Logan B. Nielsen
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Derivation of
V
a
r
(
X
ˉ
)
Var(\bar X)
Va
r
(
X
ˉ
)
By definition,
X
ˉ
=
(
X
1
+
X
2
+
⋯
+
X
n
)
/
n
\bar X = (X_1 + X_2 + \dots + X_n)/n
X
ˉ
=
(
X
1
+
X
2
+
⋯
+
X
n
)
/
n
.
Therefore:
V
a
r
(
X
ˉ
)
=
V
a
r
(
X
1
/
n
+
X
2
/
n
+
⋯
+
X
n
/
n
)
=
(
1
/
n
2
)
V
a
r
(
X
1
+
X
2
+
⋯
+
X
n
)
/
n
2
=
σ
2
/
n
\begin{align} Var(\bar X) &= Var(X_1/n + X_2/n + \dots + X_n/n) \\ &= (1/n^2) Var(X_1 + X_2 + \dots + X_n)/n^2 \\ &= \sigma^2/n \end{align}
Va
r
(
X
ˉ
)
=
Va
r
(
X
1
/
n
+
X
2
/
n
+
⋯
+
X
n
/
n
)
=
(
1/
n
2
)
Va
r
(
X
1
+
X
2
+
⋯
+
X
n
)
/
n
2
=
σ
2
/
n