<s>
In	O
probability	O
theory	O
and	O
statistics	O
,	O
kurtosis	B-Error_Name
(	O
from	O
,	O
kyrtos	O
or	O
kurtos	O
,	O
meaning	O
"	O
curved	O
,	O
arching	O
"	O
)	O
is	O
a	O
measure	O
of	O
the	O
"	O
tailedness	O
"	O
of	O
the	O
probability	O
distribution	O
of	O
a	O
real-valued	O
random	O
variable	O
.	O
</s>
<s>
Like	O
skewness	B-General_Concept
,	O
kurtosis	B-Error_Name
describes	O
a	O
particular	O
aspect	O
of	O
a	O
probability	O
distribution	O
.	O
</s>
<s>
There	O
are	O
different	O
ways	O
to	O
quantify	O
kurtosis	B-Error_Name
for	O
a	O
theoretical	O
distribution	O
,	O
and	O
there	O
are	O
corresponding	O
ways	O
of	O
estimating	O
it	O
using	O
a	O
sample	O
from	O
a	O
population	O
.	O
</s>
<s>
Different	O
measures	O
of	O
kurtosis	B-Error_Name
may	O
have	O
different	O
interpretations	O
.	O
</s>
<s>
The	O
standard	O
measure	O
of	O
a	O
distribution	O
's	O
kurtosis	B-Error_Name
,	O
originating	O
with	O
Karl	O
Pearson	O
,	O
is	O
a	O
scaled	O
version	O
of	O
the	O
fourth	O
moment	O
of	O
the	O
distribution	O
.	O
</s>
<s>
This	O
number	O
is	O
related	O
to	O
the	O
tails	O
of	O
the	O
distribution	O
,	O
not	O
its	O
peak	O
;	O
hence	O
,	O
the	O
sometimes-seen	O
characterization	O
of	O
kurtosis	B-Error_Name
as	O
"	O
peakedness	O
"	O
is	O
incorrect	O
.	O
</s>
<s>
For	O
this	O
measure	O
,	O
higher	O
kurtosis	B-Error_Name
corresponds	O
to	O
greater	O
extremity	O
of	O
deviations	B-General_Concept
(	O
or	O
outliers	O
)	O
,	O
and	O
not	O
the	O
configuration	O
of	O
data	O
near	O
the	O
mean	O
.	O
</s>
<s>
It	O
is	O
common	O
to	O
compare	O
the	O
excess	O
kurtosis	B-Error_Name
(	O
defined	O
below	O
)	O
of	O
a	O
distribution	O
to	O
0	O
,	O
which	O
is	O
the	O
excess	O
kurtosis	B-Error_Name
of	O
any	O
univariate	O
normal	O
distribution	O
.	O
</s>
<s>
Distributions	O
with	O
negative	O
excess	O
kurtosis	B-Error_Name
are	O
said	O
to	O
be	O
platykurtic	O
,	O
although	O
this	O
does	O
not	O
imply	O
the	O
distribution	O
is	O
"	O
flat-topped	O
"	O
as	O
is	O
sometimes	O
stated	O
.	O
</s>
<s>
An	O
example	O
of	O
a	O
platykurtic	B-Error_Name
distribution	I-Error_Name
is	O
the	O
uniform	O
distribution	O
,	O
which	O
does	O
not	O
produce	O
outliers	O
.	O
</s>
<s>
Distributions	O
with	O
a	O
positive	O
excess	O
kurtosis	B-Error_Name
are	O
said	O
to	O
be	O
leptokurtic	O
.	O
</s>
<s>
It	O
is	O
common	O
practice	O
to	O
use	O
excess	O
kurtosis	B-Error_Name
,	O
which	O
is	O
defined	O
as	O
Pearson	O
's	O
kurtosis	B-Error_Name
minus	O
3	O
,	O
to	O
provide	O
a	O
simple	O
comparison	O
to	O
the	O
normal	O
distribution	O
.	O
</s>
<s>
Some	O
authors	O
and	O
software	O
packages	O
use	O
"	O
kurtosis	B-Error_Name
"	O
by	O
itself	O
to	O
refer	O
to	O
the	O
excess	O
kurtosis	B-Error_Name
.	O
</s>
<s>
For	O
clarity	O
and	O
