<s>
A	O
chi-squared	B-General_Concept
test	I-General_Concept
(	O
also	O
chi-square	B-General_Concept
or	O
test	O
)	O
is	O
a	O
statistical	O
hypothesis	O
test	O
used	O
in	O
the	O
analysis	O
of	O
contingency	B-Application
tables	I-Application
when	O
the	O
sample	O
sizes	O
are	O
large	O
.	O
</s>
<s>
In	O
simpler	O
terms	O
,	O
this	O
test	O
is	O
primarily	O
used	O
to	O
examine	O
whether	O
two	O
categorical	O
variables	O
(	O
two	O
dimensions	O
of	O
the	O
contingency	B-Application
table	I-Application
)	O
are	O
independent	O
in	O
influencing	O
the	O
test	B-General_Concept
statistic	I-General_Concept
(	O
values	O
within	O
the	O
table	O
)	O
.	O
</s>
<s>
The	O
test	O
is	O
valid	O
when	O
the	O
test	B-General_Concept
statistic	I-General_Concept
is	O
chi-squared	O
distributed	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
specifically	O
Pearson	B-General_Concept
's	I-General_Concept
chi-squared	I-General_Concept
test	I-General_Concept
and	O
variants	O
thereof	O
.	O
</s>
<s>
Pearson	B-General_Concept
's	I-General_Concept
chi-squared	I-General_Concept
test	I-General_Concept
is	O
used	O
to	O
determine	O
whether	O
there	O
is	O
a	O
statistically	B-General_Concept
significant	I-General_Concept
difference	O
between	O
the	O
expected	O
frequencies	O
and	O
the	O
observed	O
frequencies	O
in	O
one	O
or	O
more	O
categories	O
of	O
a	O
contingency	B-Application
table	I-Application
.	O
</s>
<s>
For	O
contingency	B-Application
tables	I-Application
with	O
smaller	O
sample	O
sizes	O
,	O
a	O
Fisher	B-General_Concept
's	I-General_Concept
exact	I-General_Concept
test	I-General_Concept
is	O
used	O
instead	O
.	O
</s>
<s>
If	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
there	O
are	O
no	O
differences	O
between	O
the	O
classes	O
in	O
the	O
population	O
is	O
true	O
,	O
the	O
test	B-General_Concept
statistic	I-General_Concept
computed	O
from	O
the	O
observations	O
follows	O
a	O
frequency	O
distribution	O
.	O
</s>
<s>
The	O
purpose	O
of	O
the	O
test	O
is	O
to	O
evaluate	O
how	O
likely	O
the	O
observed	O
frequencies	O
would	O
be	O
assuming	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
true	O
.	O
</s>
<s>
Test	B-General_Concept
statistics	I-General_Concept
that	O
follow	O
a	O
distribution	O
occur	O
when	O
the	O
observations	O
are	O
independent	O
.	O
</s>
<s>
There	O
are	O
also	O
tests	O
for	O
testing	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
independence	O
of	O
a	O
pair	O
of	O
random	O
variables	O
based	O
on	O
observations	O
of	O
the	O
pairs	O
.	O
</s>
<s>
Chi-squared	B-General_Concept
tests	I-General_Concept
often	O
refers	O
to	O
tests	O
for	O
which	O
the	O
distribution	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
approaches	O
the	O
distribution	O
asymptotically	B-Language
,	O
meaning	O
that	O
the	O
sampling	B-General_Concept
distribution	I-General_Concept
(	O
if	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
true	O
)	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
approximates	O
a	O
chi-squared	O
distribution	O
more	O
and	O
more	O
closely	O
as	O
sample	O
sizes	O
increase	O
.	O
</s>
<s>
At	O
the	O
end	O
of	O
the	O
19th	O
century	O
,	O
Pearson	O
noticed	O
the	O
existence	O
of	O
significant	O
skewness	B-General_Concept
within	O
some	O
biological	O
observations	O
.	O
</s>
<s>
In	O
order	O
to	O
model	O
the	O
observations	O
regardless	O
of	O
being	O
normal	O
or	O
skewed	B-General_Concept
,	O
Pearson	O
,	O
in	O
a	O
series	O
of	O
articles	O
published	O
from	O
1893	O
to	O
1916	O
,	O
devised	O
the	O
Pearson	O
distribution	O
,	O
a	O
family	O
of	O
continuous	O
probability	O
distributions	O
,	O
which	O
includes	O
the	O
normal	O
distribution	O
and	O
many	O
skewed	B-General_Concept
distributions	I-General_Concept
,	O
and	O
proposed	O
a	O
method	O
of	O
statistical	O
analysis	O
consisting	O
of	O
using	O
the	O
Pearson	O
distribution	O
to	O
model	O
the	O
observation	O
and	O
performing	O
a	O
test	O
of	O
goodness	O
of	O
fit	O
to	O
determine	O
how	O
well	O
the	O
model	O
really	O
fits	O
to	O
the	O
observations	O
.	O
</s>
<s>
Suppose	O
that	O
observations	O
in	O
a	O
random	O
sample	O
from	O
a	O
population	O
are	O
classified	O
into	O
mutually	O
exclusive	O
classes	O
with	O
respective	O
observed	O
numbers	O
(	O
for	O
)	O
,	O
and	O
a	O
null	B-General_Concept
hypothesis	I-General_Concept
gives	O
