<s>
Univariate	B-General_Concept
is	O
a	O
term	O
commonly	O
used	O
in	O
statistics	O
to	O
describe	O
a	O
type	O
of	O
data	O
which	O
consists	O
of	O
observations	O
on	O
only	O
a	O
single	O
characteristic	O
or	O
attribute	O
.	O
</s>
<s>
A	O
simple	O
example	O
of	O
univariate	B-General_Concept
data	O
would	O
be	O
the	O
salaries	O
of	O
workers	O
in	O
industry	O
.	O
</s>
<s>
Like	O
all	O
the	O
other	O
data	O
,	O
univariate	B-General_Concept
data	O
can	O
be	O
visualized	O
using	O
graphs	O
,	O
images	O
or	O
other	O
analysis	O
tools	O
after	O
the	O
data	O
is	O
measured	O
,	O
collected	O
,	O
reported	O
,	O
and	O
analyzed	O
.	O
</s>
<s>
Some	O
univariate	B-General_Concept
data	O
consists	O
of	O
numbers	O
(	O
such	O
as	O
the	O
height	O
of	O
65	O
inches	O
or	O
the	O
weight	O
of	O
100	O
pounds	O
)	O
,	O
while	O
others	O
are	O
nonnumerical	O
(	O
such	O
as	O
eye	O
colors	O
of	O
brown	O
or	O
blue	O
)	O
.	O
</s>
<s>
Generally	O
,	O
the	O
terms	O
categorical	O
univariate	B-General_Concept
data	O
and	O
numerical	O
univariate	B-General_Concept
data	O
are	O
used	O
to	O
distinguish	O
between	O
these	O
types	O
.	O
</s>
<s>
Categorical	O
univariate	B-General_Concept
data	O
consists	O
of	O
non-numerical	O
observations	O
that	O
may	O
be	O
placed	O
in	O
categories	O
.	O
</s>
<s>
Categorical	O
univariate	B-General_Concept
data	O
usually	O
use	O
either	O
nominal	O
or	O
ordinal	B-Language
scale	O
of	O
measurement	O
.	O
</s>
<s>
Numerical	O
univariate	B-General_Concept
data	O
consists	O
of	O
observations	O
that	O
are	O
numbers	O
.	O
</s>
<s>
This	O
type	O
of	O
univariate	B-General_Concept
data	O
can	O
be	O
classified	O
even	O
further	O
into	O
two	O
subcategories	O
:	O
discrete	O
and	O
continuous	O
.	O
</s>
<s>
A	O
numerical	O
univariate	B-General_Concept
data	O
is	O
discrete	O
if	O
the	O
set	O
of	O
all	O
possible	O
values	O
is	O
finite	O
or	O
countably	O
infinite	O
.	O
</s>
<s>
Discrete	O
univariate	B-General_Concept
data	O
are	O
usually	O
associated	O
with	O
counting	O
(	O
such	O
as	O
the	O
number	O
of	O
books	O
read	O
by	O
a	O
person	O
)	O
.	O
</s>
<s>
A	O
numerical	O
univariate	B-General_Concept
data	O
is	O
continuous	O
if	O
the	O
set	O
of	O
all	O
possible	O
values	O
is	O
an	O
interval	O
of	O
numbers	O
.	O
</s>
<s>
Continuous	O
univariate	B-General_Concept
data	O
are	O
usually	O
associated	O
with	O
measuring	O
(	O
such	O
as	O
the	O
weights	O
of	O
people	O
)	O
.	O
</s>
<s>
Univariate	B-General_Concept
analysis	I-General_Concept
is	O
the	O
simplest	O
form	O
of	O
analyzing	O
data	O
.	O
</s>
<s>
Uni	O
means	O
"	O
one	O
"	O
,	O
so	O
the	O
data	O
has	O
only	O
one	O
variable	O
(	O
univariate	B-General_Concept
)	O
.	O
</s>
<s>
Univariate	B-General_Concept
data	O
requires	O
to	O
analyze	O
each	O
variable	O
separately	O
.	O
</s>
<s>
Univariate	B-General_Concept
data	O
does	O
not	O
answer	O
research	O
questions	O
about	O
relationships	O
between	O
variables	O
,	O
but	O
rather	O
it	O
is	O
used	O
to	O
describe	O
one	O
characteristic	O
or	O
attribute	O
that	O
varies	O
from	O
observation	O
to	O
observation	O
.	O
</s>
<s>
The	O
first	O
one	O
is	O
to	O
answer	O
a	O
research	O
question	O
with	O
descriptive	B-General_Concept
study	O
and	O
the	O
second	O
one	O
is	O
to	O
get	O
knowledge	O
about	O
how	O
attribute	O
varies	O
with	O
individual	O
effect	O
of	O
a	O
variable	O
in	O
Regression	O
analysis	O
.	O
</s>
<s>
There	O
are	O
some	O
ways	O
to	O
describe	O
patterns	O
found	O
in	O
univariate	B-General_Concept
data	O
which	O
include	O
graphical	O
methods	O
,	O
measures	O
of	O
central	O
tendency	O
and	O
measures	O
of	O
variability	O
.	O
</s>
<s>
Like	O
other	O
forms	O
of	O
statistics	O
,	O
it	O
can	O
be	O
inferential	O
or	O
descriptive	B-General_Concept
.	O
</s>
<s>
Univariate	B-General_Concept
analysis	I-General_Concept
can	O
yield	O
misleading	O
results	O
in	O
cases	O
in	O
which	O
multivariate	O
analysis	O
is	O
more	O
appropriate	O
.	O
</s>
