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Statistics for Economist
Ch. 18 Accuracy of Average
1. Sampling Distribution and Standard Error
2. Sample Average
3. What kind of Standard Error to be used?
4. Remember This
5. Measurement Error
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INDEX
STATISTICS
1
Sampling Distribution and Standard Error
2
Sample Average
3
What kind of Standard Error to be used?
4
Remember This
5
Measurement Error
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STATISTICS
1. Sampling Distribution and Standard Error
Accuracy of Sample Average
 How big the difference will be between Random Sample
Average from the Black box and Population Average?
Below is a result of random replacement sampling (25 times) drawn
out from a box containing cards that have number from 1 to 7 on each
other . And This process has done twice (Sample size 25)
 2 4 3 2 5 7 5 6 4 5 4 4 1 2 4 4 6 4 7 2 7 2 Sample Sum = 105
573
Sample Average =105/25=4.2
 51434 52177 12324 71653 66
Sample Sum = 95
334
Sample Average =94/25=3.8
 At Random Sampling
Accident
25 Numbers gonna be
changed after another
sampling
Expect Value of Sample
This would
and Standard Error
result in
Another Sample indicate the confidence
of Estimates.
Average
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1.Sampling Distribution and Standard Error
STATISTICS
Sample Average of Random replacement
Sampling
 S.D of box : When One sample got drawn out, S.D indicates the
deviation from the population average of the drawn value.
 S.E. of Sample Average : This indicates the deviation from the
population average of Drawn Sample average.
 S.D of Sample : This indicates the deviation of drawn one
Sample from the sample average. That is, This is the estimate of
S.D of Box
Expect Value of S.A. = Average of Box
S.E. of S.A. = S.E. of Sample Sum/Sample Size
= S.D of Box()/
S.D. of Sample(SD)/
Sample Size
Sample Size
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STATISTICS
1. Sampling Distribution and Standard Error
Sample Average of Random replacement Sampling
Ex1) At the 25 times random replacement sampling drawn out
from a box containing cards that have number from 1 to 7 on
each other, Calculate the Expect Value of Sample Average and
Standard Error.
Average of Box = 4  Expect Value of Sample Sum = 425 = 100
S.D of Box = 2 
S.E of Sample Sum = 2 25 = 10
 Sample Sum: 10010
 Sample Average: 40.4 (S.A is the value that is Sample sum to
sample size(25))
Expect value of S.A. = 4, S.E.= 0.4
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1. Sampling Distribution and Standard Error
STATISTICS
Sampling Distribution
 Probability Histogram of Sample Average (Sampling Distribution) :
As
Restoring Drawing Sample Average infinitely, We can get histogram of sample average
consist of infinite sample averages.
표준단위
-3
-2
-1
0
표준단위
1
2
3
50
-3
-2
-1
0
1
2
3
2.8
3.2
3.6
4
4.4
4.8
5.2
표준단위별 비율 (%)
표준단위별 비율 (%)
50
25
25
0
0
70
80
90
100
110
120
130
합
평균
- This Two Histograms have difference only in scale, over whole shape is identical.
- This Two Histogram approximate to Normal distribution asymptotically by Central Limit
Theorem.
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1. Sampling Distribution and Standard Error
STATISTICS
Sampling Distribution
Ex2) In Ex1)What is Expect value of Sample Average and Standard Error,
if We draw out sample 100 time instead of 25 times.
Average of Box = 4  Expect Value of Sample Sum = 4100 = 400
S.D of Box = 2

 Sample Sum : 400  20
S.E of Sample Sum = 2

100 = 20
Sample Average : 4  0.2
S.E. of Sample Sum=20(increased), S.E of Sample Average=0.2(decreased)
As S.E of Sample Sum is in proportional to Square root of
Sample size, Sample Average-Sample sum/sample size- is
in inverse proportional to Square root of Sample size. That is
S.E. of Sample average got decreased
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INDEX
STATISTICS
1
Sampling Distribution and Standard Error
2
Sample Average
3
What kind of Standard Error to be used?
4
Remember This
5
Measurement Error
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2. Sample Average
STATISTICS
Inference
 Inference: How to know Population Average from Sample
Average if We don’t have any information of Box?
 Ex) We got random sample of 1,000 households from The City has 20,500
households, and Sample Average of Household income is KRW32.4mil.. What is the
Average Household income of Population?
S.D of Drawn 1,000 households=KRW19mil.
S.E. of Sample Sum=KRW19mil. 1000 =KRW600mil.
S.E of Sample Average=KRW600mil./1000=KRW0.6mil.
Average Income of Whole Households is inferred to
KRW32.4mil. KRW0.6mil.
‘Observed Value of Sample Average= Average of Box + Probable Error’,
‘Probability Error  S.E of Sample Average’
We Can infer Average of Box from Observed value of Sample Average
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2. Sample Average
STATISTICS
Confidence Interval
 Confidence Level 95%: 95% of All confidence intervals include
Population Average
 95% Confidence Interval of Average
Households: Sample Average +/- 2*S.E.
income
of
25,000
 KRW32.4mil.1.2mil.  (KRW31.2mil., KRW33.6mil.)
 Why S.E. is used instead of S.D.?
S.D.: When One sample got drawn out, S.D indicates the deviation from the
population average.
S.E. of Sample Average: This indicates the deviation from the population average
of Drawn Sample average.
If We want to get Population Average from Sample Average,
We should use S.E. of Sample Average
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2. Sample Average
STATISTICS
Confidence Interval
95% confidence interval covers area － 2~+2 in Standard
Normal Distribution Curve
 Why Standard Normal Distribution Curve is used in calculating
confidence interval?
Central Limit Theorem: If A histogram of individual observed value is
not identical to Normal Distribution Curve, Shape of Probability
Histogram of Sample Average approximates to Normal Distribution
Curve.
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2. Sample Average
STATISTICS
Educational Period of Population aged 25 :

