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Basic Statistics 03
Sampling & Central Limit Theorem
Econometric Analysis
1
Sampling & C.L.T.
1. The distribution of the sample mean
2. The Central Limit Theorem
3. The application of CLT
Econometric Analysis
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1. Distribution of the Sample Means
The sampling distribution of the sample mean
is a probability distribution consisting of all
possible sample means of a given sample size
selected from a population.
nµ
1
 1
=µ
E ( X ) = E ∑ X i  = ∑ E( X i ) =
n
 n
n
1

1
Var ( X ) = Var  ∑ X i  = 2
 n
n
*in the case of finite population,
nσ 2 σ 2
∑ Var ( X i ) = n 2 = n
Var ( X ) =
Econometric Analysis
σ2 N −n
n
⋅
N −1
3
Population Distribution
µ
Econometric Analysis
Xi
4
Sampling Dist. of the Sample Means
n=N
1<n<N
n=1
µ
Econometric Analysis
Xi
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2. Central Limit Theorem
For any population with a mean µ and a
variance σ2, the sampling distribution of
the means of all possible samples of size n
will be approximately normally distributed,
with larger sample size n.
The mean of the sampling distribution equal
to µ and the variance equal to σ2/n.
Econometric Analysis
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CLT The Population Distribution
Uniform Distribution
1600
1400
1200
1000
800
600
μ＝0.5003
σ2＝0.0831
N＝250,000
400
200
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
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0.7
0.8
0.9
1.0
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CLT The Sample Distribution of Sample mean
The Normal Distribution
700
600
500
400
300
E(X) = 0.5002
Var(X) = 0.0028
n = 30
200
（σ2/30 = 0.0831/30）
sample size = 15,000
100
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
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0.8
0.9
1.0
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CLT 0
the case of X~N（µ, σ2）
If a population follows the normal
distribution, the sampling distribution of
the sample mean will also follow the
normal distribution for any sample size.
To determine the probability a sample mean
falls within a particular region, use:
X～N ( µ ,
σ2
n
)
X −µ
z=
～N (0,1)
σ n
Econometric Analysis
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CLT 1
the case of X~ non-normal dist.(µ, σ2）
If the population isn’t normally
distributed (with known σ2) and sample
size is large, the sample means will follow
the normal distribution. (See the above
figure.)
d
X
→
N (µ ,
σ2
n
)
X −µ d

→ N (0,1)
σ n
Econometric Analysis
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CLT 2 the case of X~N（µ, unknown σ2）
If the population follows the normal
distribution but σ2 is unknown, the sample
means will follow the t distribution.

But with larger sample size (at least n >30), the
sample means will follow the normal
distribution.
X −µ
t=
～t (d . f .)
s n
X −µ d
t=

→ N (0,1)
s n
Econometric Analysis
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CLT 3
the case of X~ non-N （µ, unknown σ2）
If the population isn’t normally
distributed with unknown σ2 and sample
size is large, the sample means will follow
the t distribution. (with larger sample size,
the normal distribution)
X −µ d

→ N (0,1)
s n
Econometric Analysis
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