<s>
In	O
statistics	O
,	O
Welch	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
,	O
or	O
unequal	O
variances	O
t-test	B-General_Concept
,	O
is	O
a	O
two-sample	O
location	B-General_Concept
test	I-General_Concept
which	O
is	O
used	O
to	O
test	O
the	O
(	O
null	O
)	O
hypothesis	O
that	O
two	O
populations	O
have	O
equal	O
means	O
.	O
</s>
<s>
It	O
is	O
named	O
for	O
its	O
creator	O
,	O
Bernard	O
Lewis	O
Welch	O
,	O
is	O
an	O
adaptation	O
of	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
,	O
and	O
is	O
more	O
reliable	O
when	O
the	O
two	O
samples	O
have	O
unequal	O
variances	O
and	O
possibly	O
unequal	O
sample	O
sizes	O
.	O
</s>
<s>
These	O
tests	O
are	O
often	O
referred	O
to	O
as	O
"	O
unpaired	O
"	O
or	O
"	O
independent	O
samples	O
"	O
t-tests	B-General_Concept
,	O
as	O
they	O
are	O
typically	O
applied	O
when	O
the	O
statistical	O
units	O
underlying	O
the	O
two	O
samples	O
being	O
compared	O
are	O
non-overlapping	O
.	O
</s>
<s>
Given	O
that	O
Welch	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
has	O
been	O
less	O
popular	O
than	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
and	O
may	O
be	O
less	O
familiar	O
to	O
readers	O
,	O
a	O
more	O
informative	O
name	O
is	O
"	O
Welch	O
's	O
unequal	O
variances	O
t-test	B-General_Concept
"	O
—	O
or	O
"	O
unequal	O
variances	O
t-test	B-General_Concept
"	O
for	O
brevity	O
.	O
</s>
<s>
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
assumes	O
that	O
the	O
sample	O
means	O
being	O
compared	O
for	O
two	O
populations	O
are	O
normally	O
distributed	O
,	O
and	O
that	O
the	O
populations	O
have	O
equal	O
variances	O
.	O
</s>
<s>
Welch	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
is	O
designed	O
for	O
unequal	O
population	O
variances	O
,	O
but	O
the	O
assumption	O
of	O
normality	O
is	O
maintained	O
.	O
</s>
<s>
Welch	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
is	O
an	O
approximate	O
solution	O
to	O
the	O
Behrens	B-Algorithm
–	I-Algorithm
Fisher	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Welch	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
defines	O
the	O
statistic	O
t	O
by	O
the	O
following	O
formula	O
:	O
</s>
<s>
where	O
and	O
are	O
the	O
sample	O
mean	O
and	O
its	O
standard	B-General_Concept
error	I-General_Concept
,	O
with	O
denoting	O
the	O
corrected	O
sample	O
standard	O
deviation	O
,	O
and	O
sample	O
size	O
.	O
</s>
<s>
Unlike	O
in	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
,	O
the	O
denominator	O
is	O
not	O
based	O
on	O
a	O
pooled	B-General_Concept
variance	I-General_Concept
estimate	O
.	O
</s>
<s>
Once	O
t	O
and	O
have	O
been	O
computed	O
,	O
these	O
statistics	O
can	O
be	O
used	O
with	O
the	O
t-distribution	O
to	O
test	O
one	O
of	O
two	O
possible	O
null	B-General_Concept
hypotheses	I-General_Concept
:	O
</s>
<s>
that	O
one	O
of	O
the	O
population	O
means	O
is	O
greater	O
than	O
or	O
equal	O
to	O
the	O
other	O
,	O
in	O
which	O
a	O
one-tailed	B-General_Concept
test	I-General_Concept
is	O
applied	O
.	O
</s>
<s>
Welch	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
is	O
more	O
robust	O
than	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
and	O
maintains	O
type	O
I	O
error	O
rates	O
close	O
to	O
nominal	O
for	O
unequal	O
variances	O
and	O
for	O
unequal	O
sample	O
sizes	O
under	O
normality	O
.	O
</s>
<s>
Furthermore	O
,	O
the	O
power	B-General_Concept
of	O
Welch	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
comes	O
close	O
to	O
that	O
of	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
,	O
even	O
when	O
the	O
population	O
variances	O
are	O
equal	O
and	O
sample	O
sizes	O
are	O
balanced	O
.	O
</s>
<s>
Welch	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
can	O
be	O
generalized	O
to	O
more	O
than	O
2-samples	O
,	O
which	O
is	O
more	O
robust	O
than	O
one-way	B-General_Concept
analysis	I-General_Concept
of	I-General_Concept
variance	I-General_Concept
(	O
ANOVA	O
)	O
.	O
</s>
<s>
It	O
is	O
not	O
recommended	O
to	O
pre-test	O
for	O
equal	O
variances	O
and	O
then	O
choose	O
between	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
