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Find the Sample Standard Deviation 20 , 40 , 16 , 22 , 35 , 23 , 23 , 29 , 28 , 38
20
,
40
,
16
,
22
,
35
,
23
,
23
,
29
,
28
,
38
Step 1
Find the
mean
.
The
mean
of a
set
of numbers is the
sum
divided by the number of
terms
.
‾
x
=
20
+
40
+
16
+
22
+
35
+
23
+
23
+
29
+
28
+
38
10
Simplify the
numerator
.
Add
20
and
40
.
‾
x
=
60
+
16
+
22
+
35
+
23
+
23
+
29
+
28
+
38
10
Add
60
and
16
.
‾
x
=
76
+
22
+
35
+
23
+
23
+
29
+
28
+
38
10
Add
76
and
22
.
‾
x
=
98
+
35
+
23
+
23
+
29
+
28
+
38
10
Add
98
and
35
.
‾
x
=
133
+
23
+
23
+
29
+
28
+
38
10
Add
133
and
23
.
‾
x
=
156
+
23
+
29
+
28
+
38
10
Add
156
and
23
.
‾
x
=
179
+
29
+
28
+
38
10
Add
179
and
29
.
‾
x
=
208
+
28
+
38
10
Add
208
and
28
.
‾
x
=
236
+
38
10
Add
236
and
38
.
‾
x
=
274
10
‾
x
=
274
10
Cancel the
common factor
of
274
and
10
.
Factor
2
out of
274
.
‾
x
=
2
(
137
)
10
Cancel the
common factors
.
Factor
2
out of
10
.
‾
x
=
2
⋅
137
2
⋅
5
Cancel the
common factor
.
‾
x
=
2
⋅
137
2
⋅
5
Rewrite the
expression
.
‾
x
=
137
5
‾
x
=
137
5
‾
x
=
137
5
Divide
.
‾
x
=
27.4
‾
x
=
27.4
Step 2
Simplify each value in the list.
Convert
20
to a decimal value.
20
Convert
40
to a decimal value.
40
Convert
16
to a decimal value.
16
Convert
22
to a decimal value.
22
Convert
35
to a decimal value.
35
Convert
23
to a decimal value.
23
Convert
29
to a decimal value.
29
Convert
28
to a decimal value.
28
Convert
38
to a decimal value.
38
The simplified values are
20
,
40
,
16
,
22
,
35
,
23
,
23
,
29
,
28
,
38
.
20
,
40
,
16
,
22
,
35
,
23
,
23
,
29
,
28
,
38
20
,
40
,
16
,
22
,
35
,
23
,
23
,
29
,
28
,
38
Step 3
Set
up the
formula
for
sample
standard deviation
. The
standard deviation
of a
set
of values is a
measure
of the spread of its values.
s
=
n
∑
i
=
1
√
(
x
i
-
x
a
v
g
)
2
n
-
1
Step 4
Set
up the
formula
for
standard deviation
for this
set
of numbers.
s
=
√
(
20
-
27.4
)
2
+
(
40
-
27.4
)
2
+
(
16
-
27.4
)
2
+
(
22
-
27.4
)
2
+
(
35
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
29
-
27.4
)
2
+
(
28
-
27.4
)
2
+
(
38
-
27.4
)
2
10
-
1
Step 5
Simplify the result.
Subtract
27.4
from
20
.
s
=
√
(
-
7.4
)
2
+
(
40
-
27.4
)
2
+
(
16
-
27.4
)
2
+
(
22
-
27.4
)
2
+
(
35
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
29
-
27.4
)
2
+
(
28
-
27.4
)
2
+
(
38
-
27.4
)
2
10
-
1
Raise
-
7.4
to the
power
of
2
.
s
=
√
54.76
+
(
40
-
27.4
)
2
+
(
16
-
27.4
)
2
+
(
22
-
27.4
)
2
+
(
35
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
29
-
27.4
)
2
+
(
28
-
27.4
)
2
+
(
38
-
27.4
)
2
10
-
1
Subtract
27.4
from
40
.
s
=
√
54.76
+
12.6
2
+
(
16
-
27.4
)
2
+
(
22
-
27.4
)
2
+
(
35
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
29
-
27.4
)
2
+
(
28
-
27.4
)
2
+
(
38
-
27.4
)
2
10
-
1
Raise
12.6
to the
power
of
2
.
s
=
√
54.76
+
158.76
+
(
16
-
27.4
)
2
+
(
22
-
27.4
)
2
+
(
35
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
29
-
27.4
)
2
+
(
28
-
27.4
)
2
+
(
38
-
27.4
)
2
10
-
1
Subtract
27.4
from
16
.
s
=
√
54.76
+
158.76
+
(
-
11.4
)
2
+
(
22
-
27.4
)
2
+
(
35
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
29
-
27.4
)
2
+
(
28
-
27.4
)
2
+
(
38
-
27.4
)
2
10
-
1
Raise
-
11.4
to the
power
of
2
.
s
=
√
54.76
+
158.76
+
129.96
+
(
22
-
27.4
)
2
+
(
35
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
29
-
27.4
)
2
+
(
28
-
27.4
)
2
+
(
38
-
27.4
)
2
10
-
1
Subtract
27.4
from
22
.