generality	O
,	O
however	O
,	O
this	O
article	O
explicitly	O
indicates	O
where	O
non-excess	O
kurtosis	B-Error_Name
is	O
meant	O
.	O
</s>
<s>
Alternative	O
measures	O
of	O
kurtosis	B-Error_Name
are	O
:	O
the	O
L-kurtosis	O
,	O
which	O
is	O
a	O
scaled	O
version	O
of	O
the	O
fourth	O
L-moment	O
;	O
measures	O
based	O
on	O
four	O
population	O
or	O
sample	O
quantiles	O
.	O
</s>
<s>
These	O
are	O
analogous	O
to	O
the	O
alternative	O
measures	O
of	O
skewness	B-General_Concept
that	O
are	O
not	O
based	O
on	O
ordinary	O
moments	O
.	O
</s>
<s>
where	O
μ4	O
is	O
the	O
fourth	O
central	B-General_Concept
moment	I-General_Concept
and	O
σ	O
is	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
Several	O
letters	O
are	O
used	O
in	O
the	O
literature	O
to	O
denote	O
the	O
kurtosis	B-Error_Name
.	O
</s>
<s>
Other	O
choices	O
include	O
γ2	O
,	O
to	O
be	O
similar	O
to	O
the	O
notation	O
for	O
skewness	B-General_Concept
,	O
although	O
sometimes	O
this	O
is	O
instead	O
reserved	O
for	O
the	O
excess	O
kurtosis	B-Error_Name
.	O
</s>
<s>
The	O
kurtosis	B-Error_Name
is	O
bounded	O
below	O
by	O
the	O
squared	O
skewness	B-General_Concept
plus	O
1	O
:	O
</s>
<s>
where	O
μ3	O
is	O
the	O
third	O
central	B-General_Concept
moment	I-General_Concept
.	O
</s>
<s>
There	O
is	O
no	O
upper	O
limit	O
to	O
the	O
kurtosis	B-Error_Name
of	O
a	O
general	O
probability	O
distribution	O
,	O
and	O
it	O
may	O
be	O
infinite	O
.	O
</s>
<s>
A	O
reason	O
why	O
some	O
authors	O
favor	O
the	O
excess	O
kurtosis	B-Error_Name
is	O
that	O
cumulants	O
are	O
extensive	O
.	O
</s>
<s>
Formulas	O
related	O
to	O
the	O
extensive	O
property	O
are	O
more	O
naturally	O
expressed	O
in	O
terms	O
of	O
the	O
excess	O
kurtosis	B-Error_Name
.	O
</s>
<s>
where	O
is	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
.	O
</s>
<s>
The	O
cokurtosis	O
between	O
pairs	O
of	O
variables	O
is	O
an	O
order	O
four	O
tensor	B-Device
.	O
</s>
<s>
For	O
a	O
bivariate	O
normal	O
distribution	O
,	O
the	O
cokurtosis	O
tensor	B-Device
has	O
off-diagonal	O
terms	O
that	O
are	O
neither	O
0	O
nor	O
3	O
in	O
general	O
,	O
so	O
attempting	O
to	O
"	O
correct	O
"	O
for	O
an	O
excess	O
becomes	O
confusing	O
.	O
</s>
<s>
The	O
exact	O
interpretation	O
of	O
the	O
Pearson	O
measure	O
of	O
kurtosis	B-Error_Name
(	O
or	O
excess	O
kurtosis	B-Error_Name
)	O
used	O
to	O
be	O
disputed	O
,	O
but	O
is	O
now	O
settled	O
.	O
</s>
<s>
As	O
Westfall	O
notes	O
in	O
2014	O
,	O
"	O
its	O
...	O
only	O
unambiguous	O
interpretation	O
is	O
in	O
terms	O
of	O
tail	O
extremity	O
;	O
i.e.	O
,	O
either	O
existing	O
outliers	O
(	O
for	O
the	O
sample	O
kurtosis	B-Error_Name
)	O
or	O
propensity	O
to	O
produce	O
outliers	O
(	O
for	O
the	O
kurtosis	B-Error_Name
of	O
a	O
probability	O
