<s>
In	O
descriptive	B-General_Concept
statistics	I-General_Concept
,	O
the	O
interquartile	B-General_Concept
range	I-General_Concept
(	O
IQR	B-General_Concept
)	O
is	O
a	O
measure	O
of	O
statistical	O
dispersion	O
,	O
which	O
is	O
the	O
spread	O
of	O
the	O
data	O
.	O
</s>
<s>
The	O
IQR	B-General_Concept
may	O
also	O
be	O
called	O
the	O
midspread	O
,	O
middle	B-General_Concept
50%	I-General_Concept
,	O
fourth	O
spread	O
,	O
or	O
H‑spread	O
.	O
</s>
<s>
To	O
calculate	O
the	O
IQR	B-General_Concept
,	O
the	O
data	O
set	O
is	O
divided	O
into	O
quartiles	O
,	O
or	O
four	O
rank-ordered	O
even	O
parts	O
via	O
linear	O
interpolation	O
.	O
</s>
<s>
The	O
lower	O
quartile	O
corresponds	O
with	O
the	O
25th	O
percentile	O
and	O
the	O
upper	O
quartile	O
corresponds	O
with	O
the	O
75th	O
percentile	O
,	O
so	O
IQR	B-General_Concept
=	O
Q3	O
−	O
Q1	O
.	O
</s>
<s>
The	O
IQR	B-General_Concept
is	O
an	O
example	O
of	O
a	O
trimmed	O
estimator	O
,	O
defined	O
as	O
the	O
25%	O
trimmed	O
range	B-General_Concept
,	O
which	O
enhances	O
the	O
accuracy	O
of	O
dataset	O
statistics	O
by	O
dropping	O
lower	O
contribution	O
,	O
outlying	O
points	O
.	O
</s>
<s>
It	O
is	O
also	O
used	O
as	O
a	O
robust	B-General_Concept
measure	I-General_Concept
of	I-General_Concept
scale	I-General_Concept
It	O
can	O
be	O
clearly	O
visualized	O
by	O
the	O
box	O
on	O
a	O
box	B-Application
plot	I-Application
.	O
</s>
<s>
Unlike	O
total	O
range	B-General_Concept
,	O
the	O
interquartile	B-General_Concept
range	I-General_Concept
has	O
a	O
breakdown	O
point	O
of	O
25%	O
,	O
and	O
is	O
thus	O
often	O
preferred	O
to	O
the	O
total	O
range	B-General_Concept
.	O
</s>
<s>
The	O
IQR	B-General_Concept
is	O
used	O
to	O
build	O
box	B-Application
plots	I-Application
,	O
simple	O
graphical	O
representations	O
of	O
a	O
probability	O
distribution	O
.	O
</s>
<s>
The	O
IQR	B-General_Concept
is	O
used	O
in	O
businesses	O
as	O
a	O
marker	O
for	O
their	O
income	O
rates	O
.	O
</s>
<s>
For	O
a	O
symmetric	O
distribution	O
(	O
where	O
the	O
median	O
equals	O
the	O
midhinge	B-General_Concept
,	O
the	O
average	O
of	O
the	O
first	O
and	O
third	O
quartiles	O
)	O
,	O
half	O
the	O
IQR	B-General_Concept
equals	O
the	O
median	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
(	O
MAD	O
)	O
.	O
</s>
<s>
The	O
IQR	B-General_Concept
can	O
be	O
used	O
to	O
identify	O
outliers	O
(	O
see	O
below	O
)	O
.	O
</s>
<s>
The	O
IQR	B-General_Concept
also	O
may	O
indicate	O
the	O
skewness	B-General_Concept
of	O
the	O
dataset	O
.	O
</s>
<s>
The	O
quartile	O
deviation	O
or	O
semi-interquartile	O
range	O
is	O
defined	O
as	O
half	O
the	O
IQR	B-General_Concept
.	O
</s>
<s>
The	O
IQR	B-General_Concept
of	O
a	O
set	O
of	O
values	O
is	O
calculated	O
as	O
the	O
difference	O
between	O
the	O
upper	O
and	O
lower	O
quartiles	O
,	O
Q3	O
and	O
Q1	O
.	O
</s>
<s>
For	O
the	O
data	O
in	O
this	O
table	O
the	O
interquartile	B-General_Concept
range	I-General_Concept
is	O
IQR	B-General_Concept
=	O
Q3	O
Q1	O
=	O
119	O
-	O
31	O
=	O
88	O
.	O
</s>
<s>
For	O
the	O
data	O
set	O
in	O
this	O
box	B-Application
plot	I-Application
:	O
</s>
<s>
lower	O
1.5*IQR	O
whisker	O
=	O
Q1	O
-	O
1.5	O
*	O
IQR	B-General_Concept
=	O
7	O
-	O
3	O
=	O
4	O
.	O
</s>
<s>
upper	O
1.5*IQR	O
whisker	O
=	O
Q3	O
+	O
1.5	O
*	O
IQR	B-General_Concept
=	O
9	O
+	O
3	O
=	O
12	O
.	O
</s>
<s>
This	O
means	O
the	O
1.5*IQR	O
whiskers	O
can	O
be	O
uneven	O
in	O
lengths	O
.	O
</s>
<s>
The	O
median	O
,	O
minimum	O
,	O
maximum	O
,	O
and	O
the	O
first	O
and	O
third	O
quartile	O
constitute	O
the	O
Five-number	B-Language
summary	I-Language
.	O
</s>
<s>
The	O
interquartile	B-General_Concept
range	I-General_Concept
of	O
a	O
continuous	O
distribution	O
can	O
be	O
calculated	O
by	O
integrating	O
the	O
probability	O
density	O
function	O
(	O
which	O
yields	O
the	O
cumulative	O
distribution	O
function	O
—	O
any	O
other	O
means	O
of	O
calculating	O
the	O
CDF	O
will	O
also	O
work	O
)	O
.	O
</s>
<s>
The	O
IQR	B-General_Concept
,	O
mean	O
,	O
and	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
a	O
population	O
P	O
can	O
be	O
used	O
in	O
a	O
simple	O
test	O
of	O
whether	O
or	O
not	O
P	O
is	O
normally	O
distributed	O
,	O
or	O
Gaussian	O
.	O
</s>
<s>
A	O
better	O
test	O
of	O
normality	O
,	O
such	O
as	O
Q	B-Application
–	I-Application
Q	I-Application
plot	I-Application
would	O
be	O
indicated	O
here	O
.	O
</s>
<s>
The	O
interquartile	B-General_Concept
range	I-General_Concept
is	O
often	O
used	O
to	O
find	O
outliers	O
in	O
data	O
.	O
</s>
<s>
Outliers	O
here	O
are	O
defined	O
as	O
observations	O
that	O
fall	O
below	O
Q1	O
−	O
1.5	O
IQR	B-General_Concept
or	O
above	O
Q3	O
+	O
1.5	O
IQR	B-General_Concept
.	O
</s>
<s>
In	O
a	O
boxplot	B-Application
,	O
the	O
highest	O
and	O
lowest	O
occurring	O
value	O
within	O
this	O
limit	O
are	O
indicated	O
by	O
whiskers	O
of	O
the	O
box	O
(	O
frequently	O
with	O
an	O
additional	O
bar	O
at	O
the	O
end	O
of	O
the	O
whisker	O
)	O
and	O
any	O
outliers	O
as	O
individual	O
points	O
.	O
</s>
