<s>
The	O
Wald	B-General_Concept
–	I-General_Concept
Wolfowitz	I-General_Concept
runs	I-General_Concept
test	I-General_Concept
(	O
or	O
simply	O
runs	B-General_Concept
test	I-General_Concept
)	O
,	O
named	O
after	O
statisticians	O
Abraham	O
Wald	O
and	O
Jacob	O
Wolfowitz	O
is	O
a	O
non-parametric	B-General_Concept
statistical	O
test	O
that	O
checks	O
a	O
randomness	O
hypothesis	O
for	O
a	O
two-valued	O
data	O
sequence	O
.	O
</s>
<s>
The	O
run	O
test	O
is	O
based	O
on	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
each	O
element	O
in	O
the	O
sequence	O
is	O
independently	O
drawn	O
from	O
the	O
same	O
distribution	O
.	O
</s>
<s>
Under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
the	O
number	O
of	O
runs	O
in	O
a	O
sequence	O
of	O
N	O
elements	O
is	O
a	O
random	O
variable	O
whose	O
conditional	O
distribution	O
given	O
the	O
observation	O
of	O
N+	O
positive	O
values	O
and	O
N−	O
negative	O
values	O
(	O
)	O
is	O
approximately	O
normal	O
,	O
with	O
:	O
</s>
<s>
If	O
the	O
number	O
of	O
runs	O
is	O
significantly	B-General_Concept
higher	O
or	O
lower	O
than	O
expected	O
,	O
the	O
hypothesis	O
of	O
statistical	O
independence	O
of	O
the	O
elements	O
may	O
be	O
rejected	O
.	O
</s>
<s>
Runs	B-General_Concept
tests	I-General_Concept
can	O
be	O
used	O
to	O
test	O
:	O
</s>
<s>
whether	O
a	O
function	O
fits	O
well	O
to	O
a	O
data	B-General_Concept
set	I-General_Concept
,	O
by	O
marking	O
the	O
data	O
exceeding	O
the	O
function	O
value	O
with	O
+	O
and	O
the	O
other	O
data	O
with	O
−	O
.	O
</s>
<s>
For	O
this	O
use	O
,	O
the	O
runs	B-General_Concept
test	I-General_Concept
,	O
which	O
takes	O
into	O
account	O
the	O
signs	O
but	O
not	O
the	O
distances	O
,	O
is	O
complementary	O
to	O
the	O
chi	B-General_Concept
square	I-General_Concept
test	I-General_Concept
,	O
which	O
takes	O
into	O
account	O
the	O
distances	O
but	O
not	O
the	O
signs	O
.	O
</s>
<s>
The	O
Kolmogorov	B-General_Concept
–	I-General_Concept
Smirnov	I-General_Concept
test	I-General_Concept
has	O
been	O
shown	O
to	O
be	O
more	O
powerful	O
than	O
the	O
Wald	O
–	O
Wolfowitz	O
test	O
for	O
detecting	O
differences	O
between	O
distributions	O
that	O
differ	O
solely	O
in	O
their	O
location	O
.	O
</s>
<s>
The	O
Wald	B-General_Concept
–	I-General_Concept
Wolfowitz	I-General_Concept
runs	I-General_Concept
test	I-General_Concept
has	O
been	O
extended	O
for	O
use	O
with	O
several	O
samples	O
.	O
</s>
