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Find the Variance 12 , 16 , 23 , 14 , 20 , 18 , 22 , 18 , 14 , 15
12
,
16
,
23
,
14
,
20
,
18
,
22
,
18
,
14
,
15
Step 1
The
mean
of a
set
of numbers is the
sum
divided by the number of
terms
.
‾
x
=
12
+
16
+
23
+
14
+
20
+
18
+
22
+
18
+
14
+
15
10
Step 2
Simplify the
numerator
.
Add
12
and
16
.
‾
x
=
28
+
23
+
14
+
20
+
18
+
22
+
18
+
14
+
15
10
Add
28
and
23
.
‾
x
=
51
+
14
+
20
+
18
+
22
+
18
+
14
+
15
10
Add
51
and
14
.
‾
x
=
65
+
20
+
18
+
22
+
18
+
14
+
15
10
Add
65
and
20
.
‾
x
=
85
+
18
+
22
+
18
+
14
+
15
10
Add
85
and
18
.
‾
x
=
103
+
22
+
18
+
14
+
15
10
Add
103
and
22
.
‾
x
=
125
+
18
+
14
+
15
10
Add
125
and
18
.
‾
x
=
143
+
14
+
15
10
Add
143
and
14
.
‾
x
=
157
+
15
10
Add
157
and
15
.
‾
x
=
172
10
‾
x
=
172
10
Step 3
Cancel the
common factor
of
172
and
10
.
Factor
2
out of
172
.
‾
x
=
2
(
86
)
10
Cancel the
common factors
.
Factor
2
out of
10
.
‾
x
=
2
⋅
86
2
⋅
5
Cancel the
common factor
.
‾
x
=
2
⋅
86
2
⋅
5
Rewrite the
expression
.
‾
x
=
86
5
‾
x
=
86
5
‾
x
=
86
5
Step 4
Divide
.
‾
x
=
17.2
Step 5
Set
up the
formula
for variance. The variance of a
set
of values is a
measure
of the spread of its values.
s
2
=
n
∑
i
=
1
(
x
i
-
x
a
v
g
)
2
n
-
1
Step 6
Set
up the
formula
for variance for this
set
of numbers.
s
=
(
12
-
17.2
)
2
+
(
16
-
17.2
)
2
+
(
23
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
20
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
22
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
15
-
17.2
)
2
10
-
1
Step 7
Simplify the result.
Simplify the
numerator
.
Subtract
17.2
from
12
.
s
=
(
-
5.2
)
2
+
(
16
-
17.2
)
2
+
(
23
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
20
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
22
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
15
-
17.2
)
2
10
-
1
Raise
-
5.2
to the
power
of
2
.
s
=
27.04
+
(
16
-
17.2
)
2
+
(
23
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
20
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
22
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
15
-
17.2
)
2
10
-
1
Subtract
17.2
from
16
.
s
=
27.04
+
(
-
1.2
)
2
+
(
23
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
20
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
22
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
15
-
17.2
)
2
10
-
1
Raise
-
1.2
to the
power
of
2
.
s
=
27.04
+
1.44
+
(
23
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
20
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
22
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
15
-
17.2
)
2
10
-
1
Subtract
17.2
from
23
.
s
=
27.04
+
1.44
+
5.8
2
+
(
14
-
17.2
)
2
+
(
20
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
22
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
15
-
17.2
)
2
10
-
1
Raise
5.8
to the
power
of
2
.
s
=
27.04
+
1.44
+
33.64
+
(
14
-
17.2
)
2
+
(
20
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
22
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
15
-
17.2
)
2
10
-
1
Subtract
17.2
from
14
.
s
=
27.04
+
1.44
+
33.64
+
(
-
3.2
)
2
+
(
20
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
22
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
15
-
17.2
)
2
10
-
1
Raise
-
3.2
to the
power
of
2
.
s
=
27.04
+
1.44
+
33.64
+
10.24
+
(
20
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
22
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
15
-
17.2
)
2
10
-
1
Subtract
17.2
from
20
.
s
=
27.04
+
1.44
+
33.64
+
10.24
+
2.8
2
+
(
18
-
17.2
)
2
+
(
22
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
15
-
17.2
)
2
10
-
1
Raise
2.8
to the
power
of
2
.
