Mode is the value which occurs most frequency. The mode may not exist, and even if it does, it may not be unique.

Size: px
Start display at page:

Download "Mode is the value which occurs most frequency. The mode may not exist, and even if it does, it may not be unique."

Transcription

1 1.7.4 Mode Mode s the value whch occurs most frequency. The mode may not exst, and even f t does, t may not be unque. For ungrouped data, we smply count the largest frequency of the gven value. If all are of the same frequency, no mode exts. If more than one values have the same largest frequency, then the mode s not unque. Example 7 The value for the mode of the data n Example 5 s 15 (unmodal) Example 8 {,,, 4, 5, 6, 7, 7, 7} Mode = or 7 (Bmodal) For grouped data, the mode can be found by frst dentfy the largest frequency of that class, called modal class, then apply the followng formula on the modal class: mode = d L + ( L L ) 1 1 d1+ d 1 where: L 1 s the lower class boundary of the modal class; d 1 s the dfference of the frequences of the modal class wth the prevous class and s always postve; d s the dfference of the frequences of the modal class wth the followng class and s always postve; L s the upper class boundary of the modal class. Geometrcally the mode can be represented by the followng graph and can be obtaned by usng smlar trangle propertes. The formula can be derved by nterpolaton usng second degree polynomal. 19

2 Note that the mode s ndependent of extreme values and t may be appled n qualtatve data Concluson For symmetrcally dstrbuted data, the mean, medan and mode can be used almost nterchangeably. For moderately skewed dstrbuton data, ther relatonshp can be gven by Mean - Mode 3 (Mean - Medan) Physcally, mean can be nterpreted as the center of gravty of the dstrbuton. Medan dvdes the area of the dstrbuton nto two equal parts and mode s the hghest pont of the dstrbuton. 1.8 Dsperson and Skewness Sometmes mean, medan and mode may not be able to reflect the true pcture of some data. The followng example explans the reason. 0

3 Example 9 There were two companes, Company A and Company B. Ther salares profles gven n mean, medan and mode were as follow: Company A Company B Mean $30,000 $30,000 Medan $30,000 $30,000 Mode (Nl) (Nl) However, ther detal salary ($) structures could be completely dfferent as that: Company A 5,000 15,000 5,000 35,000 45,000 55,000 Company B 5,000 5,000 5,000 55,000 55,000 55,000 Hence t s necessary to have some measures on how data are scattered. That s, we want to know what s the dsperson, or varablty n a set of data Range Range s the dfference between two extreme values. The range s easy to calculate but can not be obtaned f open ended grouped data are gven Decles, Percentle, and Fractle Decle dvdes the dstrbuton nto ten equal parts whle percentle dvdes the dstrbuton nto one hundred equal parts. There are nne decles such that 10% of the data are D 1 ; 0% of the data are D ; and so on. There are 99 percentles such that 1% of the data are P 1 ; % of the data are P ; and so on. Fractle, even more flexble, dvdes the dstrbuton nto a convenence number of parts Quartles Quartles are the most commonly used values of poston whch dvdes dstrbuton nto four equal parts such that 5% of the data are Q 1 ; 50% of the data are Q ; 75% of the data are Q 3. The frst quarter s conventonally denoted as Q 1, whle the second and thrd quarters grouped together s Q and the last quarter s Q 3. Note that Q ncludes the medan, contans half of the frequency and excludes extreme values. It s also denoted the value (Q 3 - Q 1 ) / as the Quartle Devaton, Q D, or the semnterquartle range. 1

4 1.8.4 Mean Absolute Devaton Mean absolute devaton s the mean of the absolute values of all devatons from the mean. Therefore t takes every tem nto account. Mathematcally t s gven as: f x µ f where: f s the frequency of the th tem; x s the value of the th tem or class mark; µ s the arthmetc mean Varance and Standard Devaton The varance and standard devaton are two very popular measures of varaton. Ther formulatons are categorzed nto whether to evaluate from a populaton or from a sample. The populaton varance, σ, s the mean of the square of all devatons from the mean. Mathematcally t s gven as: ( x - ) f µ f where: f s the frequency of the th tem; x s the value of the th tem or class mark; µ s the populaton arthmetc mean. The populaton standard devaton σ s defned as σ = σ. The sample varance, denoted as s gves: f ( x x) ( f ) 1 where: f s the frequency of the th tem; x s the value of the th tem or class mark; x s the sample arthmetc mean. The sample standard devaton, s, s defned as s = For ungrouped data, s. ( x x) x ( x) / Σ Σ Σ s = = n 1 n 1 n

5 For grouped data, ( Σfx) Σf Σf( x x) Σfx s = = Σf 1 Σf 1 where f = n Note that when calculatng the sample varance, we have to subtract 1 from the total frequency whch appears n the denomnator. Although when the total frequency s large, s σ, the subtracton of 1 s very mportant. Example 10 Measures of Grouped Data (Refers to the followngs Data Set) Gas Frequency Class Class fx fx Consumpton ( f ) boundary mark ( x ) x x f =, n = f n = 100 =

6 medan = = Q1 = Q = mode = (0 19) + (0 17) = 8 4. sample s.d., n( x f ) ( x f s = n( n 1) ) = = (671705) (7970) 100(100 1) Coeffcent of Varaton The coeffcent of varaton s a measure of relatve mportance. It does not depend on unt and can be used to make comparson even two samples dffer n means or relate to dfferent types of measurements. The coeffcent of varaton gves: Standard Devaton Mean 100% Example 11 x S Salesman salary $916.76/month $86.70 Clercal salary $98.50/week $0.55 4

7 86.70 CVs = 100% = 31% CVc = 100% = 1% Skewness The skewness s an abstract quantty whch shows how data pled-up. A number of measures have been suggested to determne the skewness of a gven dstrbuton. One of the smplest one s known as Pearson s measure of skewness: Skewness = Mean Mode Standard Devaton 3 (Mean Medan) Standard Devaton If the longer tal s on the rght, we say that t s skewed to the rght, and the coeffcent of skewness s postve. Skewed to the rght (postvely skewed) 5

