# A logistic approximation to the cumulative normal distribution

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 A logistic approximation to the cumulative normal distribution Shannon R. Bowling 1 ; Mohammad T. Khasawneh 2 ; Sittichai Kaewkuekool 3 ; Byung Rae Cho 4 1 Old Dominion University (USA); 2 State University of New York (USA); 3 King Mongkut s University of Technology Thonburi (THAILAND); 4 Clemson University (USA) Received February 2009 Accepted March 2009 Abstract: This paper develops a logistic approximation to the cumulative normal distribution. Although the literature contains a vast collection of approximate functions for the normal distribution, they are very complicated, not very accurate, or valid for only a limited range. This paper proposes an enhanced approximate function. When comparing the proposed function to other approximations studied in the literature, it can be observed that the proposed logistic approximation has a simpler functional form and that it gives higher accuracy, with the maximum error of less than for the entire range. This is, to the best of the authors knowledge, the lowest level of error reported in the literature. The proposed logistic approximate function may be appealing to researchers, practitioners and educators given its functional simplicity and mathematical accuracy. Keywords: normal distribution, logistic, approximation, minimax criteria 1 Introduction The most important continuous probability distribution used in engineering and science is perhaps the normal distribution. The normal distribution reasonably describes many phenomena that occur in nature. In addition, errors in measurements are extremely well approximated with the normal distribution. In 1733, DeMoivre developed the mathematical equation of the normal curve. It A logistic approximation to the cumulative normal distribution 114

2 provided a basis on which much of the theory of inductive statistics is founded. The normal distribution is often referred to as the Gaussian distribution, in honor of Karl Friedrich Gauss, who also derived its equation from a study of error in repeated measurements of an unknown quantity. The normal distribution finds numerous applications as a limiting distribution. Under certain conditions, the normal distribution provides a good approximation to binomial and hypergeometric distributions. In addition, it appears that the limiting distribution of sample averages is normal. This provides a broad base for statistical inference that proves very valuable in estimation and hypothesis testing. If a random variable X is normally distributed with mean µ and variance σ 2, its probability density function is defined as, - < x <. Equation 1. PDF of Normal Distribution. From an inspection of the normal distribution by examination of its first and second derivatives, the following properties of the distribution are known: The mode, which is the point on the horizontal axis where the curve is a maximum, occurs at x = µ. The curve is symmetric about a vertical axis through the mean µ. The curve has its points of inflection at x = µ + s. The curve is convex upward, concave upward, concave downward, convex downward, when - < x < µ-σ, µ-σ x µ, µ < x < µ+σ, and x µ+σ, respectively. The curve approaches the horizontal axis asymptotically as we proceed in either direction away from µ. The total area under the curve and above the horizontal axis is equal to one. The cumulative normal distribution is well known as A logistic approximation to the cumulative normal distribution 115

3 . Equation 2. CDF of Normal Distribution. The difficulty encountered in solving integrals of normal density functions necessitated the tabulation of normal curve areas for quick reference. However, it would be an enormous task to attempt to set up separate tables for every conceivable value of µ and σ. Fortunately, we are able to transform all the observations of any random variable X to a new set of observations of a normal random variable Z with mean zero and variance one by means of transformation z = (x-µ)/σ. The standard normal distribution is then given by Equation 3. Standard Normal Distribution. and similarly the cumulative standard normal function is defined as Equation 4. Cumulative Standard Normal Distribution. Unfortunately, there is no closed-form solution available for the above integral, and the values are usually found from the tables of the cumulative normal distribution. From a practical point of view, however, the standard normal distribution table only provides the cumulative probabilities associated with certain discrete z-values. When the z-value of interest is not available from the table, which frequently happens, practitioners often guess its probability by means of a linear interpolation using two adjacent z-values, or rely on statistical software. In order to rectify this practical inconvenience, a number of approximate functions for a cumulative normal distribution have been reported in the research community. The literature review indicates, however, that they are mathematically complicated, do not have much accuracy, and lack validity when the entire range A logistic approximation to the cumulative normal distribution 116

