O
being	O
plotted	O
by	O
making	O
the	O
width	O
of	O
the	O
box	O
proportional	O
to	O
the	O
size	O
of	O
the	O
group	O
.	O
</s>
<s>
Notched	O
box	B-Application
plots	I-Application
apply	O
a	O
"	O
notch	O
"	O
or	O
narrowing	O
of	O
the	O
box	O
around	O
the	O
median	O
.	O
</s>
<s>
The	O
height	O
of	O
the	O
notches	O
is	O
proportional	O
to	O
the	O
interquartile	B-General_Concept
range	I-General_Concept
(	O
IQR	B-General_Concept
)	O
of	O
the	O
sample	O
and	O
is	O
inversely	O
proportional	O
to	O
the	O
square	O
root	O
of	O
the	O
size	O
of	O
the	O
sample	O
.	O
</s>
<s>
The	O
width	O
of	O
the	O
notch	O
is	O
arbitrarily	O
chosen	O
to	O
be	O
visually	O
pleasing	O
,	O
and	O
should	O
be	O
consistent	O
amongst	O
all	O
box	B-Application
plots	I-Application
being	O
displayed	O
on	O
the	O
same	O
page	O
.	O
</s>
<s>
Adjusted	O
box	B-Application
plots	I-Application
are	O
intended	O
to	O
describe	O
skew	B-General_Concept
distributions	I-General_Concept
,	O
and	O
they	O
rely	O
on	O
the	O
medcouple	B-General_Concept
statistic	O
of	O
skewness	B-General_Concept
.	O
</s>
<s>
For	O
a	O
medcouple	B-General_Concept
value	O
of	O
MC	O
,	O
the	O
lengths	O
of	O
the	O
upper	O
and	O
lower	O
whiskers	O
on	O
the	O
box-plot	O
are	O
respectively	O
defined	O
to	O
be	O
:	O
</s>
<s>
For	O
a	O
symmetrical	O
data	O
distribution	O
,	O
the	O
medcouple	B-General_Concept
will	O
be	O
zero	O
,	O
and	O
this	O
reduces	O
the	O
adjusted	O
box-plot	O
to	O
the	O
Tukey	O
's	O
box-plot	O
with	O
equal	O
whisker	O
lengths	O
of	O
for	O
both	O
whiskers	O
.	O
</s>
<s>
Other	O
kinds	O
of	O
box	B-Application
plots	I-Application
,	O
such	O
as	O
the	O
violin	B-Application
plots	I-Application
and	O
the	O
bean	O
plots	O
can	O
show	O
the	O
difference	O
between	O
single-modal	O
and	O
multimodal	O
distributions	O
,	O
which	O
cannot	O
be	O
observed	O
from	O
the	O
original	O
classical	O
box-plot	O
.	O
</s>
<s>
A	O
box	B-Application
plot	I-Application
of	O
the	O
data	O
set	O
can	O
be	O
generated	O
by	O
first	O
calculating	O
five	O
relevant	O
values	O
of	O
this	O
data	O
set	O
:	O
minimum	O
,	O
maximum	O
,	O
median	O
(	O
Q2	O
)	O
,	O
first	O
quartile	O
(	O
Q1	O
)	O
,	O
and	O
third	O
quartile	O
(	O
Q3	O
)	O
.	O
</s>
<s>
The	O
interquartile	B-General_Concept
range	I-General_Concept
,	O
or	O
IQR	B-General_Concept
,	O
can	O
be	O
calculated	O
by	O
subtracting	O
the	O
first	O
quartile	O
value	O
(	O
Q1	O
)	O
from	O
the	O
third	O
quartile	O
value	O
(	O
Q3	O
)	O
:	O
</s>
<s>
1.5	O
IQR	B-General_Concept
above	O
the	O
third	O
quartile	O
is	O
:	O
</s>
<s>
1.5	O
IQR	B-General_Concept
below	O
the	O
first	O
quartile	O
is	O
:	O
</s>
<s>
The	O
upper	O
whisker	O
boundary	O
of	O
the	O
box-plot	O
is	O
the	O
largest	O
data	O
value	O
that	O
is	O
within	O
1.5	O
IQR	B-General_Concept
above	O
the	O
third	O
quartile	O
.	O
</s>
<s>
Here	O
,	O
1.5	O
IQR	B-General_Concept
above	O
the	O