8 If the longer tal s on the left, we say that s skewed to the left and the coeffcent of skewness s negatve. Skewed to the left (negatvely skewed) Example 1 We are gong to use Example 9 to evaluate the dfferent measurements of varaton. As stated above, the salary ($) scales of the two companes are: Company A: 5,000 15,000 5,000 35,000 45,000 55,000 Company B: 5,000 5,000 5,000 55,000 55,000 55,000 Range Company A: $55,000 - $5,000 = $50,000 Company B: $55,000 - $5,000 = $50,000 6

9 Mean absolute devaton Company A: $ ( 5,000-30, ,000-30, ,000-30, ,000-30, ,000-30, ,000-30,000 ) / 6 = $15,000 Company B: $ ( 5,000-30, ,000-30, ,000-30, ,000-30, ,000-30, ,000-30,000 ) / 6 = $5,000 Varance Company A: {(5,000-30,000) + (15,000-30,000) + (5,000-30,000) + (35,000-30,000) + (45,000-30,000) + (55,000-30,000) } / 6 = 91,666,667 (dollar square) Company B: {(5,000-30,000) + (5,000-30,000) + (5,000-30,000) + (55,000 - Standard devaton 30,000) + (55,000-30,000) + (55,000-30,000) } / 6 = 65,000,000 (dollar square) Company A: $ 91,666,667 = $17,078 Company B: $ 65,000,000 = $5,000 Coeffcent of varaton Company A: $17,078 / $30, % = 56.93% Company B: $5,000 / $30, % = 83.33% 7

10 Coeffcent of Skewness Pearson s 1 st coeffcent of skewness, SK 1 Mean Mode = Standard devaton Pearson s nd coeffcent of skewness SK 3(Mean Medan) = Standard devaton Chebyshev s Theorem For any set of data, the proporton of data that les between the mean plus and mnus k 1 standard devatons s at least 1 k 1.e. Pr( µ kσ x µ + kσ ) 1 k Symbols Populaton Sample Sze N n Mean µ x Standard devaton σ s Varance σ s 8

Chapter 3 Student Lecture Notes 3-1

Chapter 3 Student Lecture Notes 3-1 Chapter 3 Student Lecture otes 3-1 Busness Statstcs: A Decson-Makng Approach 6 th Edton Chapter 3 Descrbng Data Usng umercal Measures 005 Prentce-Hall, Inc. Chap 3-1 Chapter Goals After completng ths chapter,

More information

Measures of Spread IQR and Deviation. For exam X, calculate the mean, median and mode. For exam Y, calculate the mean, median and mode.

Measures of Spread IQR and Deviation. For exam X, calculate the mean, median and mode. For exam Y, calculate the mean, median and mode. Part 4 Measures of Spread IQR and Devaton In Part we learned how the three measures of center offer dfferent ways of provdng us wth a sngle representatve value for a data set. However, consder the followng

More information

Chapter 3 Descriptive Statistics: Numerical Measures Part B

Chapter 3 Descriptive Statistics: Numerical Measures Part B Sldes Prepared by JOHN S. LOUCKS St. Edward s Unversty Slde 1 Chapter 3 Descrptve Statstcs: Numercal Measures Part B Measures of Dstrbuton Shape, Relatve Locaton, and Detectng Outlers Eploratory Data Analyss

More information

UNIVERSITY OF VICTORIA Midterm June 6, 2018 Solutions

UNIVERSITY OF VICTORIA Midterm June 6, 2018 Solutions UIVERSITY OF VICTORIA Mdterm June 6, 08 Solutons Econ 45 Summer A0 08 age AME: STUDET UMBER: V00 Course ame & o. Descrptve Statstcs and robablty Economcs 45 Secton(s) A0 CR: 3067 Instructor: Betty Johnson

More information

MgtOp 215 Chapter 13 Dr. Ahn

MgtOp 215 Chapter 13 Dr. Ahn MgtOp 5 Chapter 3 Dr Ahn Consder two random varables X and Y wth,,, In order to study the relatonshp between the two random varables, we need a numercal measure that descrbes the relatonshp The covarance

More information

II. Random Variables. Variable Types. Variables Map Outcomes to Numbers

II. Random Variables. Variable Types. Variables Map Outcomes to Numbers II. Random Varables Random varables operate n much the same way as the outcomes or events n some arbtrary sample space the dstncton s that random varables are smply outcomes that are represented numercally.

More information

Tests for Two Correlations

Tests for Two Correlations PASS Sample Sze Software Chapter 805 Tests for Two Correlatons Introducton The correlaton coeffcent (or correlaton), ρ, s a popular parameter for descrbng the strength of the assocaton between two varables.

More information

02_EBA2eSolutionsChapter2.pdf 02_EBA2e Case Soln Chapter2.pdf

02_EBA2eSolutionsChapter2.pdf 02_EBA2e Case Soln Chapter2.pdf 0_EBAeSolutonsChapter.pdf 0_EBAe Case Soln Chapter.pdf Chapter Solutons: 1. a. Quanttatve b. Categorcal c. Categorcal d. Quanttatve e. Categorcal. a. The top 10 countres accordng to GDP are lsted below.

More information

OCR Statistics 1 Working with data. Section 2: Measures of location

OCR Statistics 1 Working with data. Section 2: Measures of location OCR Statstcs 1 Workng wth data Secton 2: Measures of locaton Notes and Examples These notes have sub-sectons on: The medan Estmatng the medan from grouped data The mean Estmatng the mean from grouped data

More information

Evaluating Performance

Evaluating Performance 5 Chapter Evaluatng Performance In Ths Chapter Dollar-Weghted Rate of Return Tme-Weghted Rate of Return Income Rate of Return Prncpal Rate of Return Daly Returns MPT Statstcs 5- Measurng Rates of Return

More information

Random Variables. b 2.

