<s>
The	O
partition	B-General_Concept
of	I-General_Concept
sums	I-General_Concept
of	I-General_Concept
squares	I-General_Concept
is	O
a	O
concept	O
that	O
permeates	O
much	O
of	O
inferential	O
statistics	O
and	O
descriptive	B-General_Concept
statistics	I-General_Concept
.	O
</s>
<s>
More	O
properly	O
,	O
it	O
is	O
the	O
partitioning	O
of	O
sums	O
of	O
squared	B-General_Concept
deviations	I-General_Concept
or	O
errors	O
.	O
</s>
<s>
Mathematically	O
,	O
the	O
sum	B-General_Concept
of	I-General_Concept
squared	I-General_Concept
deviations	I-General_Concept
is	O
an	O
unscaled	O
,	O
or	O
unadjusted	O
measure	O
of	O
dispersion	O
(	O
also	O
called	O
variability	O
)	O
.	O
</s>
<s>
Partitioning	O
of	O
the	O
sum	B-General_Concept
of	I-General_Concept
squared	I-General_Concept
deviations	I-General_Concept
into	O
various	O
components	O
allows	O
the	O
overall	O
variability	O
in	O
a	O
dataset	O
to	O
be	O
ascribed	O
to	O
different	O
types	O
or	O
sources	O
of	O
variability	O
,	O
with	O
the	O
relative	O
importance	O
of	O
each	O
being	O
quantified	O
by	O
the	O
size	O
of	O
each	O
component	O
of	O
the	O
overall	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
.	O
</s>
<s>
If	O
all	O
such	O
deviations	O
are	O
squared	O
,	O
then	O
summed	O
,	O
as	O
in	O
,	O
this	O
gives	O
the	O
"	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
"	O
for	O
these	O
data	O
.	O
</s>
<s>
When	O
more	O
data	O
are	O
added	O
to	O
the	O
collection	O
the	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
will	O
increase	O
,	O
except	O
in	O
unlikely	O
cases	O
such	O
as	O
the	O
new	O
data	O
being	O
equal	O
to	O
the	O
mean	O
.	O
</s>
<s>
So	O
usually	O
,	O
the	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
will	O
grow	O
with	O
the	O
size	O
of	O
the	O
data	O
collection	O
.	O
</s>
<s>
Scaling	O
(	O
also	O
known	O
as	O
normalizing	O
)	O
means	O
adjusting	O
the	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
so	O
that	O
it	O
does	O
not	O
grow	O
as	O
the	O
size	O
of	O
the	O
data	O
collection	O
grows	O
.	O
</s>
<s>
If	O
the	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
were	O
not	O
normalized	O
,	O
its	O
value	O
would	O
always	O
be	O
larger	O
for	O
the	O
sample	O
of	O
100	O
people	O
than	O
for	O
the	O
sample	O
of	O
20	O
people	O
.	O
</s>
<s>
To	O
scale	O
the	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
,	O
we	O
divide	O
it	O
by	O
the	O
degrees	O
of	O
freedom	O
,	O
i.e.	O
,	O
calculate	O
the	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
per	O
degree	O
of	O
freedom	O
,	O
or	O
variance	O
.	O
</s>
<s>
Standard	B-General_Concept
deviation	I-General_Concept
,	O
in	O
turn	O
,	O
is	O
the	O
square	O
root	O
of	O
the	O
variance	O
.	O
</s>
<s>
The	O
above	O
describes	O
how	O
the	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
is	O
used	O
in	O
descriptive	B-General_Concept
statistics	I-General_Concept
;	O
see	O
the	O
article	O
on	O
total	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
for	O
an	O
application	O
of	O
this	O
broad	O
principle	O
to	O
inferential	O
statistics	O
.	O
</s>
<s>
Given	O
a	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
including	O
a	O
constant	O
,	O
based	O
on	O
a	O
sample	O
containing	O
n	O
observations	O
,	O
the	O
total	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
can	O
be	O
partitioned	O
as	O
follows	O
into	O
the	O
explained	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
(	O
ESS	O
)	O
and	O
the	O
residual	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
(	O
RSS	O
)	O
:	O
</s>
<s>
where	O
is	O
the	O
value	O
estimated	O
by	O
the	O
regression	B-General_Concept
line	I-General_Concept
having	O
,	O
,	O
...	O
,	O
as	O
the	O
estimated	O
coefficients	O
.	O
</s>
<s>
Note	O
that	O
the	O
residual	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
can	O
be	O
further	O
partitioned	O
as	O
the	O
lack-of-fit	B-General_Concept
sum	I-General_Concept
of	I-General_Concept
squares	I-General_Concept
plus	O
the	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
due	O
to	O
pure	O
error	O
.	O
</s>