third	O
quartile	O
is	O
88.5	O
°F	O
and	O
the	O
maximum	O
is	O
81	O
°F	O
.	O
</s>
<s>
Similarly	O
,	O
the	O
lower	O
whisker	O
boundary	O
of	O
the	O
box	B-Application
plot	I-Application
is	O
the	O
smallest	O
data	O
value	O
that	O
is	O
within	O
1.5	O
IQR	B-General_Concept
below	O
the	O
first	O
quartile	O
.	O
</s>
<s>
Here	O
,	O
1.5	O
IQR	B-General_Concept
below	O
the	O
first	O
quartile	O
is	O
52.5	O
°F	O
and	O
the	O
minimum	O
is	O
57	O
°F	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
maximum	O
value	O
in	O
this	O
data	O
set	O
is	O
89	O
°F	O
,	O
and	O
1.5	O
IQR	B-General_Concept
above	O
the	O
third	O
quartile	O
is	O
88.5	O
°F	O
.	O
</s>
<s>
The	O
maximum	O
is	O
greater	O
than	O
1.5	O
IQR	B-General_Concept
plus	O
the	O
third	O
quartile	O
,	O
so	O
the	O
maximum	O
is	O
an	O
outlier	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
upper	O
whisker	O
is	O
drawn	O
at	O
the	O
greatest	O
value	O
smaller	O
than	O
1.5	O
IQR	B-General_Concept
above	O
the	O
third	O
quartile	O
,	O
which	O
is	O
79	O
°F	O
.	O
</s>
<s>
Similarly	O
,	O
the	O
minimum	O
value	O
in	O
this	O
data	O
set	O
is	O
52	O
°F	O
,	O
and	O
1.5	O
IQR	B-General_Concept
below	O
the	O
first	O
quartile	O
is	O
52.5	O
°F	O
.	O
</s>
<s>
The	O
minimum	O
is	O
smaller	O
than	O
1.5	O
IQR	B-General_Concept
minus	O
the	O
first	O
quartile	O
,	O
so	O
the	O
minimum	O
is	O
also	O
an	O
outlier	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
lower	O
whisker	O
is	O
drawn	O
at	O
the	O
smallest	O
value	O
greater	O
than	O
1.5	O
IQR	B-General_Concept
below	O
the	O
first	O
quartile	O
,	O
which	O
is	O
57	O
°F	O
.	O
</s>
<s>
Although	O
box	B-Application
plots	I-Application
may	O
seem	O
more	O
primitive	O
than	O
histograms	B-Algorithm
or	O
kernel	B-General_Concept
density	I-General_Concept
estimates	I-General_Concept
,	O
they	O
do	O
have	O
a	O
number	O
of	O
advantages	O
.	O
</s>
<s>
First	O
,	O
the	O
box	B-Application
plot	I-Application
enables	O
statisticians	O
to	O
do	O
a	O
quick	O
graphical	O
examination	O
on	O
one	O
or	O
more	O
data	O
sets	O
.	O
</s>
<s>
Lastly	O
,	O
the	O
overall	O
structure	O
of	O
histograms	B-Algorithm
and	O
kernel	B-General_Concept
density	I-General_Concept
estimate	I-General_Concept
can	O
be	O
strongly	O
influenced	O
by	O
the	O
choice	O
of	O
number	O
and	O
width	O
of	O
bins	O
techniques	O
and	O
the	O
choice	O
of	O
bandwidth	O
,	O
respectively	O
.	O
</s>
<s>
Although	O
looking	O
at	O
a	O
statistical	O
distribution	O
is	O
more	O
common	O
than	O
looking	O
at	O
a	O
box	B-Application
plot	I-Application
,	O
it	O
can	O
be	O
useful	O
to	O
compare	O
the	O
box	B-Application
plot	I-Application
against	O
the	O
probability	O
density	O
function	O
(	O
theoretical	O
histogram	B-Algorithm
)	O
for	O
a	O
normal	O
N(0,σ2 )	O
distribution	O
and	O
observe	O
their	O
characteristics	O
directly	O
(	O
as	O
shown	O
in	O
Figure	O
7	O
)	O
.	O
</s>