Random Variables. b 2. Random Varables Generally the object of an nvestgators nterest s not necessarly the acton n the sample space but rather some functon of t. Techncally a real valued functon or mappng whose doman s the sample

More information

Probability Distributions. Statistics and Quantitative Analysis U4320. Probability Distributions(cont.) Probability

Probability Distributions. Statistics and Quantitative Analysis U4320. Probability Distributions(cont.) Probability Statstcs and Quanttatve Analss U430 Dstrbutons A. Dstrbutons: How do smple probablt tables relate to dstrbutons?. What s the of gettng a head? ( con toss) Prob. Segment 4: Dstrbutons, Unvarate & Bvarate

More information

Which of the following provides the most reasonable approximation to the least squares regression line? (a) y=50+10x (b) Y=50+x (d) Y=1+50x

Which of the following provides the most reasonable approximation to the least squares regression line? (a) y=50+10x (b) Y=50+x (d) Y=1+50x Whch of the followng provdes the most reasonable approxmaton to the least squares regresson lne? (a) y=50+10x (b) Y=50+x (c) Y=10+50x (d) Y=1+50x (e) Y=10+x In smple lnear regresson the model that s begn

More information

Linear Combinations of Random Variables and Sampling (100 points)

Linear Combinations of Random Variables and Sampling (100 points) Economcs 30330: Statstcs for Economcs Problem Set 6 Unversty of Notre Dame Instructor: Julo Garín Sprng 2012 Lnear Combnatons of Random Varables and Samplng 100 ponts 1. Four-part problem. Go get some

More information

occurrence of a larger storm than our culvert or bridge is barely capable of handling? (what is The main question is: What is the possibility of

occurrence of a larger storm than our culvert or bridge is barely capable of handling? (what is The main question is: What is the possibility of Module 8: Probablty and Statstcal Methods n Water Resources Engneerng Bob Ptt Unversty of Alabama Tuscaloosa, AL Flow data are avalable from numerous USGS operated flow recordng statons. Data s usually

More information

3: Central Limit Theorem, Systematic Errors

3: Central Limit Theorem, Systematic Errors 3: Central Lmt Theorem, Systematc Errors 1 Errors 1.1 Central Lmt Theorem Ths theorem s of prme mportance when measurng physcal quanttes because usually the mperfectons n the measurements are due to several

More information

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com PhscsAndMathsTutor.com phscsandmathstutor.com June 2005 6. A scentst found that the tme taken, M mnutes, to carr out an eperment can be modelled b a normal random varable wth mean 155 mnutes and standard

More information

CHAPTER 9 FUNCTIONAL FORMS OF REGRESSION MODELS

CHAPTER 9 FUNCTIONAL FORMS OF REGRESSION MODELS CHAPTER 9 FUNCTIONAL FORMS OF REGRESSION MODELS QUESTIONS 9.1. (a) In a log-log model the dependent and all explanatory varables are n the logarthmc form. (b) In the log-ln model the dependent varable

More information

Elton, Gruber, Brown, and Goetzmann. Modern Portfolio Theory and Investment Analysis, 7th Edition. Solutions to Text Problems: Chapter 9

Elton, Gruber, Brown, and Goetzmann. Modern Portfolio Theory and Investment Analysis, 7th Edition. Solutions to Text Problems: Chapter 9 Elton, Gruber, Brown, and Goetzmann Modern Portfolo Theory and Investment Analyss, 7th Edton Solutons to Text Problems: Chapter 9 Chapter 9: Problem In the table below, gven that the rskless rate equals

More information

The Institute of Chartered Accountants of Sri Lanka

The Institute of Chartered Accountants of Sri Lanka The Insttute of Chartered Accountants of Sr Lanka Postgraduate Dploma n Accountng, Busness and Strategy Quanttatve Methods for Busness Studes Handout 0: Presentaton and Analyss of data Tables and Charts

More information

>1 indicates country i has a comparative advantage in production of j; the greater the index, the stronger the advantage. RCA 1 ij

>1 indicates country i has a comparative advantage in production of j; the greater the index, the stronger the advantage. RCA 1 ij 69 APPENDIX 1 RCA Indces In the followng we present some maor RCA ndces reported n the lterature. For addtonal varants and other RCA ndces, Memedovc (1994) and Vollrath (1991) provde more thorough revews.

More information

Chapter 5 Student Lecture Notes 5-1

Chapter 5 Student Lecture Notes 5-1 Chapter 5 Student Lecture Notes 5-1 Basc Busness Statstcs (9 th Edton) Chapter 5 Some Important Dscrete Probablty Dstrbutons 004 Prentce-Hall, Inc. Chap 5-1 Chapter Topcs The Probablty Dstrbuton of a Dscrete

More information

Capability Analysis. Chapter 255. Introduction. Capability Analysis

Capability Analysis. Chapter 255. Introduction. Capability Analysis Chapter 55 Introducton Ths procedure summarzes the performance of a process based on user-specfed specfcaton lmts. The observed performance as well as the performance relatve to the Normal dstrbuton are

More information

Survey of Math Test #3 Practice Questions Page 1 of 5

Survey of Math Test #3 Practice Questions Page 1 of 5 Test #3 Practce Questons Page 1 of 5 You wll be able to use a calculator, and wll have to use one to answer some questons. Informaton Provded on Test: Smple Interest: Compound Interest: Deprecaton: A =

More information

Introduction. Why One-Pass Statistics?

Introduction. Why One-Pass Statistics? BERKELE RESEARCH GROUP Ths manuscrpt s program documentaton for three ways to calculate the mean, varance, skewness, kurtoss, covarance, correlaton, regresson parameters and other regresson statstcs. Although

More information

Tests for Two Ordered Categorical Variables

Tests for Two Ordered Categorical Variables Chapter 253 Tests for Two Ordered Categorcal Varables Introducton Ths module computes power and sample sze for tests of ordered categorcal data such as Lkert scale data. Assumng proportonal odds, such

More information

Creating a zero coupon curve by bootstrapping with cubic splines.