or	O
Welch	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
.	O
</s>
<s>
Rather	O
,	O
Welch	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
can	O
be	O
applied	O
directly	O
and	O
without	O
any	O
substantial	O
disadvantages	O
to	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
as	O
noted	O
above	O
.	O
</s>
<s>
Welch	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
remains	O
robust	O
for	O
skewed	O
distributions	O
and	O
large	O
sample	O
sizes	O
.	O
</s>
<s>
Reliability	O
decreases	O
for	O
skewed	O
distributions	O
and	O
smaller	O
samples	O
,	O
where	O
one	O
could	O
possibly	O
perform	O
Welch	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
.	O
</s>
<s>
The	O
following	O
three	O
examples	O
compare	O
Welch	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
and	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
.	O
</s>
<s>
Samples	O
are	O
from	O
random	O
normal	O
distributions	O
using	O
the	O
R	B-Language
programming	I-Language
language	I-Language
.	O
</s>
<s>
Reference	O
p-values	O
were	O
obtained	O
by	O
simulating	O
the	O
distributions	O
of	O
the	O
t	O
statistics	O
for	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
equal	O
population	O
means	O
(	O
)	O
.	O
</s>
<s>
Welch	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
and	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
gave	O
identical	O
results	O
when	O
the	O
two	O
samples	O
have	O
similar	O
variances	O
and	O
sample	O
sizes	O
(	O
Example	O
1	O
)	O
.	O
</s>
<s>
But	O
note	O
that	O
even	O
if	O
you	O
sample	O
data	O
from	O
populations	O
with	O
identical	O
variances	O
,	O
the	O
sample	O
variances	O
will	O
differ	O
,	O
as	O
will	O
the	O
results	O
of	O
the	O
two	O
t-tests	B-General_Concept
.	O
</s>
<s>
For	O
unequal	O
variances	O
,	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
gave	O
a	O
low	O
p-value	O
when	O
the	O
smaller	O
sample	O
had	O
a	O
larger	O
variance	O
(	O
Example	O
2	O
)	O
and	O
a	O
high	O
p-value	O
when	O
the	O
larger	O
sample	O
had	O
a	O
larger	O
variance	O
(	O
Example	O
3	O
)	O
.	O
</s>
<s>
For	O
unequal	O
variances	O
,	O
Welch	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
gave	O
p-values	O
close	O
to	O
simulated	O
p-values	O
.	O
</s>
<s>
Language/Program	O
Function	O
Documentation	O
LibreOffice	B-Application
TTEST( Data1	O
;	O
Data2	O
;	O
Mode	O
;	O
Type	O
)	O
MATLAB	B-Language
ttest2(data1, data2, 'Vartype', 'unequal' )	O
Microsoft	B-Application
Excel	I-Application
pre	O
2010	O
(	O
Student	B-General_Concept
's	I-General_Concept
T	I-General_Concept
Test	I-General_Concept
)	O
TTEST(array1, array2, tails, type )	O
Microsoft	B-Application
Excel	I-Application
2010	O
and	O
later	O
(	O
Student	B-General_Concept
's	I-General_Concept
T	I-General_Concept
Test	I-General_Concept
)	O
T.TEST(array1, array2, tails, type )	O
Minitab	B-Application
Accessed	O
through	O
menu	O
Overview	O
for	O
2-Sample	O
t	O
-	O
Minitab	B-Application
:	O
—	O
official	O
documentation	O
for	O
Minitab	B-Application
version	O
18	O
.	O
</s>
<s>
SAS	B-Language
(	O
Software	O
)	O
Default	O
output	O
from	O
proc	O
ttest	B-General_Concept
(	O
labeled	O
"	O
Satterthwaite	O
"	O
)	O
Python	B-Language
(	O
through	O
3rd-party	O
library	O
SciPy	B-Application
)	O
scipy.stats.ttest_ind(a, b, equal_var=False )	O
R	B-Language
t.test(data1, data2 )	O
Haskell	B-Language
Statistics.Test.StudentT.welchTTest	O
SamplesDiffer	O
data1	O
data2	O
JMP	B-Language
Oneway( 	O
Y( YColumn )	O
,	O
X( XColumn )	O
,	O
Unequal	O
Variances( 1 )	O
)	O
;	O
Julia	B-Application
UnequalVarianceTTest(data1, data2 )	O
Statattest	O
varname1	O
==	O
varname2	O
,	O
welch	O
Google	O
SheetsTTEST(range1, range2, tails, type )	O
GraphPad	O
PrismIt	O
is	O
a	O
choice	O
on	O
the	O
t	B-General_Concept
test	I-General_Concept
dialog.IBM	O
SPSS	B-Algorithm
Statistics	O
An	O
option	O
in	O
the	O
menu	O
Jeremy	O
Miles	O
:	O
Unequal	O
variances	O
t-test	B-General_Concept
or	O
U	O
Mann-Whitney	O
test	O
?,	O
Accessed	O
2014-04-11One-Sample	O
Test	O
—	O
Official	O
documentation	O
for	O
SPSS	B-Algorithm
Statistics	O
version	O
24	O
.	O
</s>