4 of z-values is considered. In order to address these shortcomings, this paper develops a logistic approximate function for the cumulative normal distribution. The mathematical form of the proposed function is much simpler than the majority of other approximate functions studied in the literature. In fact, probabilities can be even obtained by using a calculator. Further, the accuracy of the proposed function is higher than with the other approximate functions. The remainder of the paper is organized as follows. In section 2, the existing literature on approximations to a cumulative normal distribution is discussed. Section 3 first discusses a logistic distribution and notes the similarities and differences between the logistic and normal distributions are noted. Section 3 then proposes the modified logistic approximate function by numerically identifying polynomial regression coefficients in such a way that the absolute maximum deviation between the cumulative distribution and the modified logistic function is minimized. Section 4 evaluates the accuracy of the proposed approximate function, and section 5 discusses and concludes about the results of this paper. 2 Review of Prior Research It is often beneficial to represent the distribution functions of random variables using a convenient approximation, facilitating mathematical and statistical analysis. In particular, some families of distributions have been constructed to provide approximations of a wide variety of distributions, whenever possible. Such families are frequently called systems of distributions (see Johnson and Kotz, 1969), and the particular requirements for these systems are ease of computation and facility of algebraic manipulation, while attaining a reasonable approximation. A major criterion in judging an approximate function for a certain distribution is the use of as few parameters as possible in defining the distribution. From the point of view of replacing the normal distribution by another distribution, the following can be observed: First, a lognormal distribution can give a good representation which has a small absolute value, say less than 0.25, of the coefficient of variation. Second, a particular form of logistic distribution is very close to a normal distribution. Finally, a form of the Weibull distribution with the shape parameter close to 3.25 is almost identical with the standard normal distribution. In addition to the above-mentioned general approximations, a number A logistic approximation to the cumulative normal distribution 117

5 of researchers attempted to find other approximations, which are summarized in Table 1. 3 Proposed Model 3.1 A Logistic Distribution Revisited The logistic distribution is defined as f (z) = [ ] { ( ) /β} exp {z α}/β [ ] 2, - < x < β 1+ exp z α Equation 5. Logistic Distribution. with E(z) = α, and Var(z) =. The cumulative logistic distribution is: 1 F(z) = 1+ exp {z α}/β [ ] Equation 6. Cumulative Logistic Distribution. The logistic distribution has been used extensively to represent growth functions, with z representing time. The use of the logistic curve for economic demographic purposes has been very popular from the end of nineteenth century onwards. The fact that a logistic distribution has a shape similar to that of the normal distribution makes it convenient to replace the normal by the logistic in order simplify the analysis in engineering and science. However, such substitutions must be done with care. Although there is a close similarity in shape between the normal and logistic distributions, two distinguishable differences are noted. First, the value of the kurtosis for the logistic is 4.2, considerably different from the value 3 for the normal distribution. The difference is attributed largely to the relatively longer tails of the logistic distribution. Second, the logistic distribution with mean zero and variance one specifies α = 1 and β = , and is symmetrical about α with the inflection points at whereas the standard normal curve has the inflection points at z = ±1. A logistic approximation to the cumulative normal distribution 118

6 Table 1. A summary of approximate functions of the cumulative normal distribution. A logistic approximation to the cumulative normal distribution 119

7 3.2 The Effect of γ The cumulative logistic distribution with mean zero and variance one is known as where Simulation results indicate that the parameter coefficient has a significant effect on the kurtosis and the inflection points. In order to show the effect of γ, the absolute deviation function, Equation 7. Difference between Normal and Logistic. is considered. This absolute deviation function is a function of the absolute deviation between the cumulative standard normal distribution and the cumulative logistic distribution with mean zero and variance one. Using the generalized reduced gradient algorithm (see Hillier and Liberman, 2001), the parameter γ is determined by minimizing the maximum deviation between the cumulative normal distribution and the logistic function. As can be seen in Figure 1, the maximum deviation of occurs at z = for γ = Therefore, the following equation gives the best logistic fit for the cumulative normal distribution, F(z) 1 1+ e 1.702z Equation 8. Best Fit One Parameter Logistic Equation. The functional relationship between γ and the deviation function over 4.5 z 4.5 is shown in Figure 1. Note that the absolute deviation with γ becomes relatively small at z and z = 0. As shown in Figure 2, sensitivity analysis indicates that varying the value of γ changes the values at which the deviation function intersects the z-axis. Two questions remain unanswered at this point. First, how can the effect of γ on the absolute deviations be better captured? Second, how can a consistent judgment on the degree of goodness of fit associated with γ be made in order to minimize the absolute deviations? A logistic approximation to the cumulative normal distribution 120

