Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 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

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 2/21 INDEX 1 Sampling Distribution and Standard Error 2 Sample Average 3 What kind of Standard Error to be used? 4 Remember This 5 Measurement Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 3/21 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)    Sample Sum = 105 Sample Average =105/25=4.2  Sample Sum = 95 Sample Average =94/25=3.8  Sample Sum = 105 Sample Average =105/25=4.2  Sample Sum = 95 Sample Average =94/25=3.8 Accident 25 Numbers gonna be changed after another sampling This would result in Another Sample Average At Random Sampling Expect Value of Sample and Standard Error indicate the confidence of Estimates.

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 4/21 1.Sampling Distribution and Standard Error Sample Average of Random replacement Sampling 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  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

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 5/21 Average of Box = 4  Expect Value of Sample Sum = 4  25 = 100 S.D of Box = 2  S.E of Sample Sum = 2  = 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.= 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.

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 6/21 1. Sampling Distribution and Standard Error  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. Sampling Distribution - 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.

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 7/21 S.E. of Sample Sum=20(increased), S.E of Sample Average=0.2(decreased) 1. Sampling Distribution and Standard Error 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  S.E of Sample Sum = 2  = 20  Sample Sum : 400  20  Sample Average : 4  0.2 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

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 8/21 INDEX 1 Sampling Distribution and Standard Error 2 Sample Average 3 What kind of Standard Error to be used? 4 Remember This 5 Measurement Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 9/21 2. Sample Average 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.  =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

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 10/21 If We want to get Population Average from Sample Average, We should use S.E. of Sample Average 2. Sample Average Confidence Interval  Confidence Level 95%: 95% of All confidence intervals include Population Average  95% Confidence Interval of Average income of 25,000 Households: Sample Average +/- 2*S.E.  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.

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 11/21 2. Sample Average 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.

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 12/21 2. Sample Average 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 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 25 세 인구 전체 ( 모집단 ) 400 명을 무작위추출 ( 표본 ) 표본평균의 분포

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 13/21 2. Sample Average 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.

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 14/21 INDEX 1 Sampling Distribution and Standard Error 2 Sample Average 3 What kind of Standard Error to be used? 4 Remember This 5 Measurement Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 15/21 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 3. What kind of Standard Error to be used? S.E in Box Model  S.E. of Sample Sum = S.D. of Box×  S.E. of Sample Average = S.D. of Box/  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% Sample Size 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.

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 16/21 INDEX 1 Sampling Distribution and Standard Error 2 Sample Average 3 What kind of Standard Error to be used? 4 Remember This 5 Measurement Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 17/21 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.

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 18/21 4. Remember This Statistical Inference  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).  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. Given Data Characteristic of Population Stochastic Method  Statistical Inference?

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 19/21 INDEX 1 Sampling Distribution and Standard Error 2 Sample Average 3 What Standard Error to be used? 4 Remember This 5 Measurement Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 20/21 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 (600-2)g(600+2)g 95% Confidence Interval of Real Value of weight Size of Measurement Error contained in 1 observation: About 10g (Sample Standard Deviation) Size of Measurement Error contained in Observed Average : About 1g(=10g/ )(S.E. of Sample Average)

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 21/21 5. Measurement Error Gaussian Model Box Model Repeating observation for identical object in same condition Drawing out many cards from one box  Gaussian Model: Measurement Error is identical to drawing out a error card in a single trial from a box Ex) 1 st Observation = Real Value+1 st drawing out from a error box 2 nd 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.