Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics for Economist Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 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 STATISTICS 2/31 INDEX 1 Tossing a Coin and the Normal Distribution Two Kinds of Histograms 2 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 STATISTICS 3/31 1. Tossing a Coin and the Normal Distribution  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 If 100 tosses?

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 4/31  If we toss 100 times  Total # of ways  The # of ways with exactly 50 heads  mathematical probability for the # of patterns with exactly 50 heads # of ways with 50 heads total # of ways 1. Tossing a Coin and the Normal Distribution

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 5/31  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. 1. Tossing a Coin and the Normal Distribution

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 6/31 INDEX 1 Tossing a Coin and the Normal Distribution Two Kinds of Histograms 2 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 STATISTICS 7/31 2. Two Kinds of Histograms  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 Probability Histogram and Empirical Histogram

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 8/31 INDEX 1 Tossing a Coin and the Normal Distribution Two Kinds of Histograms 2 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 STATISTICS 9/31 Empirical Histogram : Rolling a pair of dice and finding the total number of spots (a) The first 100 repetitions (b) For 1,000 repetitions (c) For 10,000 repetitions 3. The Relationship of Two Histograms

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 10/31 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. 3. The Relationship of Two Histograms

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 11/31 INDEX 1 Tossing a Coin and the Normal Distribution Two Kinds of Histograms 2 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 STATISTICS 12/31 4. Central Limit Theorem Tossing a coin 100 times : probability histogram and the normal curve of getting heads Cental Limit Theorem Expected number of heads : 50 SE of the # of the heads : 5 Very similar to the normal curve # of heads percent per sum (%) standard units Percent per standard unit (%) Tossed 100 times

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 13/31 Central Limit Theorem Tossed 100 times Tossed 400 times The histograms follow the normal curve better and better as the number of tosses goes up. 4. Central Limit Theorem

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 14/31 Central Limit Theorem Tossed 900 times The probability histogram will follow the normal curve if the number ofobservations goes up. We call it ‘ Central Limit Theorem ’. Tossed limitlessly? 4. Central Limit Theorem

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 15/31 INDEX 5 The Normal Approximation The Scope of the Normal Approximation 6 7 Conclusion

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 16/31 5. The Normal Approximation  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 Example 1

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 17/31 3. The Relationship of Two Histograms  The empirical histograms converge to the ideal probability histogram. Converge means “ Gets closer and closer to ”. Empirical Histogram and Probability Histogram Empirical Histogram Probability Histogram n → ∞ Law of average

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 18/31 a) exactly 50 heads ☞ expected value is 50, SE is 5. Using the Normal Approximation The base of the rectangle goes from 49.5 to 50.5 on the number-of-heads scale. The exact probability is 7.96%. The approximation is excellent  7.96% Example 1 5. The Normal Approximation

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 19/31 b) between 45 and 55 heads inclusive c) between 45 and 55 heads exclusive Example 1 5. The Normal Approximation

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 20/31 INDEX 5 The Normal Approximation The Scope of the Normal Approximation 6 7 Conclusion

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 21/31 6. 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. The Scope of the Normal Approximation

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 22/31 Probability Histogram of a box containg nine 0 ’ s, and one 1 25 draws 100 draws400 draws distribution of box 6. The Scope of the Normal Approximation

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 23/31 when the contents are symmetric 25 draws 50 draws % 0 6. The Scope of the Normal Approximation distribution of box

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 24/31 when the contents are asymmetric 25 draws 100 draws % 0 6. The Scope of the Normal Approximation distribution of box sum Standard unit Percent per standard unit (%) 100 draws 회 추출 sum Percent per standard unit (%) Standard unit

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 25/31  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. The histogram for a product 6. The Scope of the Normal Approximation

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 26/31 Making the magnitude of a sample larger does not make the probability histogram for a product more normal. 6. The Scope of the Normal Approximation

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 27/31 INDEX 5 The Normal Approximation The Scope of the Normal Approximation 6 7 Conclusion

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 28/31 7.Conclusion  Central Limit Theorem When drawing at random with replacement from a box the p robability 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. Central Limit Theorem

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 29/31 Expexted value and SE  When the probability histogram does follow the normal curve, it can be summarized by the expexted value and SE.  Expexted value and SE for a sum can be computed from average of box SD of box the number of draws. The expected value pins the center of the probability histogram to the horizontal axis, and the SE fixes its spread. 7.Conclusion

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 30/31 Convergence of Histogram (1) 7.Conclusion Explanation In textbook Sections 13,4-6 Examples Figure 13-1 Figures 13-3,5,6,8 Empirical Histogram → Probability Histogram Probability Histogram → Normal Distribution Sections

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 31/31 히스토그램의 수렴 (2) 7.Conclusion Convergence of Histogram (2) Conditions for convergence When the number of repetition is infinitely large When the number of draws is infinitely large.. Relevent rules Law of average (law of large numbers) Central Limit Theorem Empirical Histogram → Probability Histogram Probability Histogram → Normal Distribution