8 0,015 0,010 0,005 D(z) 0,000-0,005-0,010-0,015-4,50-4,00-3,50-3,00-2,50-2,00-1,50-1,00-0,50 0,00 0,50 1,00 1,50 2,00 2,50 3,00 3,50 4,00 4,50 z Figure 1. The difference between the logistic function and cumulative normal distribution. γ = γ = γ =1. 65 γ = γ = 1.8 Figure 2. The variation of the deviation function for different values of γ. A logistic approximation to the cumulative normal distribution 121

9 3.3 A Polynomial Logistic Regression Function In order to address the first question, a functional relationship between γ values and their corresponding non-origin intersection points on the x-axis,, is established, as shown in Figure 3. This is to ensure that the effect of γ on the absolute deviations is better captured. In this particular case, a second-order regression function turns out to be an appropriate function for. For example, a polynomial regression without considering the absolute deviations is obtained as = 0.061z , as shown in Figure 3. Based on this experimental observation, Notice that the equation does not include the firstorder z term. This is due to the fact that the function is symmetric about the y- axis. By incorporating the standard normal distribution function, the modified absolute deviation function is then given by: Equation 9. Modified Absolute Deviation Function. 2,9 2,7 2,5 γ 2,3 2,1 1,9 1,7 1, z Figure 3. The variation of γ as a function of z. A logistic approximation to the cumulative normal distribution 122

10 3.4 Minimax Decision Criterion In order to address the second question raised in section 3.2 and to make the analysis on a consistent basis, the minimax decision criterion, which minimizes the maximum deviation, is employed. The minimax criterion is widely used in decision theory. Using the generalized reduced gradient algorithm, the parameters and are determined by minimizing the maximum deviation using equation 9. As a result, the minimum of the maximum deviation with occurs at z = ± 3.16 for = and = Therefore, the following equation gives the best approximation to the standard cumulative normal distribution, while minimizing the maximum deviation: 1 F(z) 1+ e ( *z *z) Equation 10. Best Fit Two Parameter Logistic Function. 4 Accuracy of the Proposed Logistic Approximation As it is the case for any approximate function, the proposed model should be evaluated in order to ensure that it is an appropriate approximation in terms of accuracy and range of validity. Table 2 compares the proposed logistic approximation with two popular approximations developed by Hamaker (1978) and Lin (1988), as well as the cumulative normal distribution. As it can be clearly seen, Hamaker s and Lin s approximations are valid for only one half of the cumulative normal distribution (i.e., the tail probabilities). Figure 4 illustrates the deviation between the cumulative standard normal distribution, and the approximate functions. The proposed approximation is more accurate when -2.2 < x < The proposed approximate function is validated by comparing it to the actual standard normal distribution table in order to check the accuracy and range of validity. As shown in Table 3, the accuracy is quite high. Most of the probabilities using the proposed approximation coincide to the probabilities shown in the standard normal table. The maximum error is less than 1.4x10-4. It is also noted that the approximate function is relatively simple, compared to others available in the literature. A logistic approximation to the cumulative normal distribution 123

11 Z Standard Normal Hamaker (1978) Lin (1988) Proposed Table 2. Illustration of the proposed logistic approximation vs. the cumulative normal distribution and that of Hamaker (1978) and Lin (1988). Figure 4. Graphical illustration of the deviation of the proposed logistic approximation, Hamaker s (1978), and Lin s (1988) to the cumulative standard normal distribution. A logistic approximation to the cumulative normal distribution 124

12 Table 3. Standard Normal Table vs. Proposed Approximation (Approximation Values). A logistic approximation to the cumulative normal distribution 125