Creating a zero coupon curve by bootstrapping with cubic splines. MMA 708 Analytcal Fnance II Creatng a zero coupon curve by bootstrappng wth cubc splnes. erg Gryshkevych Professor: Jan R. M. Röman 0.2.200 Dvson of Appled Mathematcs chool of Educaton, Culture and Communcaton

More information

Analysis of Variance and Design of Experiments-II

Analysis of Variance and Design of Experiments-II Analyss of Varance and Desgn of Experments-II MODULE VI LECTURE - 4 SPLIT-PLOT AND STRIP-PLOT DESIGNS Dr. Shalabh Department of Mathematcs & Statstcs Indan Insttute of Technology Kanpur An example to motvate

More information

15-451/651: Design & Analysis of Algorithms January 22, 2019 Lecture #3: Amortized Analysis last changed: January 18, 2019

15-451/651: Design & Analysis of Algorithms January 22, 2019 Lecture #3: Amortized Analysis last changed: January 18, 2019 5-45/65: Desgn & Analyss of Algorthms January, 09 Lecture #3: Amortzed Analyss last changed: January 8, 09 Introducton In ths lecture we dscuss a useful form of analyss, called amortzed analyss, for problems

More information

Risk and Return: The Security Markets Line

Risk and Return: The Security Markets Line FIN 614 Rsk and Return 3: Markets Professor Robert B.H. Hauswald Kogod School of Busness, AU 1/25/2011 Rsk and Return: Markets Robert B.H. Hauswald 1 Rsk and Return: The Securty Markets Lne From securtes

More information

TCOM501 Networking: Theory & Fundamentals Final Examination Professor Yannis A. Korilis April 26, 2002

TCOM501 Networking: Theory & Fundamentals Final Examination Professor Yannis A. Korilis April 26, 2002 TO5 Networng: Theory & undamentals nal xamnaton Professor Yanns. orls prl, Problem [ ponts]: onsder a rng networ wth nodes,,,. In ths networ, a customer that completes servce at node exts the networ wth

More information

Sampling Distributions of OLS Estimators of β 0 and β 1. Monte Carlo Simulations

Sampling Distributions of OLS Estimators of β 0 and β 1. Monte Carlo Simulations Addendum to NOTE 4 Samplng Dstrbutons of OLS Estmators of β and β Monte Carlo Smulatons The True Model: s gven by the populaton regresson equaton (PRE) Y = β + β X + u = 7. +.9X + u () where β = 7. and

More information

Copyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Dr. Wayne A. Taylor

Copyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Dr. Wayne A. Taylor Taylor Enterprses, Inc. ormalzed Indvduals (I ) Chart Copyrght 07 by Taylor Enterprses, Inc., All Rghts Reserved. ormalzed Indvduals (I) Control Chart Dr. Wayne A. Taylor Abstract: The only commonly used

More information

Spatial Variations in Covariates on Marriage and Marital Fertility: Geographically Weighted Regression Analyses in Japan

Spatial Variations in Covariates on Marriage and Marital Fertility: Geographically Weighted Regression Analyses in Japan Spatal Varatons n Covarates on Marrage and Martal Fertlty: Geographcally Weghted Regresson Analyses n Japan Kenj Kamata (Natonal Insttute of Populaton and Socal Securty Research) Abstract (134) To understand

More information

4. Greek Letters, Value-at-Risk

4. Greek Letters, Value-at-Risk 4 Greek Letters, Value-at-Rsk 4 Value-at-Rsk (Hull s, Chapter 8) Math443 W08, HM Zhu Outlne (Hull, Chap 8) What s Value at Rsk (VaR)? Hstorcal smulatons Monte Carlo smulatons Model based approach Varance-covarance

More information

Notes on experimental uncertainties and their propagation

Notes on experimental uncertainties and their propagation Ed Eyler 003 otes on epermental uncertantes and ther propagaton These notes are not ntended as a complete set of lecture notes, but nstead as an enumeraton of some of the key statstcal deas needed to obtan

More information

The Integration of the Israel Labour Force Survey with the National Insurance File

The Integration of the Israel Labour Force Survey with the National Insurance File The Integraton of the Israel Labour Force Survey wth the Natonal Insurance Fle Natale SHLOMO Central Bureau of Statstcs Kanfey Nesharm St. 66, corner of Bach Street, Jerusalem Natales@cbs.gov.l Abstact:

More information

Economic Design of Short-Run CSP-1 Plan Under Linear Inspection Cost

Economic Design of Short-Run CSP-1 Plan Under Linear Inspection Cost Tamkang Journal of Scence and Engneerng, Vol. 9, No 1, pp. 19 23 (2006) 19 Economc Desgn of Short-Run CSP-1 Plan Under Lnear Inspecton Cost Chung-Ho Chen 1 * and Chao-Yu Chou 2 1 Department of Industral

More information

Multifactor Term Structure Models

Multifactor Term Structure Models 1 Multfactor Term Structure Models A. Lmtatons of One-Factor Models 1. Returns on bonds of all maturtes are perfectly correlated. 2. Term structure (and prces of every other dervatves) are unquely determned

More information

ECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE)

ECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE) ECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE) May 17, 2016 15:30 Frst famly name: Name: DNI/ID: Moble: Second famly Name: GECO/GADE: Instructor: E-mal: Queston 1 A B C Blank Queston 2 A B C Blank Queston

More information

Standardization. Stan Becker, PhD Bloomberg School of Public Health

Standardization. Stan Becker, PhD Bloomberg School of Public Health Ths work s lcensed under a Creatve Commons Attrbuton-NonCommercal-ShareAlke Lcense. Your use of ths materal consttutes acceptance of that lcense and the condtons of use of materals on ths ste. Copyrght