Distribution of Sample resembles Population Distribution.
Why? The Law of Large Number – Empirical Histogram approximates to
Probability Histogram

If Sample size were enough, Sample Average infers Population Average and
Sample Standard Deviation infer Population Standard Deviation

25세 인구 전체 (모집단)
400명을 무작위추출 (표본)
표본평균의 분포
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2. Sample Average
STATISTICS
Sampling Distribution
 There is great difference between Distribution of Population and
Normal Distribution. But Probability Histogram of Sample Average
(Sample Distribution of Sample Average) resembles Normal
Distribution Curve
In Simple Random Sample of Sized 400 people, What is the probability of event that
Sample Average is in 11.2(year) and 12(year)
= Area of (－2, 2) section in Standard Normal Distribution Curve = 95%
 95% confidence interval of Average Education Period of Population = (11.2(year),
12.0(year))
A individual observed value does not follow Normal
Distribution Curve, Shape of Probability Histogram of
Sample Average resembles to Normal Distribution Curve.
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INDEX
STATISTICS
1
Sampling Distribution and Standard Error
2
Sample Average
3
What kind of Standard Error to be used?
4
Remember This
5
Measurement Error
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3. What kind of Standard Error to be used?
STATISTICS
S.E in Box Model
 Inferring Sum

S.E. of Sample Sum
 Inferring Average

S.E. of Sample Average
 Inferring Number

S.E. of Sample Number
 Inferring Ration

S.E. of Sample Ratio
 S.E. of Sample Sum = S.D. of Box×
 S.E. of Sample Average = S.D. of Box/
Sample Size
Sample Size
 S.E. of Sample Size = S.E. of 0-1 Box Sample Sum
 S.E. of Sample Ratio = (S.E. of Sample Number/Sample Size)×100%
In General, We don’t know what the S.D. of Box is, We replace
S.D. of Box to known Sample S.D. to calculate S.E.
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INDEX
STATISTICS
1
Sampling Distribution and Standard Error
2
Sample Average
3
What kind of Standard Error to be used?
4
Remember This
5
Measurement Error
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STATISTICS
4. Remember This
Box Model
 There is difference between Sample Average and Population
Average and that is Probable Error
 Less Probable error means More confidence level, More Probable
Error means Less confidence level.
 General Size of Probable Error is inferred from S.E.
 S.E of Sample Average is inferred by dividing Sample S.E by
Square root of Sample size.
Random Sampling let us calculate S.E. of Sample Average
directly from only Sample size and Sample S.D.
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STATISTICS
4. Remember This
Statistical Inference
 Statistical Inference?
Given Data
Stochastic
Method
Characteristic
of Population
 Many Method learned until now is meaningless in model that is
not conversed to Box model. To do Statistical inference, We
need a stochastic model like a box model.

Statistical Inference is based on Stochastic Method.

It is meaningless to calculate S.E. in a model that is not
conversed to the Box model (ex. Including trend or intention).
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INDEX
STATISTICS
1
Sampling Distribution and Standard Error
2
Sample Average
3
What Standard Error to be used?
4
Remember This
5
Measurement Error
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STATISTICS
5. Measurement Error
Measurement Error
Measurement Error : Not in Object itself, There are Stochastic Errors in
Measuring process, A Observed Value has measurement error compared to
the Real Value.
Repeating more and more and then Averaging reduces a Measurement Error
(inverse proportional to square root of number of repetition), increases 정
Confidence of inference (proportional to square root of repetition).
Ex) 100 times repetition of measuring weight of 1 gobbet of meat
Average = 600g, Standard Deviation=10g
•Size of Measurement Error contained in 1 observation:
About 10g (Sample Standard Deviation)
•Size of Measurement Error contained in Observed
Average : About 1g(=10g/ 100 )(S.E. of Sample
Average)
95% Confidence Interval of
Real Value of weight
(600-2)g
(600+2)g
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5. Measurement Error
STATISTICS
Gaussian Model
Gaussian Model
Box Model
Drawing out many cards
from one box
Repeating observation for
identical object in same condition
Gaussian Model: Measurement Error is identical to drawing out a
error card in a single trial from a box
Ex) 1st Observation = Real Value+1st drawing out from a error box
2nd Observation = Real Value+2nd drawing out from a error box
……
:
100th Observation = Real Value+100th drawing out from a error box
S.D. of
observations of
repetition
= S.D. of
Probable Errors
In Gaussian Model, S.D. of Observations in repetition is a estimate of S.D. of a Error box.
As Sample size get larger, Confidence level of a Estimate increases.
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