13 References Cadwell, J. H. (1951). The Bivariate Normal Integral. Biometrika, 38, Hamaker, H. C. (1978). Approximating the Cumulative Normal Distribution and its Inverse. Applied Statistics, 27, Hart, R. G. A. (1963). Close Approximation Related to the Error Function. Mathematics of Computation, 20, Hiller, F. S., & Liberman, G. J. (2001). Introduction to Operations Research. 7 th Ed., New York: McGraw-Hill. Hoyt, J. P. A. (1968). Simple Approximation to the Standard Normal Probability Density Function. American Statistician, 22(3), Johnson, N. L., & Kotz, S. (1969). Distributions in Statistics: Continuous Univariate Distributions. Houghton Mifflin Company: Boston. Lin, J. T. (1988). Alternative to Hamaker s Approximations to the Cumulative Normal Distribution and its Inverse. Applied Statistics, 37, Lin, J. T. (1989). Approximating the Normal Tail Probability and its Inverse for use on a Pocket-Calculator. Applied Statistics, 38, Lin, J. T. (1990). A Simpler Logistic Approximation to the Normal Tail Probability and its Inverse. Applied Statistics, 39, Polya, G. (1945). Remarks on Computing the Probability Integral in One and Two Dimensions. Proceedings of the 1 st Berkeley Symposium on Mathematical Statistics and Probability, Schucany, W. R.. & Gray, H. L. (1968). A New Approximation Related to the Error Function. Mathematics of Computation, 22, Zelen, M., & Severo, N. C. (1966). Probability Functions. In Handbook of Mathematical Functions, Ed. Abramowitz and I. A. Stegun, Washington D.C.: Department of US Government. A logistic approximation to the cumulative normal distribution 126

14 Journal of Industrial Engineering and Management, 2009 ( Article's contents are provided on a Attribution-Non Commercial 3.0 Creative commons license. Readers are allowed to copy, distribute and communicate article's contents, provided the author's and Journal of Industrial Engineering and Management's names are included. It must not be used for commercial purposes. To see the complete license contents, please visit A logistic approximation to the cumulative normal distribution 127

### Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

### 8. THE NORMAL DISTRIBUTION

8. THE NORMAL DISTRIBUTION The normal distribution with mean μ and variance σ 2 has the following density function: The normal distribution is sometimes called a Gaussian Distribution, after its inventor,

### z 3. 4 which is the domain often used in normal tables. We recommend the last of these new formulas.

Approximations of the Standard Normal Distribution B. Zogheib and M. Hlynka Division of Mathematics, Science and Technology, Nova Southeastern University 0 College Ave., Fort Lauderdale-Davie, FL, -7796,

### 2.0 Lesson Plan. Answer Questions. Summary Statistics. Histograms. The Normal Distribution. Using the Standard Normal Table

2.0 Lesson Plan Answer Questions 1 Summary Statistics Histograms The Normal Distribution Using the Standard Normal Table 2. Summary Statistics Given a collection of data, one needs to find representations

### Probability Calculator

Chapter 95 Introduction Most statisticians have a set of probability tables that they refer to in doing their statistical wor. This procedure provides you with a set of electronic statistical tables that

### Chapter 3 RANDOM VARIATE GENERATION

Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.

### Mathematics. Probability and Statistics Curriculum Guide. Revised 2010

Mathematics Probability and Statistics Curriculum Guide Revised 2010 This page is intentionally left blank. Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction

### Probability and Statistics

CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

### Maximum Likelihood Estimation

Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for

### Determining distribution parameters from quantiles

Determining distribution parameters from quantiles John D. Cook Department of Biostatistics The University of Texas M. D. Anderson Cancer Center P. O. Box 301402 Unit 1409 Houston, TX 77230-1402 USA cook@mderson.org

### Review of Random Variables

Chapter 1 Review of Random Variables Updated: January 16, 2015 This chapter reviews basic probability concepts that are necessary for the modeling and statistical analysis of financial data. 1.1 Random

### Session 7 Bivariate Data and Analysis

Session 7 Bivariate Data and Analysis Key Terms for This Session Previously Introduced mean standard deviation New in This Session association bivariate analysis contingency table co-variation least squares

### Lecture 8: More Continuous Random Variables

Lecture 8: More Continuous Random Variables 26 September 2005 Last time: the eponential. Going from saying the density e λ, to f() λe λ, to the CDF F () e λ. Pictures of the pdf and CDF. Today: the Gaussian

### In this chapter, you will learn to use descriptive statistics to organize, summarize, analyze, and interpret data for contract pricing.