More information

Appendix - Normally Distributed Admissible Choices are Optimal

Appendix - Normally Distributed Admissible Choices are Optimal Appendx - Normally Dstrbuted Admssble Choces are Optmal James N. Bodurtha, Jr. McDonough School of Busness Georgetown Unversty and Q Shen Stafford Partners Aprl 994 latest revson September 00 Abstract

More information

Appendix for Solving Asset Pricing Models when the Price-Dividend Function is Analytic

Appendix for Solving Asset Pricing Models when the Price-Dividend Function is Analytic Appendx for Solvng Asset Prcng Models when the Prce-Dvdend Functon s Analytc Ovdu L. Caln Yu Chen Thomas F. Cosmano and Alex A. Hmonas January 3, 5 Ths appendx provdes proofs of some results stated n our

More information

Hewlett Packard 10BII Calculator

Hewlett Packard 10BII Calculator Hewlett Packard 0BII Calculator Keystrokes for the HP 0BII are shown n the tet. However, takng a mnute to revew the Quk Start secton, below, wll be very helpful n gettng started wth your calculator. Note:

More information

Physics 4A. Error Analysis or Experimental Uncertainty. Error

Physics 4A. Error Analysis or Experimental Uncertainty. Error Physcs 4A Error Analyss or Expermental Uncertanty Slde Slde 2 Slde 3 Slde 4 Slde 5 Slde 6 Slde 7 Slde 8 Slde 9 Slde 0 Slde Slde 2 Slde 3 Slde 4 Slde 5 Slde 6 Slde 7 Slde 8 Slde 9 Slde 20 Slde 2 Error n

More information

/ Computational Genomics. Normalization

/ Computational Genomics. Normalization 0-80 /02-70 Computatonal Genomcs Normalzaton Gene Expresson Analyss Model Computatonal nformaton fuson Bologcal regulatory networks Pattern Recognton Data Analyss clusterng, classfcaton normalzaton, mss.

More information

Principles of Finance

Principles of Finance Prncples of Fnance Grzegorz Trojanowsk Lecture 6: Captal Asset Prcng Model Prncples of Fnance - Lecture 6 1 Lecture 6 materal Requred readng: Elton et al., Chapters 13, 14, and 15 Supplementary readng:

More information

Simple Regression Theory II 2010 Samuel L. Baker

Simple Regression Theory II 2010 Samuel L. Baker SIMPLE REGRESSIO THEORY II Smple Regresson Theory II 00 Samuel L. Baker Assessng how good the regresson equaton s lkely to be Assgnment A gets nto drawng nferences about how close the regresson lne mght

More information

ECE 586GT: Problem Set 2: Problems and Solutions Uniqueness of Nash equilibria, zero sum games, evolutionary dynamics

ECE 586GT: Problem Set 2: Problems and Solutions Uniqueness of Nash equilibria, zero sum games, evolutionary dynamics Unversty of Illnos Fall 08 ECE 586GT: Problem Set : Problems and Solutons Unqueness of Nash equlbra, zero sum games, evolutonary dynamcs Due: Tuesday, Sept. 5, at begnnng of class Readng: Course notes,

More information

Random Variables. 8.1 What is a Random Variable? Announcements: Chapter 8

Random Variables. 8.1 What is a Random Variable? Announcements: Chapter 8 Announcements: Quz starts after class today, ends Monday Last chance to take probablty survey ends Sunday mornng. Next few lectures: Today, Sectons 8.1 to 8. Monday, Secton 7.7 and extra materal Wed, Secton

More information

Applications of Myerson s Lemma

Applications of Myerson s Lemma Applcatons of Myerson s Lemma Professor Greenwald 28-2-7 We apply Myerson s lemma to solve the sngle-good aucton, and the generalzaton n whch there are k dentcal copes of the good. Our objectve s welfare

More information

OPERATIONS RESEARCH. Game Theory

OPERATIONS RESEARCH. Game Theory OPERATIONS RESEARCH Chapter 2 Game Theory Prof. Bbhas C. Gr Department of Mathematcs Jadavpur Unversty Kolkata, Inda Emal: bcgr.umath@gmal.com 1.0 Introducton Game theory was developed for decson makng

More information

- contrast so-called first-best outcome of Lindahl equilibrium with case of private provision through voluntary contributions of households

- contrast so-called first-best outcome of Lindahl equilibrium with case of private provision through voluntary contributions of households Prvate Provson - contrast so-called frst-best outcome of Lndahl equlbrum wth case of prvate provson through voluntary contrbutons of households - need to make an assumpton about how each household expects

More information

Elements of Economic Analysis II Lecture VI: Industry Supply

Elements of Economic Analysis II Lecture VI: Industry Supply Elements of Economc Analyss II Lecture VI: Industry Supply Ka Hao Yang 10/12/2017 In the prevous lecture, we analyzed the frm s supply decson usng a set of smple graphcal analyses. In fact, the dscusson

More information

EDC Introduction

EDC Introduction .0 Introducton EDC3 In the last set of notes (EDC), we saw how to use penalty factors n solvng the EDC problem wth losses. In ths set of notes, we want to address two closely related ssues. What are, exactly,

More information

Real Exchange Rate Fluctuations, Wage Stickiness and Markup Adjustments

Real Exchange Rate Fluctuations, Wage Stickiness and Markup Adjustments Real Exchange Rate Fluctuatons, Wage Stckness and Markup Adjustments Yothn Jnjarak and Kanda Nakno Nanyang Technologcal Unversty and Purdue Unversty January 2009 Abstract Motvated by emprcal evdence on

More information

Calibration Methods: Regression & Correlation. Calibration Methods: Regression & Correlation

