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Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics for Economist Ch. 13 The Normal Approximation for Probability Histograms 1. Tossing a Coin and the Normal Distribution 2. Two Kinds of Histograms 3. The Relationship of Two Histograms 4. Central Limit Theorem 5. The Normal Approximation 6. The Scope of the Normal Approximation 7. Conclusion Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics INDEX STATISTICS 1 Tossing a Coin and the Normal Distribution 2 Two Kinds of Histograms 3 The Relationship of Two Histograms 4 Central Limit Theorem Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 2/31 1. Tossing a Coin and the Normal Distribution STATISTICS When a coin is tossed a large number of times, the percentage of heads will be closed to 50%. Ex) 5 Tosses # of heads # of ways 0 1 1 5 2 10 3 10 4 5 5 1 If 100 tosses? Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 3/31 1. Tossing a Coin and the Normal Distribution STATISTICS If we toss 100 times 1.Total # of ways 2100 1.27 1030 2. The # of ways with exactly 50 heads 100! 100 99 51 1.01 1029 50 ! 50! 50 49 1 3. mathematical probability for the # of patterns with exactly 50 heads 1.01 10 29 0.08 8% 30 1.27 10 # of ways with 50 heads total # of ways Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 4/31 1. Tossing a Coin and the Normal Distribution STATISTICS To calculate in the previous way the probability of getting exactly 50 heads from 100 tosses is inconvenient. Any alternatives? When you toss lots of coins, the distribution of the number or ratio of getting heads among all tosses is well approximated by the normal curve. We will solve this problem later using the normal curve. The answer is 7.96%. It is almost same with the previous answer 8% acquired from the binomial distribution. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 5/31 INDEX STATISTICS 1 Tossing a Coin and the Normal Distribution 2 Two Kinds of Histograms 3 The Relationship of Two Histograms 4 Central Limit Theorem Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 6/31 STATISTICS 2. Two Kinds of Histograms Probability Histogram and Empirical Histogram Probability Histograms A kind of graph which represents probability, not data. It is made up of rectangles and the base of each rectangle is centered at a possible value for the sum of draws. The area of the rectangles equals a probability of getting the value. Empirical Histograms A kind of histogram which represents experimentally acquired probability from observed data. The areas of each rectangle represents empirical density. To get density one should divide the data into intervals, count the frequencies. The Relationship of the two: Empirical histograms converge to a probability of histogram Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 7/31 INDEX STATISTICS 1 Tossing a Coin and the Normal Distribution 2 Two Kinds of Histograms 3 The Relationship of Two Histograms 4 Central Limit Theorem Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 8/31 3. The Relationship of Two Histograms STATISTICS Empirical Histogram : Rolling a pair of dice and finding the total number of spots 0.20 0.15 0.10 (a) The first 100 repetitions 0.05 0.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0.20 0.15 0.10 (b) For 1,000 repetitions 0.05 0.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0.20 0.15 0.10 (c) For 10,000 repetitions 0.05 0.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 9/31 STATISTICS 3. The Relationship of Two Histograms Probability Histogram Probability Histogram of sum : the probability histogram can be found through tossing limitless times and getting the limit of empirical histograms In case of rolling a pair of dice we can calculate the ideal probability easily. Probability Histogram 0.20 0.15 0.10 0.05 0.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 10/31 INDEX STATISTICS 1 Tossing a Coin and the Normal Distribution 2 Two Kinds of Histograms 3 The Relationship of Two Histograms 4 Central Limit Theorem Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 11/31 4. Central Limit Theorem STATISTICS Cental Limit Theorem standard units -3 -2 -1 0 1 10 2 3 50 5 25 Percent per standard unit (%) percent per sum (%) Tossed 100 times Expected number of heads : 50 SE of the # of the0 heads : 5 0 35 40 45 50 # of heads 55 Very similar to the normal curve. Tossing a coin 100 times : probability histogram and the normal curve of getting heads Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 12/31 4. Central Limit Theorem STATISTICS Central Limit Theorem 9 8 7 6 5 4 3 2 1 0 45 40 35 30 25 20 15 10 5 0 Tossed 100 times 35 40 45 50 55 60 65 4.5 The histograms follow the normal curve better and better as the number of tosses goes up. 4 3.5 3 2.5 2 1.5 1 0.5 0 45 40 35 30 25 20 15 10 5 0 Tossed 400 times 170 180 190 200 210 220 230 Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 13/31 4. Central Limit Theorem STATISTICS Central Limit Theorem 3 45 40 35 30 25 20 15 10 5 0 Tossed 900 times 2.5 2 1.5 1 0.5 0 405 420 435 450 465 480 Tossed limitlessly? 495 The probability histogram will follow the normal curve if the number of observations goes up. We call it ‘Central Limit Theorem’. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 14/31 INDEX STATISTICS 5 The Normal Approximation 6 The Scope of the Normal Approximation 7 Conclusion Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 15/31 STATISTICS 5. The Normal Approximation Example 1 A coin will be tossed 100 times. Estimate the probability of getting a) exactly 50 heads b) between 45 and 55 heads inclusive c) between 45 and 55 heads exclusive Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 16/31 STATISTICS 3. The Relationship of Two Histograms Empirical Histogram and Probability Histogram The empirical histograms converge to the ideal probability histogram. Converge means “Gets closer and closer to”. Empirical Histogram Law of average n→∞ Probability Histogram Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 17/31 STATISTICS 5. The Normal Approximation Example 1 a) exactly 50 heads ☞ expected value is 50, SE is 5. 49.5 50 0.01 , 50 49.5 50 50.5 50.5 50 0.01 50 The base of the rectangle goes from 49.5 to 50.5 on the number-of-heads scale. Using the Normal Approximation 7.96% The exact probability is 7.96%. The approximation is excellent. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 18/31 STATISTICS 5. The Normal Approximation Example 1 b) between 45 and 55 heads inclusive c) between 45 and 55 heads exclusive 44.5 50 -1.1 0 55.5 45.5 1.1 -0.9 50 54.5 0 0.9 63.18 % 72.87 % Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 19/31 INDEX STATISTICS 5 The Normal Approximation 6 The Scope of the Normal Approximation 7 Conclusion Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 20/31 6. The Scope of the Normal Approximation STATISTICS The Scope of the Normal Approximation With regards to drawing from a box, the normal approximation works perfectly well, so long as you remember one thing. The more the histogram of the numbers in the box differs from the normal curve. The more draws are needed before the approximation takes hold. However, the speed of approximation differs according to the contents contained in the box. When we represent the contents through a histogram, the more similar to the normal curve the form is, the faster it follows the normal curve. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 21/31 6. The Scope of the Normal Approximation STATISTICS Probability Histogram of a box containg nine 0’s, and one 1 100 50 0 -2 -1 0 1 2 3 4 5 6 0 1 distribution of box 7 25 draws 1 4 7 10 13 16 19 100 draws 22 28 34 40 46 52 58 400 draws Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 22/31 6. The Scope of the Normal Approximation STATISTICS when the contents are symmetric 50 % 0 1 2 3 distribution of box 35 40 45 50 55 60 65 25 draws 80 85 90 95 100 105 110 115 120 50 draws Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 23/31 6. The Scope of the Normal Approximation STATISTICS when the contents are asymmetric 50 % -2.25 -1.12 0.00 1.12 2.25 3.37 25회 추출 0 1 25 2 3 4 5 6 7 distribution of box 8 Standard unit -3.51 50 0 40 50 60 70 80 90 100 110 120 130 140 150 160 sum Percent per standard unit (%) Percent per standard unit (%) Standard unit -3.37 50 -2.11 -0.70 0.70 2.11 3.51 100 draws 25 25 draws 100 draws 0 275 300 325 350 375 400 sum 425 450 475 500 525 Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 24/31 6. The Scope of the Normal Approximation STATISTICS The histogram for a product Central Limit Theorem does not apply to a product. ☞ The probability histogram for a product will usually be quite different from normal. Making the number of rolls larger does not make the histogram more normal. The histogram for a product does not convergo to the normal curve. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 25/31 6. The Scope of the Normal Approximation STATISTICS Making the magnitude of a sample larger does not make the probability histogram for a product more normal. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 26/31 INDEX STATISTICS 5 The Normal Approximation 6 The Scope of the Normal Approximation 7 Conclusion Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 27/31 STATISTICS 7.Conclusion Central Limit Theorem Central Limit Theorem When drawing at random with replacement from a box the probability histogram for the sum will follow the normal curve. Even if the contents of the box do not. But when we approximate a probability histogram to the normal curve, the number of draws needed to approximate changes. If the distribution of the box is similar to the normal curve, the number of draws needed is small, but otherwise the number should be larger. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 28/31 STATISTICS 7.Conclusion Expexted value and SE When the probability histogram does follow the normal curve, it can be summarized by the expexted value and SE. The expected value pins the center of the probability histogram to the horizontal axis, and the SE fixes its spread. • • • Expexted value and SE for a sum can be computed from average of box SD of box the number of draws. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 29/31 7.Conclusion STATISTICS Convergence of Histogram (1) Empirical Histogram → Probability Histogram Probability Histogram → Normal Distribution Explanation In textbook Sections 13.2-3 Sections 13,4-6 Examples Figure 13-1 Figures 13-3,5,6,8 Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 30/31 STATISTICS 7.Conclusion Convergence 히스토그램의 수렴 of Histogram (2) (2) Empirical Histogram → Probability Histogram Conditions for convergence When the number of repetition is infinitely large Probability Histogram → Normal Distribution When the number of draws . is infinitely large . Relevent rules Law of average Central Limit Theorem (law of large numbers) Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 31/31