3.0 - Chapter Introduction In this chapter, you will learn to use descriptive statistics to organize, summarize, analyze, and interpret data for contract pricing. Categories of Statistics. Statistics is

### Equations, Inequalities & Partial Fractions

Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities

### Chapter 10 Monte Carlo Methods

411 There is no result in nature without a cause; understand the cause and you will have no need for the experiment. Leonardo da Vinci (1452-1519) Chapter 10 Monte Carlo Methods In very broad terms one

### Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in

### Mathematics Chapter 8 and 10 Test Summary 10M2

Quadratic expressions and equations Expressions such as x 2 + 3x, a 2 7 and 4t 2 9t + 5 are called quadratic expressions because the highest power of the variable is 2. The word comes from the Latin quadratus

### UNIT I: RANDOM VARIABLES PART- A -TWO MARKS

UNIT I: RANDOM VARIABLES PART- A -TWO MARKS 1. Given the probability density function of a continuous random variable X as follows f(x) = 6x (1-x) 0

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### Helpsheet. Giblin Eunson Library LINEAR EQUATIONS. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:

Helpsheet Giblin Eunson Library LINEAR EQUATIONS Use this sheet to help you: Solve linear equations containing one unknown Recognize a linear function, and identify its slope and intercept parameters Recognize

### Continuous Random Variables and Probability Distributions. Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage

4 Continuous Random Variables and Probability Distributions Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage Continuous r.v. A random variable X is continuous if possible values

### A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution

A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 4: September

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

### PROPERTIES OF THE SAMPLE CORRELATION OF THE BIVARIATE LOGNORMAL DISTRIBUTION

PROPERTIES OF THE SAMPLE CORRELATION OF THE BIVARIATE LOGNORMAL DISTRIBUTION Chin-Diew Lai, Department of Statistics, Massey University, New Zealand John C W Rayner, School of Mathematics and Applied Statistics,

### Time series Forecasting using Holt-Winters Exponential Smoothing

Time series Forecasting using Holt-Winters Exponential Smoothing Prajakta S. Kalekar(04329008) Kanwal Rekhi School of Information Technology Under the guidance of Prof. Bernard December 6, 2004 Abstract

### Continuous Distributions

MAT 2379 3X (Summer 2012) Continuous Distributions Up to now we have been working with discrete random variables whose R X is finite or countable. However we will have to allow for variables that can take

### SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation

SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu September 19, 2015 Outline

### Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

### BNG 202 Biomechanics Lab. Descriptive statistics and probability distributions I

BNG 202 Biomechanics Lab Descriptive statistics and probability distributions I Overview The overall goal of this short course in statistics is to provide an introduction to descriptive and inferential

Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse

### 1 The Pareto Distribution

Estimating the Parameters of a Pareto Distribution Introducing a Quantile Regression Method Joseph Lee Petersen Introduction. A broad approach to using correlation coefficients for parameter estimation

### Geostatistics Exploratory Analysis

Instituto Superior de Estatística e Gestão de Informação Universidade Nova de Lisboa Master of Science in Geospatial Technologies Geostatistics Exploratory Analysis Carlos Alberto Felgueiras cfelgueiras@isegi.unl.pt

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### 4. Factor polynomials over complex numbers, describe geometrically, and apply to real-world situations. 5. Determine and apply relationships among syn

I The Real and Complex Number Systems 1. Identify subsets of complex numbers, and compare their structural characteristics. 2. Compare and contrast the properties of real numbers with the properties of

### Simple formulas to option pricing and hedging in the Black Scholes model

Simple formulas to option pricing and hedging in the Black Scholes model Paolo Pianca Department of Applied Mathematics University Ca Foscari of Venice Dorsoduro 385/E, 3013 Venice, Italy pianca@unive.it