the	O
probability	O
that	O
an	O
observation	O
falls	O
into	O
the	O
th	O
class	O
.	O
</s>
<s>
Pearson	O
proposed	O
that	O
,	O
under	O
the	O
circumstance	O
of	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
being	O
correct	O
,	O
as	O
the	O
limiting	O
distribution	O
of	O
the	O
quantity	O
given	O
below	O
is	O
the	O
distribution	O
.	O
</s>
<s>
One	O
test	B-General_Concept
statistic	I-General_Concept
that	O
follows	O
a	O
chi-squared	O
distribution	O
exactly	O
is	O
the	O
test	O
that	O
the	O
variance	O
of	O
a	O
normally	O
distributed	O
population	O
has	O
a	O
given	O
value	O
based	O
on	O
a	O
sample	O
variance	O
.	O
</s>
<s>
For	O
an	O
exact	O
test	O
used	O
in	O
place	O
of	O
the	O
2	O
×	O
2	O
chi-squared	B-General_Concept
test	I-General_Concept
for	O
independence	O
,	O
see	O
Fisher	B-General_Concept
's	I-General_Concept
exact	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
For	O
an	O
exact	O
test	O
used	O
in	O
place	O
of	O
the	O
2	O
×	O
1	O
chi-squared	B-General_Concept
test	I-General_Concept
for	O
goodness	O
of	O
fit	O
,	O
see	O
binomial	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
Cochran	B-General_Concept
–	I-General_Concept
Mantel	I-General_Concept
–	I-General_Concept
Haenszel	I-General_Concept
chi-squared	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
Likelihood-ratio	B-General_Concept
tests	I-General_Concept
in	O
general	O
statistical	O
modelling	O
,	O
for	O
testing	O
whether	O
there	O
is	O
evidence	O
of	O
the	O
need	O
to	O
move	O
from	O
a	O
simple	O
model	O
to	O
a	O
more	O
complicated	O
one	O
(	O
where	O
the	O
simple	O
model	O
is	O
nested	O
within	O
the	O
complicated	O
one	O
)	O
.	O
</s>
<s>
Using	O
the	O
chi-squared	O
distribution	O
to	O
interpret	O
Pearson	B-General_Concept
's	I-General_Concept
chi-squared	I-General_Concept
statistic	I-General_Concept
requires	O
one	O
to	O
assume	O
that	O
the	O
discrete	O
probability	O
of	O
observed	O
binomial	O
frequencies	O
in	O
the	O
table	O
can	O
be	O
approximated	O
by	O
the	O
continuous	O
chi-squared	O
distribution	O
.	O
</s>
<s>
To	O
reduce	O
the	O
error	O
in	O
approximation	O
,	O
Frank	O
Yates	O
suggested	O
a	O
correction	B-General_Concept
for	I-General_Concept
continuity	I-General_Concept
that	O
adjusts	O
the	O
formula	O
for	O
Pearson	B-General_Concept
's	I-General_Concept
chi-squared	I-General_Concept
test	I-General_Concept
by	O
subtracting	O
0.5	O
from	O
the	O
absolute	O
difference	O
between	O
each	O
observed	O
value	O
and	O
its	O
expected	O
value	O
in	O
a	O
contingency	B-Application
table	I-Application
.	O
</s>
<s>
This	O
reduces	O
the	O
chi-squared	O
value	O
obtained	O
and	O
thus	O
increases	O
its	O
p-value	B-General_Concept
.	O
</s>
<s>
The	O
test	B-General_Concept
statistic	I-General_Concept
in	O
this	O
instance	O
could	O
be	O
set	O
to	O
be	O
the	O
sum	O
of	O
squares	O
about	O
the	O
sample	O
mean	O
,	O
divided	O
by	O
the	O
nominal	O
value	O
for	O
the	O
variance	O
(	O
i.e.	O
</s>
<s>
For	O
example	O
,	O
if	O
the	O
sample	O
size	O
is	O
21	O
,	O
the	O
acceptance	O
region	O
for	O
with	O
a	O
significance	B-General_Concept
level	I-General_Concept
of	O
5%	O
is	O
between	O
9.59	O
and	O
34.17	O
.	O
</s>
<s>
The	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
that	O
each	O
person	O
's	O
neighborhood	O
of	O
residence	O
is	O
independent	O
of	O
the	O
person	O
's	O
occupational	O
classification	O
.	O
</s>
<s>
The	O
sum	O
of	O
these	O
quantities	O
over	O
all	O
of	O
the	O
cells	O
is	O
the	O
test	B-General_Concept
statistic	I-General_Concept
;	O
in	O
this	O
case	O
,	O
.	O
</s>
<s>
If	O
the	O
test	B-General_Concept
statistic	I-General_Concept
is	O
improbably	O
large	O
according	O
to	O
that	O
chi-squared	O
distribution	O
,	O
then	O
one	O
rejects	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
independence	O
.	O
</s>
<s>
In	O
cryptanalysis	O
,	O
the	O
chi-squared	B-General_Concept
test	I-General_Concept
is	O
used	O
to	O
compare	O
the	O
distribution	O
of	O
plaintext	O
and	O
(	O
possibly	O
)	O
decrypted	O
ciphertext	O
.	O
</s>
<s>
In	O
bioinformatics	O
,	O
the	O
chi-squared	B-General_Concept
test	I-General_Concept
is	O
used	O
to	O
compare	O
the	O
distribution	O
of	O
certain	O
properties	O
of	O
genes	O
(	O
e.g.	O
,	O
genomic	O
content	O
,	O
mutation	O
rate	O
,	O
interaction	O
network	O
clustering	O
,	O
etc	O
.	O
)	O
</s>
