Exploring Lognormal Incomes V1D 20 Nov, 2014 1D 1D 2014 NNN+ - PDF document

Exploring Lognormal Incomes V1D 20 Nov, 2014 1D 1D 2014 NNN+ 1 2014 NNN+ 2 Exploring Log-Normal Distributions Lognormal Incomes A Log-Normal distribution is generated from a normal with Milo Schield mu = Ln(Median) and sigma = Sqrt[2*Ln(Mean/Median)]. Augsburg College The lognormal is always positive and right-skewed. Editor: www.StatLit.org Examples: US Rep: International Statistical Literacy Project • Incomes (bottom 97%), assets, size of cities • Weight and blood pressure of humans (by gender) Benefit: www.StatLit.org/ • calculate the share of total income held by the top X% pdf/2014-Schield-Explore-LogNormal-Incomes-Slides.pdf • calculate share of total income held by the ‘above-average’ XLS/Create-LogNormal-Incomes-Excel2013.xlsx • explore effects of change in mean-median ratio. 1D 1D 2014 NNN+ 3 2014 NNN+ 4 Log-Normal Distributions Lognormal and Excel Use Excel to focus on the model and the results. “In many ways, it [the Log-Normal] has remained the Cinderella of distributions, the interest of writers in the Excel has two Log-Normal functions: learned journals being curiously sporadic and that of the  Standard: =LOGNORM.DIST(X, mu, sigma, k) authors of statistical test-books but faintly aroused.” k=0 for PDF; k=1 for CDF.  Inverse: =LOGNORM.INV(X, mu, sigma) “We … state our belief that the lognormal is as fundamental a Use Standard to calculate/graph the PDF and CDF. distribution in statistics as is the normal, despite the stigma of Use Inverse to find cutoffs: quartiles, to 1%, etc. the derivative nature of its name.” Use Excel to create graphs that show comparisons. Aitchison and Brown (1957). P 1. 1D 1D 2014 NNN+ 5 2014 NNN+ 6 Log-Normal Bibliography Distribution of Units . . Theoretical Distribution of Units by Income Mode: 20K 100% Cumulative Distribution Function (CDF): Percentage of Units with Incomes below price 75% 50% Units can be individuals, households or families 25% Probability Distribution Function (PDF): as a percentage of the Modal PDF 0% 0 50 100 150 200 250 300 350 400 450 500 Incomes ($1,000) LogNormal Dist of Units Median=50K; Mean=80K 2014-Schield-Explore-LogNormal-Incomes-Slides.pdf 1

Exploring Lognormal Incomes V1D 20 Nov, 2014 1D 1D 2014 NNN+ 7 2014 NNN+ 8 Distribution of Households Paired Distributions and Total Income by Income For anything that is distributed by X, there are Suppose the distribution of households by income always two distributions: is log-normal with normal parameters mu# and sigma#. 1. Distribution of subjects by X 2. Distribution of total X by X. Then the distribution of total income by amount has Sometime we ignore the 2 nd : height or weight. a log-normal distribution with these parameters: mu$ = mu# + sigma#^2; sigma$ = sigma#. Sometimes we care about the 2 nd : income or assets. Surprise: If the 1 st is lognormal, so is the 2 nd . See Aitchison and Brown (1963) p. 158. Special thanks to Mohammod Irfan (Denver University) for his help on this topic. 1D 1D 2014 NNN+ 9 2014 NNN+ 10 Distribution of Distribution of Total Income Households and Total Income Distribution of Total Income . Distribution of Households by Income; by Income per Household Distribution of Total Income by Amount Mode: 50K 100% 100% Distribution of Total Income by Median: Amount of Income Percentage of Maximum 128K 75% Mode: $50K 75% Cumulative Distribution Function (CDF): Median: $128K Percentage of Total Income below price Ave $205K 50% 50% Probability Distribution Function (PDF): 25% Households by Income 25% as a percentage of the Modal PDF Mode: $20K; Median: $50K Mean=$80K 0% 0% 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 Income ($1,000) Unit Incomes ($1,000) Log Normal Distribution of Households by Income Income/House: Mean=80K; Median=50K LogNormal Dist of Units by Income Median=50K; Mean=80K 1D 1D 2014 NNN+ 11 2014 NNN+ 12 Lorenz Curve and Champagne-Glass Gini Coefficient Distribution Pctg of Households vs. Pctg of Income Pctg of Income vs. Pctg. of Households . The Gini coefficient 100% 100% is determined by the Top 50% (above $50k): 83% of total Income Bottom ‐ Up Top 10% (above $175k: 38% of total Income 80% 80% Mean#/Median# ratio. Top 1% (above $475k): 8.7% of total Income Gini = 0.507 Percentage of Income Percentage of Households Top 0.1% (above $1M): 1.7% of total Income 60% 60% The bigger this ratio 40% 40% Gini Coefficient: the bigger the Gini Top 50% (above $50k) have 83% of total Income 0.507 Top 10% (above $175k) have 38% of total Income 20% Bigger means 20% Top 1% (above $475k) have 8.7% of total Income coefficient and the Top 0.1% (above $1M) have 1.7% of total Income more unequal 0% greater the economic 0% 0% 20% 40% 60% 80% 100% 0% 20% 40% 60% 80% 100% inequality. Percentage of Income Percentage of Households Log Normal Distribution of Households by Income Income/House: Mean=80K; Median=50K Log Normal Distribution of Households by Income Income/House: Mean=80K; Median=50K 2014-Schield-Explore-LogNormal-Incomes-Slides.pdf 2