Calibration Methods: Regression & Correlation. Calibration Methods: Regression & Correlation Calbraton Methods: Regresson & Correlaton Calbraton A seres of standards run (n replcate fashon) over a gven concentraton range. Standards Comprsed of analte(s) of nterest n a gven matr composton. Matr

More information

Available online: 20 Dec 2011

Available online: 20 Dec 2011 Ths artcle was downloaded by: [UVA Unverstetsbblotheek SZ] On: 16 May 212, At: 6:32 Publsher: Taylor & Francs Informa Ltd Regstered n England and Wales Regstered Number: 172954 Regstered offce: Mortmer

More information

AS MATHEMATICS HOMEWORK S1

AS MATHEMATICS HOMEWORK S1 Name Teacher AS MATHEMATICS HOMEWORK S1 Mathematcs Department September 015 Verson 1.0 Contents Contents... AS Maths Homework S1 014... 3 HW1 Data1 dscrete data, bo plots, stem and leaf dagrams... 4 HW

More information

Technological inefficiency and the skewness of the error component in stochastic frontier analysis

Technological inefficiency and the skewness of the error component in stochastic frontier analysis Economcs Letters 77 (00) 101 107 www.elsever.com/ locate/ econbase Technologcal neffcency and the skewness of the error component n stochastc fronter analyss Martn A. Carree a,b, * a Erasmus Unversty Rotterdam,

More information

Microeconomics: BSc Year One Extending Choice Theory

Microeconomics: BSc Year One Extending Choice Theory mcroeconomcs notes from http://www.economc-truth.co.uk by Tm Mller Mcroeconomcs: BSc Year One Extendng Choce Theory Consumers, obvously, mostly have a choce of more than two goods; and to fnd the favourable

More information

Midterm Exam. Use the end of month price data for the S&P 500 index in the table below to answer the following questions.

Midterm Exam. Use the end of month price data for the S&P 500 index in the table below to answer the following questions. Unversty of Washngton Summer 2001 Department of Economcs Erc Zvot Economcs 483 Mdterm Exam Ths s a closed book and closed note exam. However, you are allowed one page of handwrtten notes. Answer all questons

More information

A Bootstrap Confidence Limit for Process Capability Indices

A Bootstrap Confidence Limit for Process Capability Indices A ootstrap Confdence Lmt for Process Capablty Indces YANG Janfeng School of usness, Zhengzhou Unversty, P.R.Chna, 450001 Abstract The process capablty ndces are wdely used by qualty professonals as an

More information

Elton, Gruber, Brown and Goetzmann. Modern Portfolio Theory and Investment Analysis, 7th Edition. Solutions to Text Problems: Chapter 4

Elton, Gruber, Brown and Goetzmann. Modern Portfolio Theory and Investment Analysis, 7th Edition. Solutions to Text Problems: Chapter 4 Elton, Gruber, Brown and Goetzmann Modern ortfolo Theory and Investment Analyss, 7th Edton Solutons to Text roblems: Chapter 4 Chapter 4: roblem 1 A. Expected return s the sum of each outcome tmes ts assocated

More information

Introduction. Chapter 7 - An Introduction to Portfolio Management

Introduction. Chapter 7 - An Introduction to Portfolio Management Introducton In the next three chapters, we wll examne dfferent aspects of captal market theory, ncludng: Brngng rsk and return nto the pcture of nvestment management Markowtz optmzaton Modelng rsk and

More information

COMPARISON OF THE ANALYTICAL AND NUMERICAL SOLUTION OF A ONE-DIMENSIONAL NON-STATIONARY COOLING PROBLEM. László Könözsy 1, Mátyás Benke 2

COMPARISON OF THE ANALYTICAL AND NUMERICAL SOLUTION OF A ONE-DIMENSIONAL NON-STATIONARY COOLING PROBLEM. László Könözsy 1, Mátyás Benke 2 COMPARISON OF THE ANALYTICAL AND NUMERICAL SOLUTION OF A ONE-DIMENSIONAL NON-STATIONARY COOLING PROBLEM László Könözsy 1, Mátyás Benke Ph.D. Student 1, Unversty Student Unversty of Mskolc, Department of

More information

Using Conditional Heteroskedastic

Using Conditional Heteroskedastic ITRON S FORECASTING BROWN BAG SEMINAR Usng Condtonal Heteroskedastc Varance Models n Load Research Sample Desgn Dr. J. Stuart McMenamn March 6, 2012 Please Remember» Phones are Muted: In order to help

More information

Measures of Dispersion (Range, standard deviation, standard error) Introduction

Measures of Dispersion (Range, standard deviation, standard error) Introduction Measures of Dispersion (Range, standard deviation, standard error) Introduction We have already learnt that frequency distribution table gives a rough idea of the distribution of the variables in a sample

More information

Evaluation of the Factors Affecting Initial Public offering Underpricing by Newly-accepted Companies into Tehran Stock Exchange

Evaluation of the Factors Affecting Initial Public offering Underpricing by Newly-accepted Companies into Tehran Stock Exchange Internatonal Research Journal of Appled and Basc Scences 4 Avalable onlne at www.rjabs.com ISSN 5-8X / Vol, 8 (7): 873-88 Scence Explorer Publcatons Evaluaton of the Factors Affectng Intal Publc offerng

More information

An Application of Alternative Weighting Matrix Collapsing Approaches for Improving Sample Estimates

An Application of Alternative Weighting Matrix Collapsing Approaches for Improving Sample Estimates Secton on Survey Research Methods An Applcaton of Alternatve Weghtng Matrx Collapsng Approaches for Improvng Sample Estmates Lnda Tompkns 1, Jay J. Km 2 1 Centers for Dsease Control and Preventon, atonal

More information

Midterm Version 2 Solutions

Midterm Version 2 Solutions Econ 45 Fall 07 age UIVERSITY OF VICTORIA Mdterm Verson Solutons October 07 AME: STUDET UMBER: V00 Course ame & o. Descrve Statstcs and robably Secton(s) Economcs 45 A0 CR: 098 Instructor: Betty Johnson