distribution	O
)	O
.	O
"	O
</s>
<s>
The	O
logic	O
is	O
simple	O
:	O
Kurtosis	B-Error_Name
is	O
the	O
average	O
(	O
or	O
expected	O
value	O
)	O
of	O
the	O
standardized	O
data	O
raised	O
to	O
the	O
fourth	O
power	O
.	O
</s>
<s>
Standardized	O
values	O
that	O
are	O
less	O
than	O
1	O
(	O
i.e.	O
,	O
data	O
within	O
one	O
standard	B-General_Concept
deviation	I-General_Concept
of	O
the	O
mean	O
,	O
where	O
the	O
"	O
peak	O
"	O
would	O
be	O
)	O
contribute	O
virtually	O
nothing	O
to	O
kurtosis	B-Error_Name
,	O
since	O
raising	O
a	O
number	O
that	O
is	O
less	O
than	O
1	O
to	O
the	O
fourth	O
power	O
makes	O
it	O
closer	O
to	O
zero	O
.	O
</s>
<s>
The	O
only	O
data	O
values	O
(	O
observed	O
or	O
observable	O
)	O
that	O
contribute	O
to	O
kurtosis	B-Error_Name
in	O
any	O
meaningful	O
way	O
are	O
those	O
outside	O
the	O
region	O
of	O
the	O
peak	O
;	O
i.e.	O
,	O
the	O
outliers	O
.	O
</s>
<s>
Therefore	O
,	O
kurtosis	B-Error_Name
measures	O
outliers	O
only	O
;	O
it	O
measures	O
nothing	O
about	O
the	O
"	O
peak	O
"	O
.	O
</s>
<s>
Many	O
incorrect	O
interpretations	O
of	O
kurtosis	B-Error_Name
that	O
involve	O
notions	O
of	O
peakedness	O
have	O
been	O
given	O
.	O
</s>
<s>
One	O
is	O
that	O
kurtosis	B-Error_Name
measures	O
both	O
the	O
"	O
peakedness	O
"	O
of	O
the	O
distribution	O
and	O
the	O
heaviness	O
of	O
its	O
tail	O
.	O
</s>
<s>
Various	O
other	O
incorrect	O
interpretations	O
have	O
been	O
suggested	O
,	O
such	O
as	O
"	O
lack	O
of	O
shoulders	O
"	O
(	O
where	O
the	O
"	O
shoulder	O
"	O
is	O
defined	O
vaguely	O
as	O
the	O
area	O
between	O
the	O
peak	O
and	O
the	O
tail	O
,	O
or	O
more	O
specifically	O
as	O
the	O
area	O
about	O
one	O
standard	B-General_Concept
deviation	I-General_Concept
from	O
the	O
mean	O
)	O
or	O
"	O
bimodality	O
"	O
.	O
</s>
<s>
Balanda	O
and	O
MacGillivray	O
assert	O
that	O
the	O
standard	O
definition	O
of	O
kurtosis	B-Error_Name
"	O
is	O
a	O
poor	O
measure	O
of	O
the	O
kurtosis	B-Error_Name
,	O
peakedness	O
,	O
or	O
tail	O
weight	O
of	O
a	O
distribution	O
"	O
and	O
instead	O
propose	O
to	O
"	O
define	O
kurtosis	B-Error_Name
vaguely	O
as	O
the	O
location	O
-	O
and	O
scale-free	O
movement	O
of	O
probability	O
mass	O
from	O
the	O
shoulders	O
of	O
a	O
distribution	O
into	O
its	O
center	O
and	O
tails	O
"	O
.	O
</s>
<s>
In	O
1986	O
Moors	O
gave	O
an	O
interpretation	O
of	O
kurtosis	B-Error_Name
.	O
</s>
<s>
where	O
X	O
is	O
a	O
random	O
variable	O
,	O
μ	O
is	O
the	O
mean	O
and	O
σ	O
is	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
The	O
kurtosis	B-Error_Name
can	O
now	O
be	O
seen	O
as	O
a	O
measure	O
of	O
the	O
dispersion	O
of	O
Z2	O
around	O
its	O
expectation	O
.	O
</s>
<s>
In	O
terms	O
of	O
the	O
original	O
variable	O
X	O
,	O
the	O
kurtosis	B-Error_Name
is	O
