<s>
Grouped	B-General_Concept
data	I-General_Concept
are	O
data	O
formed	O
by	O
aggregating	O
individual	O
observations	O
of	O
a	O
variable	O
into	O
groups	O
,	O
so	O
that	O
a	O
frequency	O
distribution	O
of	O
these	O
groups	O
serves	O
as	O
a	O
convenient	O
means	O
of	O
summarizing	O
or	O
analyzing	B-General_Concept
the	O
data	O
.	O
</s>
<s>
There	O
are	O
two	O
major	O
types	O
of	O
grouping	O
:	O
data	B-General_Concept
binning	I-General_Concept
of	O
a	O
single-dimensional	O
variable	O
,	O
replacing	O
individual	O
numbers	O
by	O
counts	O
in	O
bins	O
;	O
and	O
grouping	O
multi-dimensional	O
variables	O
by	O
some	O
of	O
the	O
dimensions	O
(	O
especially	O
by	O
independent	O
variables	O
)	O
,	O
obtaining	O
the	O
distribution	O
of	O
ungrouped	O
dimensions	O
(	O
especially	O
the	O
dependent	O
variables	O
)	O
.	O
</s>
<s>
The	O
idea	O
of	O
grouped	B-General_Concept
data	I-General_Concept
can	O
be	O
illustrated	O
by	O
considering	O
the	O
following	O
raw	O
dataset	O
:	O
</s>
<s>
For	O
example	O
,	O
suppose	O
in	O
the	O
above	O
example	O
,	O
there	O
are	O
three	O
types	O
of	O
students	O
:	O
1	O
)	O
Below	O
normal	O
,	O
if	O
the	O
response	O
time	O
is	O
5	O
to	O
14	O
seconds	O
,	O
2	O
)	O
normal	O
if	O
it	O
is	O
between	O
15	O
and	O
24	O
seconds	O
,	O
and	O
3	O
)	O
above	O
normal	O
if	O
it	O
is	O
25	O
seconds	O
or	O
more	O
,	O
then	O
the	O
grouped	B-General_Concept
data	I-General_Concept
looks	O
like	O
:	O
</s>
<s>
The	O
grouped	B-General_Concept
data	I-General_Concept
looks	O
like	O
:	O
</s>
<s>
An	O
estimate	O
,	O
,	O
of	O
the	O
mean	O
of	O
the	O
population	O
from	O
which	O
the	O
data	O
are	O
drawn	O
can	O
be	O
calculated	O
from	O
the	O
grouped	B-General_Concept
data	I-General_Concept
as	O
:	O
</s>
<s>
Note	O
that	O
the	O
result	O
of	O
this	O
will	O
be	O
different	O
from	O
the	O
sample	O
mean	O
of	O
the	O
ungrouped	B-General_Concept
data	I-General_Concept
.	O
</s>
<s>
The	O
mean	O
for	O
the	O
grouped	B-General_Concept
data	I-General_Concept
in	O
the	O
above	O
example	O
,	O
can	O
be	O
calculated	O
as	O
follows	O
:	O
</s>
<s>
The	O
mean	O
for	O
the	O
grouped	B-General_Concept
data	I-General_Concept
in	O
example	O
4	O
above	O
can	O
be	O
calculated	O
as	O
follows	O
:	O
</s>