More information

Taxation and Externalities. - Much recent discussion of policy towards externalities, e.g., global warming debate/kyoto

Taxation and Externalities. - Much recent discussion of policy towards externalities, e.g., global warming debate/kyoto Taxaton and Externaltes - Much recent dscusson of polcy towards externaltes, e.g., global warmng debate/kyoto - Increasng share of tax revenue from envronmental taxaton 6 percent n OECD - Envronmental

More information

A Comparison of Statistical Methods in Interrupted Time Series Analysis to Estimate an Intervention Effect

A Comparison of Statistical Methods in Interrupted Time Series Analysis to Estimate an Intervention Effect Transport and Road Safety (TARS) Research Joanna Wang A Comparson of Statstcal Methods n Interrupted Tme Seres Analyss to Estmate an Interventon Effect Research Fellow at Transport & Road Safety (TARS)

More information

Price and Quantity Competition Revisited. Abstract

Price and Quantity Competition Revisited. Abstract rce and uantty Competton Revsted X. Henry Wang Unversty of Mssour - Columba Abstract By enlargng the parameter space orgnally consdered by Sngh and Vves (984 to allow for a wder range of cost asymmetry,

More information

Fourth report on the consistency of risk weighted assets

Fourth report on the consistency of risk weighted assets 11 June 2014 Fourth report on the consstency of rsk weghted assets Resdental mortgages drll-down analyss Contents Executve summary 4 1. Introducton 5 2. Defntons 7 General 7 By varable 7 Loan-to-Value

More information

Interval Estimation for a Linear Function of. Variances of Nonnormal Distributions. that Utilize the Kurtosis

Interval Estimation for a Linear Function of. Variances of Nonnormal Distributions. that Utilize the Kurtosis Appled Mathematcal Scences, Vol. 7, 013, no. 99, 4909-4918 HIKARI Ltd, www.m-hkar.com http://dx.do.org/10.1988/ams.013.37366 Interval Estmaton for a Lnear Functon of Varances of Nonnormal Dstrbutons that

More information

Number of women 0.15

Number of women 0.15 . Grouped Data (a Mdponts Trmester (months Number o women Relatve Frequency Densty.5 [0, 3 40 40/400 = 0.60 0.60/3 = 0. 4.5 [3, 6 60 60/400 = 0.5 0.5/3 = 0.05 7.5 [6, 9 00 00/400 = 0.5 0.5/3 = 0.0833 0.60

More information

Lecture Note 2 Time Value of Money

Lecture Note 2 Time Value of Money Seg250 Management Prncples for Engneerng Managers Lecture ote 2 Tme Value of Money Department of Systems Engneerng and Engneerng Management The Chnese Unversty of Hong Kong Interest: The Cost of Money

More information

4: SPOT MARKET MODELS

4: SPOT MARKET MODELS 4: SPOT MARKET MODELS INCREASING COMPETITION IN THE BRITISH ELECTRICITY SPOT MARKET Rchard Green (1996) - Journal of Industral Economcs, Vol. XLIV, No. 2 PEKKA SULAMAA The obect of the paper Dfferent polcy

More information

ISyE 512 Chapter 9. CUSUM and EWMA Control Charts. Instructor: Prof. Kaibo Liu. Department of Industrial and Systems Engineering UW-Madison

ISyE 512 Chapter 9. CUSUM and EWMA Control Charts. Instructor: Prof. Kaibo Liu. Department of Industrial and Systems Engineering UW-Madison ISyE 512 hapter 9 USUM and EWMA ontrol harts Instructor: Prof. Kabo Lu Department of Industral and Systems Engneerng UW-Madson Emal: klu8@wsc.edu Offce: Room 317 (Mechancal Engneerng Buldng) ISyE 512 Instructor:

More information

arxiv: v1 [q-fin.pm] 13 Feb 2018

arxiv: v1 [q-fin.pm] 13 Feb 2018 WHAT IS THE SHARPE RATIO, AND HOW CAN EVERYONE GET IT WRONG? arxv:1802.04413v1 [q-fn.pm] 13 Feb 2018 IGOR RIVIN Abstract. The Sharpe rato s the most wdely used rsk metrc n the quanttatve fnance communty

More information

Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator.

Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator. UNIVERSITY OF EAST ANGLIA School of Economcs Man Seres PG Examnaton 2016-17 BANKING ECONOMETRICS ECO-7014A Tme allowed: 2 HOURS Answer ALL FOUR questons. Queston 1 carres a weght of 30%; queston 2 carres

More information

Risk Reduction and Real Estate Portfolio Size

Risk Reduction and Real Estate Portfolio Size Rsk Reducton and Real Estate Portfolo Sze Stephen L. Lee and Peter J. Byrne Department of Land Management and Development, The Unversty of Readng, Whteknghts, Readng, RG6 6AW, UK. A Paper Presented at

More information

3/3/2014. CDS M Phil Econometrics. Vijayamohanan Pillai N. Truncated standard normal distribution for a = 0.5, 0, and 0.5. CDS Mphil Econometrics

3/3/2014. CDS M Phil Econometrics. Vijayamohanan Pillai N. Truncated standard normal distribution for a = 0.5, 0, and 0.5. CDS Mphil Econometrics Lmted Dependent Varable Models: Tobt an Plla N 1 CDS Mphl Econometrcs Introducton Lmted Dependent Varable Models: Truncaton and Censorng Maddala, G. 1983. Lmted Dependent and Qualtatve Varables n Econometrcs.

More information

Problem Set 6 Finance 1,

Problem Set 6 Finance 1, Carnege Mellon Unversty Graduate School of Industral Admnstraton Chrs Telmer Wnter 2006 Problem Set 6 Fnance, 47-720. (representatve agent constructon) Consder the followng two-perod, two-agent economy.

