Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics for Economist 1.Unit Transformation 2.The Normal Distribution Curve 3.Areas under the Normal Curve 4.The Normal Approximation for Data 5.Percentiles 6.Interquartile and Box Plot 7.Percentiles and the Normal Curve Chap 4. The Normal Approximation

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 2/25 INDEX 1 Unit Transformation 2 The Normal Distribution Curve 3 Areas under the Normal Curve 4 The Normal Approximation for Data 5 Percentiles 6 Interquartile and Box Plot 7 Percentiles and the Normal Curve

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 3/25  A transformation which adds a constant or multiply by a constant to a measurement value.  Standardization is a kind of unit transformation which subtracts average and divide by SD. Unit Transformation 1. Unit Transformation

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 4/25 Standardization 1. Unit Transformation – 1 – Unit Transformation The shape of distribution is indifferent to the unit.

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 5/25 INDEX 1 Unit Transformation 2 The Normal Distribution Curve 3 Areas under the Normal Curve 4 The Normal Approximation for Data 5 Percentiles 6 Interquartile and Box Plot 7 Percentiles and the Normal Curve

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 6/25  The normal distribution is an ideal histogram. This is a mathematical model approximating the distribution of the real data and is a distribution of population. Normal Distribution Curve  is population mean and  is population standard deviation 2. The Normal Distribution Curve

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 7/25 the area under the standard normal curve between -1 and +1 : about 68% between -2 and +2 : about 95% between -3 and +3 : about 99.7% Standard normal distribution  Among normal distributions, satisfying E(X) =0, SD(X) =1 2. The Normal Distribution Curve

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 8/25 2. The Normal Distribution Curve A histogram for heights of women compared to the normal curve

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 9/25 INDEX 1 Unit Transformation 2 The Normal Distribution Curve 3 Areas under the Normal Curve 4 The Normal Approximation for Data 5 Percentiles 6 Interquartile and Box Plot 7 Percentiles and the Normal Curve

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 10/25 Finding areas under the normal curve (I) z Standard normal 0 z The area between 0 and 1 is 34.13%  Use the standard normal distribution table. 3. Areas under the Normal Curve

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 11/25 3. Areas under the Normal Curve  Find the area between -2 and 1 under the standard normal distribution curve. Finding areas under the normal curve(II) ☞ the area between -2 and 0 is the same as the area between 0 and 2, by symmetry ☞ the area of it is about 48% and the area between 0 and 1 is about 34% ☞ so the area between -2 and 1 is about 48%+34%=82% -2 =

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 12/25 INDEX 1 Unit Transformation 2 The Normal Distribution Curve 3 Areas under the Normal Curve 4 The Normal Approximation for Data 5 Percentiles 6 Interquartile and Box Plot 7 Percentiles and the Normal Curve

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 13/25  By the center and the spread around the center, the average and SD summarize a histogram which follows the normal curve. The Average and SD The average and SD are good summary for a histogram. 4. Normal Approximation for Data

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 14/ Normal Approximation for Data ☞ convert to standard units (from -2 to 1) ☞ find the area above the shaded standard-units interval ☞ about 82% (exactly 83.2%) 150.4cm 176cm167.5cm standardization Sketch in the standard normal curve Normal approximation Ex) What is the ratio of the men whose heights are between 150.4cm and 176cm ?

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 15/25 INDEX 1 Unit Transformation 2 The Normal Distribution Curve 3 Areas under the Normal Curve 4 The Normal Approximation for Data 5 Percentiles 6 Interquartile and Box Plot 7 Percentiles and the Normal Curve

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 16/ ,0004,0006,0008,00012,00010,000  Not all the histograms follow the normal curve. No one has negative incomes in the income histogram. But the normal approximation suggests that about 8% of the families had negative incomes. The histogram does not follow the normal curve at all well. Histogram for families by income 5. Percentiles

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 17/25 5. Percentiles  Percentiles are good summaries for such a histogram having a long tail. percentileincome($10/year) , , , , ,540 10% of families had incomes of $7,436 or less, and 90% were above. Percentiles for family income

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 18/25 INDEX 1 Unit Transformation 2 The Normal Distribution Curve 3 Areas under the Normal Curve 4 The Normal Approximation for Data 5 Percentiles 6 Interquartile and Box Plot 7 Percentiles and the Normal Curve

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 19/25  interquartile the 25 th, 50 th and 75 th percentiles are the 1 st, 2 nd and 3 rd quartile. Especially, the 50 th percentile is just the median and the 2 nd quartile.  Interquartile Range interquartile (interquartile range)=(3 rd quartile)-(1 st quartile) 6. Interquartile and box plot

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 20/25  Five number summary for data: minimum, the 1 st quartile, 2 nd quartile, 3 rd quartile, and the maximum  Box plot describes the five number summary  Two vertical lines of the box indicates the minimum and the maximum.  Three horizontal lines in the box indicates three quartiles.  Sometimes the 10% and 90% are used instead. 6. Interquartile and box plot

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 21/25 Box plot (family income) 6. Interquartile and box plot

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 22/ : KOSPI 2 : 삼성전자 3 : SK 텔레콤 4: 한국전력 5 : 포항제철 6 : 현대차 7 : 기아차 8 : 신한은행 9 : 삼성전기 10: 삼성증권 종목 Weekly profitability(%) Weekly profitability of firms in Interquartile and box plot

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 23/25 6. Interquartile and box plot Firm Weekly profitability(%) POSCO Samsung Comparison of Samsung Electronics and POSCO Samsung Electronics : from the minimum -26% to the maximum 37% The range of POSCO is smaller than that of Samsung

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 24/25 INDEX 1 Unit Transformation 2 The Normal Distribution Curve 3 Areas under the Normal Curve 4 The Normal Approximation for Data 5 Percentiles 6 Interquartile and Box Plot 7 Percentiles and the Normal Curve

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 25/25 7. Percentiles and Normal Curve  Ex) Estimate the score of the upper 5% at the midterm of statistics. ( E(X)= 27.93, Var(X) = ) Midterm score Standard unit Finding Percentiles ☞ when z =1.65, the size of [0,1.65] is 45% ☞ 1.65 * 8.52 = ☞ = 41.99