<s>
Central	O
tendency	O
is	O
one	O
of	O
the	O
most	O
common	O
numerical	O
descriptive	B-General_Concept
measures	O
.	O
</s>
<s>
It	O
's	O
used	O
to	O
estimate	O
the	O
central	O
location	O
of	O
the	O
univariate	B-General_Concept
data	O
by	O
the	O
calculation	O
of	O
mean	O
,	O
median	O
and	O
mode	O
.	O
</s>
<s>
However	O
,	O
if	O
the	O
data	O
is	O
numerical	O
in	O
nature	O
(	O
ordinal	B-Language
or	O
interval/ratio	O
)	O
then	O
the	O
mode	O
,	O
median	O
,	O
or	O
mean	O
can	O
all	O
be	O
used	O
to	O
describe	O
the	O
data	O
.	O
</s>
<s>
Using	O
more	O
than	O
one	O
of	O
these	O
measures	O
provides	O
a	O
more	O
accurate	O
descriptive	B-General_Concept
summary	O
of	O
central	O
tendency	O
for	O
the	O
univariate	B-General_Concept
.	O
</s>
<s>
A	O
measure	O
of	O
variability	O
or	O
dispersion	O
(	O
deviation	O
from	O
the	O
mean	O
)	O
of	O
a	O
univariate	B-General_Concept
data	O
set	O
can	O
reveal	O
the	O
shape	O
of	O
a	O
univariate	B-General_Concept
data	O
distribution	O
more	O
sufficiently	O
.	O
</s>
<s>
The	O
three	O
most	O
frequently	O
used	O
measures	O
of	O
variability	O
are	O
range	B-General_Concept
,	O
variance	O
and	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
If	O
the	O
distribution	O
of	O
data	O
is	O
symmetrical	O
,	O
then	O
the	O
measures	O
of	O
variability	O
are	O
usually	O
the	O
variance	O
and	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
However	O
,	O
if	O
the	O
data	O
are	O
skewed	B-General_Concept
,	O
then	O
the	O
measure	O
of	O
variability	O
that	O
would	O
be	O
appropriate	O
for	O
that	O
data	O
set	O
is	O
the	O
range	B-General_Concept
.	O
</s>
<s>
Descriptive	B-General_Concept
statistics	I-General_Concept
describe	O
a	O
sample	O
or	O
population	O
.	O
</s>
<s>
They	O
can	O
be	O
part	O
of	O
exploratory	B-General_Concept
data	I-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
For	O
ordinal	B-Language
variables	O
the	O
median	O
can	O
be	O
calculated	O
as	O
a	O
measure	O
of	O
central	O
tendency	O
and	O
the	O
range	B-General_Concept
(	O
and	O
variations	O
of	O
it	O
)	O
as	O
a	O
measure	O
of	O
dispersion	O
.	O
</s>
<s>
For	O
interval	O
level	O
variables	O
,	O
the	O
arithmetic	O
mean	O
(	O
average	O
)	O
and	O
standard	B-General_Concept
deviation	I-General_Concept
are	O
added	O
to	O
the	O
toolbox	O
and	O
,	O
for	O
ratio	O
level	O
variables	O
,	O
we	O
add	O
the	O
geometric	O
mean	O
and	O
harmonic	O
mean	O
as	O
measures	O
of	O
central	O
tendency	O
and	O
the	O
coefficient	O
of	O
variation	O
as	O
a	O
measure	O
of	O
dispersion	O
.	O
</s>
<s>
For	O
interval	O
and	O
ratio	O
level	O
data	O
,	O
further	O
descriptors	O
include	O
the	O
variable	O
's	O
skewness	B-General_Concept
and	O
kurtosis	B-Error_Name
.	O
</s>
<s>
For	O
interval	O
and	O
ratio	O
level	O
data	O
,	O
a	O
one-sample	B-General_Concept
t-test	I-General_Concept
can	O
let	O
us	O
infer	O
whether	O
the	O
mean	O
in	O
our	O
sample	O
matches	O
some	O
proposed	O
number	O
(	O
typically	O
0	O
)	O
.	O
</s>
<s>
Other	O
available	O
tests	O
of	O
location	O
include	O
the	O
one-sample	O
sign	B-General_Concept
test	I-General_Concept
and	O
Wilcoxon	B-General_Concept
signed	I-General_Concept
rank	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
The	O
most	O
frequently	O
used	O
graphical	O
illustrations	O
for	O
univariate	B-General_Concept
data	O
are	O
:	O
</s>
<s>
Histograms	B-Algorithm
are	O
used	O
to	O
estimate	O
distribution	O
of	O
the	O
data	O
,	O
with	O
the	O
frequency	O
of	O
values	O
assigned	O
to	O
a	O
value	O
range	B-General_Concept
called	O
a	O
bin	B-General_Concept
.	O
</s>
<s>
Univariate	B-General_Concept
distribution	O
is	O
a	O
dispersal	O
type	O
of	O
a	O
single	O
random	O
variable	O
described	O
either	O
with	O
a	O
probability	O
mass	O
function	O
(	O
pmf	O
)	O
for	O
discrete	O
probability	O
distribution	O
,	O
or	O
probability	O
density	O
function	O
(	O
pdf	O
)	O
for	O
continuous	O
probability	O
distribution	O
.	O
</s>