a	O
measure	O
of	O
the	O
dispersion	O
of	O
X	O
around	O
the	O
two	O
values	O
μ±σ	O
.	O
</s>
<s>
The	O
excess	O
kurtosis	B-Error_Name
is	O
defined	O
as	O
kurtosis	B-Error_Name
minus	O
3	O
.	O
</s>
<s>
Distributions	O
with	O
zero	O
excess	O
kurtosis	B-Error_Name
are	O
called	O
mesokurtic	O
,	O
or	O
mesokurtotic	O
.	O
</s>
<s>
The	O
most	O
prominent	O
example	O
of	O
a	O
mesokurtic	B-Error_Name
distribution	I-Error_Name
is	O
the	O
normal	O
distribution	O
family	O
,	O
regardless	O
of	O
the	O
values	O
of	O
its	O
parameters	O
.	O
</s>
<s>
A	O
distribution	O
with	O
positive	O
excess	O
kurtosis	B-Error_Name
is	O
called	O
leptokurtic	O
,	O
or	O
leptokurtotic	O
.	O
</s>
<s>
A	O
distribution	O
with	O
negative	O
excess	O
kurtosis	B-Error_Name
is	O
called	O
platykurtic	O
,	O
or	O
platykurtotic	O
.	O
</s>
<s>
In	O
terms	O
of	O
shape	O
,	O
a	O
platykurtic	B-Error_Name
distribution	I-Error_Name
has	O
thinner	O
tails	O
.	O
</s>
<s>
Examples	O
of	O
platykurtic	B-Error_Name
distributions	I-Error_Name
include	O
the	O
continuous	O
and	O
discrete	O
uniform	O
distributions	O
,	O
and	O
the	O
raised	O
cosine	O
distribution	O
.	O
</s>
<s>
The	O
most	O
platykurtic	B-Error_Name
distribution	I-Error_Name
of	O
all	O
is	O
the	O
Bernoulli	O
distribution	O
with	O
p	O
=	O
1/2	O
(	O
for	O
example	O
the	O
number	O
of	O
times	O
one	O
obtains	O
"	O
heads	O
"	O
when	O
flipping	O
a	O
coin	O
once	O
,	O
a	O
coin	O
toss	O
)	O
,	O
for	O
which	O
the	O
excess	O
kurtosis	B-Error_Name
is	O
−2	O
.	O
</s>
<s>
The	O
effects	O
of	O
kurtosis	B-Error_Name
are	O
illustrated	O
using	O
a	O
parametric	O
family	O
of	O
distributions	O
whose	O
kurtosis	B-Error_Name
can	O
be	O
adjusted	O
while	O
their	O
lower-order	O
moments	O
and	O
cumulants	O
remain	O
constant	O
.	O
</s>
<s>
For	O
the	O
kurtosis	B-Error_Name
to	O
exist	O
,	O
we	O
require	O
m>5	O
/2	O
.	O
</s>
<s>
Then	O
the	O
mean	O
and	O
skewness	B-General_Concept
exist	O
and	O
are	O
both	O
identically	O
zero	O
.	O
</s>
<s>
Then	O
the	O
only	O
free	O
parameter	O
is	O
m	O
,	O
which	O
controls	O
the	O
fourth	O
moment	O
(	O
and	O
cumulant	O
)	O
and	O
hence	O
the	O
kurtosis	B-Error_Name
.	O
</s>
<s>
One	O
can	O
reparameterize	O
with	O
,	O
where	O
is	O
the	O
excess	O
kurtosis	B-Error_Name
as	O
defined	O
above	O
.	O
</s>
<s>
This	O
yields	O
a	O
one-parameter	O
leptokurtic	O
family	O
with	O
zero	O
mean	O
,	O
unit	O
variance	O
,	O
zero	O
skewness	B-General_Concept
,	O
and	O
arbitrary	O
non-negative	O
excess	O
kurtosis	B-Error_Name
.	O
</s>
<s>
In	O
the	O
images	O
on	O
the	O
right	O
,	O
the	O
blue	O
curve	O
represents	O
the	O
density	O
with	O
excess	O
kurtosis	B-Error_Name
of	O
2	O
.	O
</s>
<s>
One	O
can	O
see	O
that	O
the	O
normal	O
density	O
allocates	O
little	O
probability	O
