find standard deviation for the observation 1,2,3,4,5.
Answers
Answer:
The standard deviation of
{
1
,
2
,
3
,
4
,
5
}
=
[
5
2
−
1
12
]
1
2
=
√
2
Explanation:
Explanation:
Let's develop a general formula then as a particular you get standard deviation of
1
,
2
,
3
,
4
and
5
. If we have
{
1
,
2
,
3
,
...
.
,
n
}
and we need to find the standard deviation of this numbers.
Note that
Var
(
X
)
=
1
n
n
∑
i
=
1
x
2
i
−
(
1
n
n
∑
i
=
1
x
i
)
2
⇒
Var
(
X
)
=
1
n
n
∑
i
=
1
i
2
−
(
1
n
n
∑
i
=
1
i
)
2
⇒
Var
(
X
)
=
1
n
⋅
n
(
n
+
1
)
(
2
n
+
1
)
6
−
(
1
n
⋅
n
(
n
+
1
)
2
)
2
⇒
Var
(
X
)
=
(
n
+
1
)
(
2
n
+
1
)
6
−
(
n
+
1
2
)
2
⇒
Var
(
X
)
=
n
+
1
2
[
2
n
+
1
3
−
n
+
1
2
]
⇒
Var
(
X
)
=
n
+
1
2
⋅
n
−
1
6
⇒
Var
(
X
)
=
n
2
−
1
12
So, Standard deviation of
{
1
,
2
,
3
,
...
.
,
n
}
is
[
Var
(
X
)
]
1
2
=
[
n
2
−
1
12
]
1
2
In particular, your case the standard deviation of
{
1
,
2
,
3
,
4
,
5
}
=
[
5
2
−
1
12
]
1
2
=
√
2
.