### Week 1: Functions and Equations

Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.1-2.2, and Chapter

### Models for Count Data With Overdispersion

Models for Count Data With Overdispersion Germán Rodríguez November 6, 2013 Abstract This addendum to the WWS 509 notes covers extra-poisson variation and the negative binomial model, with brief appearances

### 3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

### 6.4 Normal Distribution

Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

### http://www.jstor.org This content downloaded on Tue, 19 Feb 2013 17:28:43 PM All use subject to JSTOR Terms and Conditions

A Significance Test for Time Series Analysis Author(s): W. Allen Wallis and Geoffrey H. Moore Reviewed work(s): Source: Journal of the American Statistical Association, Vol. 36, No. 215 (Sep., 1941), pp.

### APPROXIMATING THE CUMULATIVE DISTRIBUTION FUNCTION OF THE NORMAL DISTRIBUTION

Journal of Statistical Research 2007, Vol. 41, No. 1, pp. 59 67 Bangladesh ISSN 0256-422 X APPROXIMATING THE CUMULATIVE DISTRIBUTION FUNCTION OF THE NORMAL DISTRIBUTION Amit Choudhury Department of Statistics

### EMPIRICAL FREQUENCY DISTRIBUTION

INTRODUCTION TO MEDICAL STATISTICS: Mirjana Kujundžić Tiljak EMPIRICAL FREQUENCY DISTRIBUTION observed data DISTRIBUTION - described by mathematical models 2 1 when some empirical distribution approximates

### Review of Fundamental Mathematics

Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

Graphing Quadratic Functions In our consideration of polynomial functions, we first studied linear functions. Now we will consider polynomial functions of order or degree (i.e., the highest power of x

### 1 Sufficient statistics

1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =

### South Carolina College- and Career-Ready (SCCCR) Probability and Statistics

South Carolina College- and Career-Ready (SCCCR) Probability and Statistics South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR)

### Hypothesis Testing for Beginners

Hypothesis Testing for Beginners Michele Piffer LSE August, 2011 Michele Piffer (LSE) Hypothesis Testing for Beginners August, 2011 1 / 53 One year ago a friend asked me to put down some easy-to-read notes

### Research Variables. Measurement. Scales of Measurement. Chapter 4: Data & the Nature of Measurement

Chapter 4: Data & the Nature of Graziano, Raulin. Research Methods, a Process of Inquiry Presented by Dustin Adams Research Variables Variable Any characteristic that can take more than one form or value.

### The Big 50 Revision Guidelines for S1

The Big 50 Revision Guidelines for S1 If you can understand all of these you ll do very well 1. Know what is meant by a statistical model and the Modelling cycle of continuous refinement 2. Understand

### South Carolina College- and Career-Ready (SCCCR) Algebra 1

South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process

### HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate

### Univariate Regression

Univariate Regression Correlation and Regression The regression line summarizes the linear relationship between 2 variables Correlation coefficient, r, measures strength of relationship: the closer r is

### Normality Testing in Excel

Normality Testing in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com

### A Coefficient of Variation for Skewed and Heavy-Tailed Insurance Losses. Michael R. Powers[ 1 ] Temple University and Tsinghua University

A Coefficient of Variation for Skewed and Heavy-Tailed Insurance Losses Michael R. Powers[ ] Temple University and Tsinghua University Thomas Y. Powers Yale University [June 2009] Abstract We propose a

### An Introduction to Modeling Stock Price Returns With a View Towards Option Pricing

An Introduction to Modeling Stock Price Returns With a View Towards Option Pricing Kyle Chauvin August 21, 2006 This work is the product of a summer research project at the University of Kansas, conducted

### Module 3: Correlation and Covariance

Using Statistical Data to Make Decisions Module 3: Correlation and Covariance Tom Ilvento Dr. Mugdim Pašiƒ University of Delaware Sarajevo Graduate School of Business O ften our interest in data analysis

### Content DESCRIPTIVE STATISTICS. Data & Statistic. Statistics. Example: DATA VS. STATISTIC VS. STATISTICS

Content DESCRIPTIVE STATISTICS Dr Najib Majdi bin Yaacob MD, MPH, DrPH (Epidemiology) USM Unit of Biostatistics & Research Methodology School of Medical Sciences Universiti Sains Malaysia. Introduction

Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express

### Permutation Tests for Comparing Two Populations

Permutation Tests for Comparing Two Populations Ferry Butar Butar, Ph.D. Jae-Wan Park Abstract Permutation tests for comparing two populations could be widely used in practice because of flexibility of

### Multivariate Normal Distribution

Multivariate Normal Distribution Lecture 4 July 21, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Lecture #4-7/21/2011 Slide 1 of 41 Last Time Matrices and vectors Eigenvalues

### Probability. Distribution. Outline

7 The Normal Probability Distribution Outline 7.1 Properties of the Normal Distribution 7.2 The Standard Normal Distribution 7.3 Applications of the Normal Distribution 7.4 Assessing Normality 7.5 The

### Outline. Random Variables. Examples. Random Variable

Outline Random Variables M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno Random variables. CDF and pdf. Joint random variables. Correlated, independent, orthogonal. Correlation,

### Interpretation of Somers D under four simple models

Interpretation of Somers D under four simple models Roger B. Newson 03 September, 04 Introduction Somers D is an ordinal measure of association introduced by Somers (96)[9]. It can be defined in terms

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### CHAPTER 2 Estimating Probabilities

CHAPTER 2 Estimating Probabilities Machine Learning Copyright c 2016. Tom M. Mitchell. All rights reserved. *DRAFT OF January 24, 2016* *PLEASE DO NOT DISTRIBUTE WITHOUT AUTHOR S PERMISSION* This is a

### Introduction to General and Generalized Linear Models

Introduction to General and Generalized Linear Models General Linear Models - part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby

### Dongfeng Li. Autumn 2010

Autumn 2010 Chapter Contents Some statistics background; ; Comparing means and proportions; variance. Students should master the basic concepts, descriptive statistics measures and graphs, basic hypothesis

### 1. How different is the t distribution from the normal?

Statistics 101 106 Lecture 7 (20 October 98) c David Pollard Page 1 Read M&M 7.1 and 7.2, ignoring starred parts. Reread M&M 3.2. The effects of estimated variances on normal approximations. t-distributions.

### 1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number

1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x - x) B. x 3 x C. 3x - x D. x - 3x 2) Write the following as an algebraic expression

### Getting Correct Results from PROC REG

Getting Correct Results from PROC REG Nathaniel Derby, Statis Pro Data Analytics, Seattle, WA ABSTRACT PROC REG, SAS s implementation of linear regression, is often used to fit a line without checking

### ALGEBRA I / ALGEBRA I SUPPORT

Suggested Sequence: CONCEPT MAP ALGEBRA I / ALGEBRA I SUPPORT August 2011 1. Foundations for Algebra 2. Solving Equations 3. Solving Inequalities 4. An Introduction to Functions 5. Linear Functions 6.

### Bracken County Schools Curriculum Guide Math

Unit 1: Expressions and Equations (Ch. 1-3) Suggested Length: Semester Course: 4 weeks Year Course: 8 weeks Program of Studies Core Content 1. How do you use basic skills and operands to create and solve

### Correlation key concepts:

CORRELATION Correlation key concepts: Types of correlation Methods of studying correlation a) Scatter diagram b) Karl pearson s coefficient of correlation c) Spearman s Rank correlation coefficient d)

### PowerTeaching i3: Algebra I Mathematics

PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I Key Ideas and

### Lecture 3: Continuous distributions, expected value & mean, variance, the normal distribution

Lecture 3: Continuous distributions, expected value & mean, variance, the normal distribution 8 October 2007 In this lecture we ll learn the following: 1. how continuous probability distributions differ

### Sample Size Determination

Sample Size Determination Population A: 10,000 Population B: 5,000 Sample 10% Sample 15% Sample size 1000 Sample size 750 The process of obtaining information from a subset (sample) of a larger group (population)

### Nonparametric adaptive age replacement with a one-cycle criterion

Nonparametric adaptive age replacement with a one-cycle criterion P. Coolen-Schrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK e-mail: Pauline.Schrijner@durham.ac.uk

### Common Core Unit Summary Grades 6 to 8

Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations

### Curve Sketching GUIDELINES FOR SKETCHING A CURVE: A. Domain. B. Intercepts: x- and y-intercepts.

Curve Sketching GUIDELINES FOR SKETCHING A CURVE: A. Domain. B. Intercepts: x- and y-intercepts. C. Symmetry: even (f( x) = f(x)) or odd (f( x) = f(x)) function or neither, periodic function. ( ) ( ) D.