mass	O
to	O
the	O
regions	O
far	O
from	O
the	O
mean	O
(	O
"	O
has	O
thin	O
tails	O
"	O
)	O
,	O
compared	O
with	O
the	O
blue	O
curve	O
of	O
the	O
leptokurtic	O
Pearson	O
type	O
VII	O
density	O
with	O
excess	O
kurtosis	B-Error_Name
of	O
2	O
.	O
</s>
<s>
Each	O
has	O
a	O
mean	O
and	O
skewness	B-General_Concept
of	O
zero	O
.	O
</s>
<s>
C	O
:	O
raised	O
cosine	O
distribution	O
,	O
cyan	O
curve	O
,	O
excess	O
kurtosis	B-Error_Name
=	O
−	O
593762	O
...	O
</s>
<s>
U	O
:	O
uniform	O
distribution	O
,	O
magenta	O
curve	O
(	O
shown	O
for	O
clarity	O
as	O
a	O
rectangle	O
in	O
both	O
images	O
)	O
,	O
excess	O
kurtosis	B-Error_Name
=	O
−	O
1.2	O
.	O
</s>
<s>
Note	O
that	O
in	O
these	O
cases	O
the	O
platykurtic	O
densities	O
have	O
bounded	O
support	O
,	O
whereas	O
the	O
densities	O
with	O
positive	O
or	O
zero	O
excess	O
kurtosis	B-Error_Name
are	O
supported	O
on	O
the	O
whole	O
real	O
line	O
.	O
</s>
<s>
One	O
cannot	O
infer	O
that	O
high	O
or	O
low	O
kurtosis	B-Error_Name
distributions	O
have	O
the	O
characteristics	O
indicated	O
by	O
these	O
examples	O
.	O
</s>
<s>
where	O
m4	O
is	O
the	O
fourth	O
sample	O
moment	B-General_Concept
about	I-General_Concept
the	I-General_Concept
mean	I-General_Concept
,	O
m2	O
is	O
the	O
second	O
sample	O
moment	B-General_Concept
about	I-General_Concept
the	I-General_Concept
mean	I-General_Concept
(	O
that	O
is	O
,	O
the	O
sample	O
variance	O
)	O
,	O
xi	O
is	O
the	O
ith	O
value	O
,	O
and	O
is	O
the	O
sample	O
mean	O
.	O
</s>
<s>
where	O
the	O
values	O
are	O
the	O
standardized	O
data	O
values	O
using	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
defined	O
using	O
n	O
rather	O
than	O
n−1	O
in	O
the	O
denominator	O
.	O
</s>
<s>
The	O
average	O
of	O
these	O
values	O
is	O
18.05	O
and	O
the	O
excess	O
kurtosis	B-Error_Name
is	O
thus	O
18.05	O
−3	O
=	O
15.05	O
.	O
</s>
<s>
This	O
example	O
makes	O
it	O
clear	O
that	O
data	O
near	O
the	O
"	O
middle	O
"	O
or	O
"	O
peak	O
"	O
of	O
the	O
distribution	O
do	O
not	O
contribute	O
to	O
the	O
kurtosis	B-Error_Name
statistic	O
,	O
hence	O
kurtosis	B-Error_Name
does	O
not	O
measure	O
"	O
peakedness	O
"	O
.	O
</s>
<s>
Given	O
a	O
sub-set	O
of	O
samples	O
from	O
a	O
population	O
,	O
the	O
sample	O
excess	O
kurtosis	B-Error_Name
above	O
is	O
a	O
biased	O
estimator	O
of	O
the	O
population	O
excess	O
kurtosis	B-Error_Name
.	O
</s>
<s>
An	O
alternative	O
estimator	O
of	O
the	O
population	O
excess	O
kurtosis	B-Error_Name
,	O
which	O
is	O
unbiased	O
in	O
random	O
samples	O
of	O
a	O
normal	O
distribution	O
,	O
is	O
defined	O
as	O
follows	O
:	O
</s>
<s>
where	O
k4	O
is	O
the	O
unique	O
symmetric	O
unbiased	O
estimator	O
of	O
the	O
fourth	O
cumulant	O
,	O
k2	O
is	O
the	O
unbiased	O
estimate	O
of	O
the	O
second	O
