<s>
Tukey	B-General_Concept
's	I-General_Concept
range	I-General_Concept
test	I-General_Concept
,	O
also	O
known	O
as	O
Tukey	O
's	O
test	O
,	O
Tukey	O
method	O
,	O
Tukey	O
's	O
honest	O
significance	O
test	O
,	O
or	O
Tukey	B-General_Concept
's	I-General_Concept
HSD	I-General_Concept
(	O
honestly	O
significant	B-General_Concept
difference	I-General_Concept
)	O
test	O
,	O
is	O
a	O
single-step	O
multiple	B-General_Concept
comparison	I-General_Concept
procedure	O
and	O
statistical	O
test	O
.	O
</s>
<s>
It	O
can	O
be	O
used	O
to	O
find	O
means	O
that	O
are	O
significantly	B-General_Concept
different	O
from	O
each	O
other	O
.	O
</s>
<s>
Named	O
after	O
John	O
Tukey	O
,	O
it	O
compares	O
all	O
possible	O
pairs	O
of	O
means	O
,	O
and	O
is	O
based	O
on	O
a	O
studentized	O
range	O
distribution	O
(	O
q	O
)	O
(	O
this	O
distribution	O
is	O
similar	O
to	O
the	O
distribution	O
of	O
t	O
from	O
the	O
t-test	B-General_Concept
.	O
</s>
<s>
and	O
identifies	O
any	O
difference	O
between	O
two	O
means	O
that	O
is	O
greater	O
than	O
the	O
expected	O
standard	B-General_Concept
error	I-General_Concept
.	O
</s>
<s>
A	O
common	O
mistaken	O
belief	O
is	O
that	O
Tukey	B-General_Concept
's	I-General_Concept
HSD	I-General_Concept
should	O
only	O
be	O
used	O
following	O
a	O
significant	O
ANOVA	O
.	O
</s>
<s>
This	O
test	O
is	O
often	O
followed	O
by	O
the	O
Compact	B-Application
Letter	I-Application
Display	I-Application
(	O
CLD	O
)	O
statistical	O
procedure	O
to	O
render	O
the	O
output	O
of	O
this	O
test	O
more	O
transparent	O
to	O
non-statistician	O
audiences	O
.	O
</s>
<s>
There	O
is	O
equal	O
within-group	O
variance	O
across	O
the	O
groups	O
associated	O
with	O
each	O
mean	O
in	O
the	O
test	O
(	O
homogeneity	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
)	O
.	O
</s>
<s>
where	O
is	O
the	O
larger	O
of	O
the	O
two	O
means	O
being	O
compared	O
,	O
is	O
the	O
smaller	O
of	O
the	O
two	O
means	O
being	O
compared	O
,	O
and	O
SE	O
is	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
sum	O
of	O
the	O
means	O
.	O
</s>
<s>
Since	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
for	O
Tukey	O
's	O
test	O
states	O
that	O
all	O
means	O
being	O
compared	O
are	O
from	O
the	O
same	O
population	O
(	O
i.e.	O
</s>
<s>
Suppose	O
that	O
we	O
take	O
a	O
sample	O
of	O
size	O
n	O
from	O
each	O
of	O
k	O
populations	O
with	O
the	O
same	O
normal	O
distribution	O
N(μ, σ2 )	O
and	O
suppose	O
that	O
min	O
is	O
the	O
smallest	O
of	O
these	O
sample	O
means	O
and	O
max	O
is	O
the	O
largest	O
of	O
these	O
sample	O
means	O
,	O
and	O
suppose	O
S2	O
is	O
the	O
pooled	B-General_Concept
sample	I-General_Concept
variance	I-General_Concept
from	O
these	O
samples	O
.	O
</s>
<s>
In	O
addition	O
,	O
R	B-Language
offers	O
a	O
cumulative	O
distribution	O
function	O
(	O
ptukey	O
)	O
and	O
a	O
quantile	O
function	O
(	O
qtukey	O
)	O
forq	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
one	O
has	O
to	O
calculate	O
the	O
estimated	O
standard	O
deviation	O
for	O
each	O
pairwise	O
comparison	O
as	O
formalized	O
by	O
Clyde	O
Kramer	O
in	O
1956	O
,	O
so	O
the	O
procedure	O
for	O
unequal	O
sample	O
sizes	O
is	O
sometimes	O
referred	O
to	O
as	O
the	O
Tukey	B-General_Concept
–	I-General_Concept
Kramer	I-General_Concept
method	I-General_Concept
which	O
is	O
as	O
follows	O
:	O
</s>
<s>
It	O
is	O
possible	O
to	O
generate	O
a	O
set	O
of	O
pseudorandom	O
samples	O
of	O
strictly	O
positive	O
measure	O
such	O
that	O
hypothesis	O
=	O
is	O
rejected	O
at	O
significance	B-General_Concept
level	I-General_Concept
while	O
=	O
=	O
is	O
not	O
rejected	O
even	O
at	O
.	O
</s>