s
=
27.04
+
1.44
+
33.64
+
10.24
+
7.84
+
(
18
-
17.2
)
2
+
(
22
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
15
-
17.2
)
2
10
-
1
Subtract
17.2
from
18
.
s
=
27.04
+
1.44
+
33.64
+
10.24
+
7.84
+
0.8
2
+
(
22
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
15
-
17.2
)
2
10
-
1
Raise
0.8
to the
power
of
2
.
s
=
27.04
+
1.44
+
33.64
+
10.24
+
7.84
+
0.64
+
(
22
-
17.2
)
2
+
(
18
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
15
-
17.2
)
2
10
-
1
Subtract
17.2
from
22
.
s
=
27.04
+
1.44
+
33.64
+
10.24
+
7.84
+
0.64
+
4.8
2
+
(
18
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
15
-
17.2
)
2
10
-
1
Raise
4.8
to the
power
of
2
.
s
=
27.04
+
1.44
+
33.64
+
10.24
+
7.84
+
0.64
+
23.04
+
(
18
-
17.2
)
2
+
(
14
-
17.2
)
2
+
(
15
-
17.2
)
2
10
-
1
Subtract
17.2
from
18
.
s
=
27.04
+
1.44
+
33.64
+
10.24
+
7.84
+
0.64
+
23.04
+
0.8
2
+
(
14
-
17.2
)
2
+
(
15
-
17.2
)
2
10
-
1
Raise
0.8
to the
power
of
2
.
s
=
27.04
+
1.44
+
33.64
+
10.24
+
7.84
+
0.64
+
23.04
+
0.64
+
(
14
-
17.2
)
2
+
(
15
-
17.2
)
2
10
-
1
Subtract
17.2
from
14
.
s
=
27.04
+
1.44
+
33.64
+
10.24
+
7.84
+
0.64
+
23.04
+
0.64
+
(
-
3.2
)
2
+
(
15
-
17.2
)
2
10
-
1
Raise
-
3.2
to the
power
of
2
.
s
=
27.04
+
1.44
+
33.64
+
10.24
+
7.84
+
0.64
+
23.04
+
0.64
+
10.24
+
(
15
-
17.2
)
2
10
-
1
Subtract
17.2
from
15
.
s
=
27.04
+
1.44
+
33.64
+
10.24
+
7.84
+
0.64
+
23.04
+
0.64
+
10.24
+
(
-
2.2
)
2
10
-
1
Raise
-
2.2
to the
power
of
2
.
s
=
27.04
+
1.44
+
33.64
+
10.24
+
7.84
+
0.64
+
23.04
+
0.64
+
10.24
+
4.84
10
-
1
Add
27.04
and
1.44
.
s
=
28.48
+
33.64
+
10.24
+
7.84
+
0.64
+
23.04
+
0.64
+
10.24
+
4.84
10
-
1
Add
28.48
and
33.64
.
s
=
62.12
+
10.24
+
7.84
+
0.64
+
23.04
+
0.64
+
10.24
+
4.84
10
-
1
Add
62.12
and
10.24
.
s
=
72.36
+
7.84
+
0.64
+
23.04
+
0.64
+
10.24
+
4.84
10
-
1
Add
72.36
and
7.84
.
s
=
80.2
+
0.64
+
23.04
+
0.64
+
10.24
+
4.84
10
-
1
Add
80.2
and
0.64
.
s
=
80.84
+
23.04
+
0.64
+
10.24
+
4.84
10
-
1
Add
80.84
and
23.04
.
s
=
103.88
+
0.64
+
10.24
+
4.84
10
-
1
Add
103.88
and
0.64
.
s
=
104.52
+
10.24
+
4.84
10
-
1
Add
104.52
and
10.24
.
s
=
114.76
+
4.84
10
-
1
Add
114.76
and
4.84
.
s
=
119.6
10
-
1
s
=
119.6
10
-
1
Simplify the
expression
.
Subtract
1
from
10
.
s
=
119.6
9
Divide
119.6
by
9
.
s
=
13.2
‾
8
s
=
13.2
‾
8
s
=
13.2
‾
8
Step 8
Approximate
the result.
s
2
≈
13.2889
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