s
=
√
54.76
+
158.76
+
129.96
+
(
-
5.4
)
2
+
(
35
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
29
-
27.4
)
2
+
(
28
-
27.4
)
2
+
(
38
-
27.4
)
2
10
-
1
Raise
-
5.4
to the
power
of
2
.
s
=
√
54.76
+
158.76
+
129.96
+
29.16
+
(
35
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
29
-
27.4
)
2
+
(
28
-
27.4
)
2
+
(
38
-
27.4
)
2
10
-
1
Subtract
27.4
from
35
.
s
=
√
54.76
+
158.76
+
129.96
+
29.16
+
7.6
2
+
(
23
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
29
-
27.4
)
2
+
(
28
-
27.4
)
2
+
(
38
-
27.4
)
2
10
-
1
Raise
7.6
to the
power
of
2
.
s
=
√
54.76
+
158.76
+
129.96
+
29.16
+
57.76
+
(
23
-
27.4
)
2
+
(
23
-
27.4
)
2
+
(
29
-
27.4
)
2
+
(
28
-
27.4
)
2
+
(
38
-
27.4
)
2
10
-
1
Subtract
27.4
from
23
.
s
=
√
54.76
+
158.76
+
129.96
+
29.16
+
57.76
+
(
-
4.4
)
2
+
(
23
-
27.4
)
2
+
(
29
-
27.4
)
2
+
(
28
-
27.4
)
2
+
(
38
-
27.4
)
2
10
-
1
Raise
-
4.4
to the
power
of
2
.
s
=
√
54.76
+
158.76
+
129.96
+
29.16
+
57.76
+
19.36
+
(
23
-
27.4
)
2
+
(
29
-
27.4
)
2
+
(
28
-
27.4
)
2
+
(
38
-
27.4
)
2
10
-
1
Subtract
27.4
from
23
.
s
=
√
54.76
+
158.76
+
129.96
+
29.16
+
57.76
+
19.36
+
(
-
4.4
)
2
+
(
29
-
27.4
)
2
+
(
28
-
27.4
)
2
+
(
38
-
27.4
)
2
10
-
1
Raise
-
4.4
to the
power
of
2
.
s
=
√
54.76
+
158.76
+
129.96
+
29.16
+
57.76
+
19.36
+
19.36
+
(
29
-
27.4
)
2
+
(
28
-
27.4
)
2
+
(
38
-
27.4
)
2
10
-
1
Subtract
27.4
from
29
.
s
=
√
54.76
+
158.76
+
129.96
+
29.16
+
57.76
+
19.36
+
19.36
+
1.6
2
+
(
28
-
27.4
)
2
+
(
38
-
27.4
)
2
10
-
1
Raise
1.6
to the
power
of
2
.
s
=
√
54.76
+
158.76
+
129.96
+
29.16
+
57.76
+
19.36
+
19.36
+
2.56
+
(
28
-
27.4
)
2
+
(
38
-
27.4
)
2
10
-
1
Subtract
27.4
from
28
.
s
=
√
54.76
+
158.76
+
129.96
+
29.16
+
57.76
+
19.36
+
19.36
+
2.56
+
0.6
2
+
(
38
-
27.4
)
2
10
-
1
Raise
0.6
to the
power
of
2
.
s
=
√
54.76
+
158.76
+
129.96
+
29.16
+
57.76
+
19.36
+
19.36
+
2.56
+
0.36
+
(
38
-
27.4
)
2
10
-
1
Subtract
27.4
from
38
.
s
=
√
54.76
+
158.76
+
129.96
+
29.16
+
57.76
+
19.36
+
19.36
+
2.56
+
0.36
+
10.6
2
10
-
1
Raise
10.6
to the
power
of
2
.
s
=
√
54.76
+
158.76
+
129.96
+
29.16
+
57.76
+
19.36
+
19.36
+
2.56
+
0.36
+
112.36
10
-
1
Add
54.76
and
158.76
.
s
=
√
213.52
+
129.96
+
29.16
+
57.76
+
19.36
+
19.36
+
2.56
+
0.36
+
112.36
10
-
1
Add
213.52
and
129.96
.
s
=
√
343.48
+
29.16
+
57.76
+
19.36
+
19.36
+
2.56
+
0.36
+
112.36
10
-
1
Add
343.48
and
29.16
.
s
=
√
372.64
+
57.76
+
19.36
+
19.36
+
2.56
+
0.36
+
112.36
10
-
1
Add
372.64
and
57.76
.
s
=
√
430.4
+
19.36
+
19.36
+
2.56
+
0.36
+
112.36
10
-
1
Add
430.4
and
19.36
.
s
=
√
449.76
+
19.36
+
2.56
+
0.36
+
112.36
10
-
1
Add
449.76
and
19.36
.
s
=
√
469.12
+
2.56
+
0.36
+
112.36
10
-
1
Add
469.12
and
2.56
.
s
=
√
471.68
+
0.36
+
112.36
10
-
1
Add
471.68
and
0.36
.
s
=
√
472.04
+
112.36
10
-
1
Add
472.04
and
112.36
.
s
=
√
584.4
10
-
1
Subtract
1
from
10
.
s
=
√
584.4
9
Divide
584.4
by
9
.
s
=
√
64.9
‾
3
s
=
√
64.9
‾
3
Step 6
The
standard deviation
should be rounded to one more decimal place than the original
data
. If the original
data
were mixed, round to one decimal place more than the least precise.
8.1
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