### Sampling Distributions and the Central Limit Theorem

135 Part 2 / Basic Tools of Research: Sampling, Measurement, Distributions, and Descriptive Statistics Chapter 10 Sampling Distributions and the Central Limit Theorem In the previous chapter we explained

### NPTEL STRUCTURAL RELIABILITY

NPTEL Course On STRUCTURAL RELIABILITY Module # 02 Lecture 6 Course Format: Web Instructor: Dr. Arunasis Chakraborty Department of Civil Engineering Indian Institute of Technology Guwahati 6. Lecture 06:

### MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 2004-2012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................

### LOGIT AND PROBIT ANALYSIS

LOGIT AND PROBIT ANALYSIS A.K. Vasisht I.A.S.R.I., Library Avenue, New Delhi 110 012 amitvasisht@iasri.res.in In dummy regression variable models, it is assumed implicitly that the dependent variable Y

### Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

### Temperature Scales. The metric system that we are now using includes a unit that is specific for the representation of measured temperatures.

Temperature Scales INTRODUCTION The metric system that we are now using includes a unit that is specific for the representation of measured temperatures. The unit of temperature in the metric system is

### Supplement to Call Centers with Delay Information: Models and Insights

Supplement to Call Centers with Delay Information: Models and Insights Oualid Jouini 1 Zeynep Akşin 2 Yves Dallery 1 1 Laboratoire Genie Industriel, Ecole Centrale Paris, Grande Voie des Vignes, 92290

### Advanced Algebra 2. I. Equations and Inequalities

Advanced Algebra 2 I. Equations and Inequalities A. Real Numbers and Number Operations 6.A.5, 6.B.5, 7.C.5 1) Graph numbers on a number line 2) Order real numbers 3) Identify properties of real numbers

### Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

### Chapter 10. Key Ideas Correlation, Correlation Coefficient (r),

Chapter 0 Key Ideas Correlation, Correlation Coefficient (r), Section 0-: Overview We have already explored the basics of describing single variable data sets. However, when two quantitative variables

### Treatment and analysis of data Applied statistics Lecture 3: Sampling and descriptive statistics

Treatment and analysis of data Applied statistics Lecture 3: Sampling and descriptive statistics Topics covered: Parameters and statistics Sample mean and sample standard deviation Order statistics and

### Chapter 17: Aggregation

Chapter 17: Aggregation 17.1: Introduction This is a technical chapter in the sense that we need the results contained in it for future work. It contains very little new economics and perhaps contains

### Appendix C: Graphs. Vern Lindberg

Vern Lindberg 1 Making Graphs A picture is worth a thousand words. Graphical presentation of data is a vital tool in the sciences and engineering. Good graphs convey a great deal of information and can

### Interactive Math Glossary Terms and Definitions

Terms and Definitions Absolute Value the magnitude of a number, or the distance from 0 on a real number line Additive Property of Area the process of finding an the area of a shape by totaling the areas

### 7. Normal Distributions

7. Normal Distributions A. Introduction B. History C. Areas of Normal Distributions D. Standard Normal E. Exercises Most of the statistical analyses presented in this book are based on the bell-shaped

### Masconomet Regional High School Curriculum Guide

Masconomet Regional High School Curriculum Guide COURSE TITLE: Algebra 2 COURSE NUMBER: 1322 DEPARTMENT: Mathematics GRADE LEVEL(S) & PHASE: 10 12, CP LENGTH OF COURSE: Full Year Course Description: This

### COGNITIVE TUTOR ALGEBRA

COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,

### Piecewise Cubic Splines

280 CHAP. 5 CURVE FITTING Piecewise Cubic Splines The fitting of a polynomial curve to a set of data points has applications in CAD (computer-assisted design), CAM (computer-assisted manufacturing), and