Exploring Lognormal Incomes V1D 20 Nov, 2014 1D 1D 2014 NNN+ 13 2014 NNN+ 14 As Mean-Median Ratio ↑ Log-Normal Balance Conjecture Rich get Richer (relatively) Conjecture: If household (HH) income is distributed log-normally Log-normal distribution. Median HH income: $50K. and X% of households have below-average incomes, then Top 5% Top 1% X% of all income is earned by HH with above-average incomes. Mean# Min$ %Income Min$ %Income Gini Example: If 60% of HH have below-average incomes, then 60% 55 103 11% 138 2.9% 0.24 of total income is earned by HH having above-average incomes. 60 135 15% 204 4.2% 0.33 Evidence using Excel spreadsheet: 65 165 18% 270 5.5% 0.39 Suppose Mean# = 50K and Median# = 80K. 70 193 20% 337 6.6% 0.44 • 68.61%: Percentage of HH having below-average income 75 220 23% 406 7.7% 0.48 • 68.61%: Percentage of total income that is associated with 80 246 25% 477 8.7% 0.51 HH having above-average incomes. QED 85 272 27% 549 9.7% 0.53 90 298 29% 623 10.7% 0.56 1D 1D 2014 NNN+ 15 2014 NNN+ 16 Minimum Income Which parameters best model Minimum Income for Top 5% and top 1% versus Mean Income US household incomes? 900 Minimum Income ($,1000) 800 y = 5.4 x 700 US Median Income (Table 691*) 600 . • $46,089 in 1970; $50,303 in 2008 500 400 300 Share of Total Income by Top 5% (Table 693*) y = 2.93 x 200 • 16.6% in 1970; 21.5% in 2008 100 0 Best log-normal fits: 60 70 80 90 100 110 120 130 140 150 • 1970 Median 46K, Mean 53K: Ratio = 1.15 Mean Income ($,1000) • 2008 Median 50K, Mean 73K ; Ratio = 1.46 Log Normal Distribution of Households by Income Median Income: 50K * 2011 US Statistical Abstract (2008 dollars). 1D 1D 2014 NNN+ 17 2014 NNN+ 18 Conclusion Bibliography Aitchison J and JAC Brown (1957). The Log-normal Distribution . Cambridge Using the LogNormal distributions provides a (UK): Cambridge University Press. Searchable copy at Google Books: principled way students can explore a plausible http://books.google.com/books?id=Kus8AAAAIAAJ Cobham, Alex and Andy Sumner (2014). Is inequality all about the tails?: The distribution of incomes. Palma measure of income inequality. Significance. Volume 11 Issue 1. www.significancemagazine.org/details/magazine/5871201/Is-inequality- all-about-the-tails-The-Palma-measure-of-income-inequality.html Limpert, E., W.A. Stahel and M. Abbt (2001). Log-normal Distributions across Allows students to explore the difference between the Sciences: Keys and Clues. Bioscience 51, No 5, May 2001, 342-352. part and whole when using percentage grammar. Copy at http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf Schield, Milo (2013) Creating a Log-Normal Distribution using Excel 2013. www.statlit.org/pdf/Create-LogNormal-Excel2013-Demo-6up.pdf Stahel, Werner (2014). Website: http://stat.ethz.ch/~stahel Univ. Denver (2014). Using the LogNormal Distribution. Copy at http://www.du.edu/ifs/help/understand/economy/poverty/lognormal.html Wikipedia. LogNormal Distribution. 2014-Schield-Explore-LogNormal-Incomes-Slides.pdf 3

1D 2014 NNN+ 1 Exploring Lognormal Incomes Milo Schield Augsburg College Editor: www.StatLit.org US Rep: International Statistical Literacy Project www.StatLit.org/ pdf/2014-Schield-Explore-LogNormal-Incomes-Slides.pdf XLS/Create-LogNormal-Incomes-Excel2013.xlsx

1D 2014 NNN+ 2 Log-Normal Distributions A Log-Normal distribution is generated from a normal with mu = Ln(Median) and sigma = Sqrt[2*Ln(Mean/Median)]. The lognormal is always positive and right-skewed. Examples: • Incomes (bottom 97%), assets, size of cities • Weight and blood pressure of humans (by gender) Benefit: • calculate the share of total income held by the top X% • calculate share of total income held by the ‘above-average’ • explore effects of change in mean-median ratio.

1D 2014 NNN+ 3 Log-Normal Distributions “In many ways, it [the Log-Normal] has remained the Cinderella of distributions, the interest of writers in the learned journals being curiously sporadic and that of the authors of statistical test-books but faintly aroused.” “We … state our belief that the lognormal is as fundamental a distribution in statistics as is the normal, despite the stigma of the derivative nature of its name.” Aitchison and Brown (1957). P 1.