cumulant	O
(	O
identical	O
to	O
the	O
unbiased	O
estimate	O
of	O
the	O
sample	O
variance	O
)	O
,	O
m4	O
is	O
the	O
fourth	O
sample	O
moment	B-General_Concept
about	I-General_Concept
the	I-General_Concept
mean	I-General_Concept
,	O
m2	O
is	O
the	O
second	O
sample	O
moment	B-General_Concept
about	I-General_Concept
the	I-General_Concept
mean	I-General_Concept
,	O
xi	O
is	O
the	O
ith	O
value	O
,	O
and	O
is	O
the	O
sample	O
mean	O
.	O
</s>
<s>
This	O
adjusted	O
Fisher	O
–	O
Pearson	O
standardized	O
moment	O
coefficient	O
is	O
the	O
version	O
found	O
in	O
Excel	B-Application
and	O
several	O
statistical	O
packages	O
including	O
Minitab	B-Application
,	O
SAS	B-Language
,	O
and	O
SPSS	B-Algorithm
.	O
</s>
<s>
where	O
is	O
the	O
corresponding	O
sample	O
skewness	B-General_Concept
.	O
</s>
<s>
The	O
sample	O
kurtosis	B-Error_Name
is	O
a	O
useful	O
measure	O
of	O
whether	O
there	O
is	O
a	O
problem	O
with	O
outliers	O
in	O
a	O
data	O
set	O
.	O
</s>
<s>
Larger	O
kurtosis	B-Error_Name
indicates	O
a	O
more	O
serious	O
outlier	O
problem	O
,	O
and	O
may	O
lead	O
the	O
researcher	O
to	O
choose	O
alternative	O
statistical	O
methods	O
.	O
</s>
<s>
Agostino	B-General_Concept
's	I-General_Concept
K-squared	I-General_Concept
test	I-General_Concept
is	O
a	O
goodness-of-fit	O
normality	B-General_Concept
test	I-General_Concept
based	O
on	O
a	O
combination	O
of	O
the	O
sample	O
skewness	B-General_Concept
and	O
sample	O
kurtosis	B-Error_Name
,	O
as	O
is	O
the	O
Jarque	B-General_Concept
–	I-General_Concept
Bera	I-General_Concept
test	I-General_Concept
for	O
normality	O
.	O
</s>
<s>
For	O
non-normal	O
samples	O
,	O
the	O
variance	O
of	O
the	O
sample	O
variance	O
depends	O
on	O
the	O
kurtosis	B-Error_Name
;	O
for	O
details	O
,	O
please	O
see	O
variance	O
.	O
</s>
<s>
Pearson	O
's	O
definition	O
of	O
kurtosis	B-Error_Name
is	O
used	O
as	O
an	O
indicator	O
of	O
intermittency	O
in	O
turbulence	O
.	O
</s>
<s>
Assume	O
a	O
random	O
variable	O
has	O
expectation	O
,	O
variance	O
and	O
kurtosis	B-Error_Name
.	O
</s>
<s>
In	O
other	O
words	O
:	O
If	O
the	O
kurtosis	B-Error_Name
is	O
large	O
,	O
we	O
might	O
see	O
a	O
lot	O
values	O
either	O
all	O
below	O
or	O
above	O
the	O
mean	O
.	O
</s>
<s>
Applying	O
band-pass	O
filters	O
to	O
digital	B-Algorithm
images	I-Algorithm
,	O
kurtosis	B-Error_Name
values	O
tend	O
to	O
be	O
uniform	O
,	O
independent	O
of	O
the	O
range	O
of	O
the	O
filter	O
.	O
</s>
<s>
This	O
behavior	O
,	O
termed	O
kurtosis	B-Error_Name
convergence	O
,	O
can	O
be	O
used	O
to	O
detect	O
image	O
splicing	O
in	O
forensic	O
analysis	O
.	O
</s>
<s>
A	O
different	O
measure	O
of	O
"	O
kurtosis	B-Error_Name
"	O
is	O
provided	O
by	O
using	O
L-moments	O
instead	O
of	O
the	O
ordinary	O
moments	O
.	O
</s>