More information

Correlations and Copulas

Correlations and Copulas Correlatons and Copulas Chapter 9 Rsk Management and Fnancal Insttutons, Chapter 6, Copyrght John C. Hull 2006 6. Coeffcent of Correlaton The coeffcent of correlaton between two varables V and V 2 s defned

More information

A Utilitarian Approach of the Rawls s Difference Principle

A Utilitarian Approach of the Rawls s Difference Principle 1 A Utltaran Approach of the Rawls s Dfference Prncple Hyeok Yong Kwon a,1, Hang Keun Ryu b,2 a Department of Poltcal Scence, Korea Unversty, Seoul, Korea, 136-701 b Department of Economcs, Chung Ang Unversty,

More information

Chapter 4 Calculation of the weight (W0)

Chapter 4 Calculation of the weight (W0) Chapter 4 Calculaton of the weght (W0) 1 Scope of the Famly Income and Expendture Survey (FIES) tems adopted for the weghts In the FIES, lvng expendtures are categorzed as follows: Dsbursements Expendtures

More information

3.1 Measures of Central Tendency

3.1 Measures of Central Tendency 3.1 Measures of Central Tendency n Summation Notation x i or x Sum observation on the variable that appears to the right of the summation symbol. Example 1 Suppose the variable x i is used to represent

More information

Explaining and Comparing

Explaining and Comparing ACES EU CENTERS OF EXCELLENCE GRANT AY2011-12 DELIVERABLE GWU Explanng and Comparng AY 2011-12 Practcal Modfed Gn Index Amr Shoham (wth M Malul, Danel Shapra) Practcal Modfed Gn Index M Malul, Danel Shapra

More information

EXTENSIVE VS. INTENSIVE MARGIN: CHANGING PERSPECTIVE ON THE EMPLOYMENT RATE. and Eliana Viviano (Bank of Italy)

EXTENSIVE VS. INTENSIVE MARGIN: CHANGING PERSPECTIVE ON THE EMPLOYMENT RATE. and Eliana Viviano (Bank of Italy) EXTENSIVE VS. INTENSIVE MARGIN: CHANGING PERSPECTIVE ON THE EMPLOYMENT RATE Andrea Brandoln and Elana Vvano (Bank of Italy) 2 European User Conference for EU-LFS and EU-SILC, Mannhem 31 March 1 Aprl, 2011

More information

Consumption Based Asset Pricing

Consumption Based Asset Pricing Consumpton Based Asset Prcng Mchael Bar Aprl 25, 208 Contents Introducton 2 Model 2. Prcng rsk-free asset............................... 3 2.2 Prcng rsky assets................................ 4 2.3 Bubbles......................................

More information

Chapter 6 Risk, Return, and the Capital Asset Pricing Model

Chapter 6 Risk, Return, and the Capital Asset Pricing Model Whch s better? (1) 6% return wth no rsk, or (2) 20% return wth rsk. Chapter 6 Rsk, Return, and the Captal Asset Prcng Model Cannot say - need to know how much rsk comes wth the 20% return. What do we know

More information

Domestic Savings and International Capital Flows

Domestic Savings and International Capital Flows Domestc Savngs and Internatonal Captal Flows Martn Feldsten and Charles Horoka The Economc Journal, June 1980 Presented by Mchael Mbate and Chrstoph Schnke Introducton The 2 Vews of Internatonal Captal

More information

Quiz on Deterministic part of course October 22, 2002

Quiz on Deterministic part of course October 22, 2002 Engneerng ystems Analyss for Desgn Quz on Determnstc part of course October 22, 2002 Ths s a closed book exercse. You may use calculators Grade Tables There are 90 ponts possble for the regular test, or

More information

MEASURES OF DISPERSION, RELATIVE STANDING AND SHAPE. Dr. Bijaya Bhusan Nanda,

MEASURES OF DISPERSION, RELATIVE STANDING AND SHAPE. Dr. Bijaya Bhusan Nanda, MEASURES OF DISPERSION, RELATIVE STANDING AND SHAPE Dr. Bijaya Bhusan Nanda, CONTENTS What is measures of dispersion? Why measures of dispersion? How measures of dispersions are calculated? Range Quartile

More information

Survey of Math: Chapter 22: Consumer Finance Borrowing Page 1

Survey of Math: Chapter 22: Consumer Finance Borrowing Page 1 Survey of Math: Chapter 22: Consumer Fnance Borrowng Page 1 APR and EAR Borrowng s savng looked at from a dfferent perspectve. The dea of smple nterest and compound nterest stll apply. A new term s the

More information

4.4 Doob s inequalities

4.4 Doob s inequalities 34 CHAPTER 4. MARTINGALES 4.4 Doob s nequaltes The frst nterestng consequences of the optonal stoppng theorems are Doob s nequaltes. If M n s a martngale, denote M n =max applen M. Theorem 4.8 If M n s

More information

THIS PAPER SHOULD NOT BE OPENED UNTIL PERMISSION HAS BEEN GIVEN BY THE INVIGILATOR.

THIS PAPER SHOULD NOT BE OPENED UNTIL PERMISSION HAS BEEN GIVEN BY THE INVIGILATOR. UNVERSTY OF SWAZLAND FACULTY OF SOCAL SCENCES DEPARTMENT OF STATSTCS AND DEMOGRAPHY MAN EXAMNATON 2016 TTTLE OF PAPER: DEMOGRAPHC METHODS 1 COURSE NUMBER: DEM 201 TME ALLOWED: 2 Hours NSTRUCTONS: ANSWER

More information

UWB Indoor Delay Profile Model For Residential and Commercial Environments

UWB Indoor Delay Profile Model For Residential and Commercial Environments UWB Indoor Delay Profle Model For Resdental and Commercal Envronments S.S. Ghassemzadeh 1, L. J. Greensten, A. Kavčć 3, T. Svensson 3, V. Tarokh 3 Abstract We present a statstcal model for the delay profle

More information