Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Number of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day"

Transcription

1 LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the we will go o to the case of a frequecy distributio. The first thig to ote is that, whereas the rage as well as the quartile deviatio are two such measures of dispersio which are NOT based o all the values, the mea deviatio ad the stadard deviatio are two such measures of dispersio that ivolve each ad every data-value i their computatio. You must have oted that the rage was measurig the dispersio of the data-set aroud the mid-rage, whereas the quartile deviatio was measurig the dispersio of the data-set aroud the media. How are we to decide upo the amout of dispersio roud the arithmetic mea? It would seem reasoable to compute the DISTANCE of each observed value i the series from the arithmetic mea of the series. Let us do this for a simple data-set show below: The Number of Fatalities i Motorway Accidets i oe Week: Day Number of fatalities Suday 4 Moday 6 Tuesday Wedesday 0 Thursday 3 Friday 5 Saturday 8 Total 8 Let us do this for a simple data-set show below: The Number of Fatalities i Motorway Accidets i oe Week: Day Number of fatalities Suday 4 Moday 6 Tuesday Wedesday 0 Thursday 3 Friday 5 Saturday 8 Total 8

2 The arithmetic mea umber of fatalities per day is I order to determie the distaces of the data-values from the mea, we subtract our value of the arithmetic mea from each daily figure, ad this gives us the deviatios that occur i the third colum of the table below: Day Number of fatalities Suday 4 0 Moday 6 + Tuesday Wedesday 0 4 Thursday 3 1 Friday Saturday TOTAL 8 0 The deviatios are egative whe the daily figure is less tha the mea (4 accidets) ad positive whe the figure is higher tha the mea. It does seem, however, that our efforts for computig the dispersio of this data set have bee i vai, for we fid that the total amout of dispersio obtaied by summig the (x x) colum comes out to be zero! I fact, this should be o surprise, for it is a basic property of the arithmetic mea that:the sum of the deviatios of the values from the mea is zero. The questio arises: How will we measure the dispersio that is actually preset i our data-set? Our problem might at first sight seem irresolvable, for by this criterio it appears that o series has ay dispersio. Yet we kow that this is absolutely icorrect, ad we must thik of some other way of hadlig this situatio. Surely, we might look at the umerical differece betwee the mea ad the daily fatality figures without cosiderig whether these are positive or egative. Let us deote these absolute differeces by modulus of d or mod d. This is evidet from the third colum of the table below: d d Total 14

3 By igorig the sig of the deviatios we have achieved a o-zero sum i our secod colum. Averagig these absolute differeces, we obtai a measure of dispersio kow as the mea deviatio. I other words, the mea deviatio is give by the formula: MEAN DEVIATION: M.D. d i As we are averagig the absolute deviatios of the observatios from their mea, therefore the complete ame of this measure is mea absolute deviatio --- but geerally we just say mea deviatio. Applyig this formula i our example, we fid that: The mea deviatio of the umber of fatalities is 14 M.D.. 7 The formula that we have just cosidered is valid i the case of raw data. I case of grouped data i.e. a frequecy distributio, the formula becomes MEAN DEVIATION FOR GROUPED DATA: M.D. f i x i x fi d i As far as the graphical represetatio of the mea deviatio is cocered, it ca be depicted by a horizotal lie segmet draw below the -axis o the graph of the frequecy distributio, as show below:

4 f Mea Deviatio The approach which we have adopted i the cocept of the mea deviatio is both quick ad simple. But the problem is that we itroduce a kid of artificiality i its calculatio by igorig the algebraic sigs of the deviatios. I problems ivolvig descriptios ad comparisos aloe, the mea deviatio ca validly be applied; but because the egative sigs have bee discarded, further theoretical developmet or applicatio of the cocept is impossible. Mea deviatio is a absolute measure of dispersio. Its relative measure, kow as the co-efficiet of mea deviatio, is obtaied by dividig the mea deviatio by the average used i the calculatio of deviatios i.e. the arithmetic mea. Thus Co-efficiet of M.D: Sometimes, the mea deviatio is computed by averagig the absolute deviatios of the datavalues from the media i.e. M. D. Mea x x~ Mea deviatio Ad whe will we have a situatio whe we will be usig the media istead of the mea?as discussed earlier, the media will be more appropriate tha the mea i those cases where our data-set cotais a few very high or very low values.i such a situatio, the coefficiet of mea deviatio is give by: Co-efficiet of M.D: M.D. Media Let us ow cosider the stadard deviatio --- that statistic which is the most importat ad the most widely used measure of dispersio. The poit that made earlier that from the mathematical poit of view, it is ot very preferable to take the absolute values of the deviatios, This problem is overcome by computig the stadard deviatio. I order to compute the stadard deviatio, rather tha takig the absolute values of the deviatios, we square the deviatios. Averagig these squared deviatios, we obtai a statistic that is kow as the variace.

5 Variace ( x x) Let us compute this quatity for the data of the above example. Our -values were: Takig the deviatios of the -values from their mea, ad the squarig these deviatios, we obtai: ( x x ) ( x x ) Obviously, both ( ) ad () equal 4, both ( 4) ad (4) equal 16, ad both ( 1) ad (1) 1.

6 Hece (x x) 4 is ow positive, ad this positive value has bee achieved without bedig the rules of mathematics. Averagig these squared deviatios, the variace is give by: Variace: ( x x) The variace is frequetly employed i statistical work, but it should be oted that the figure achieved is i squared uits of measuremet. I the example that we have just cosidered, the variace has come out to be 6 squared fatalities, which does ot seem to make much sese! I order to obtai a aswer which is i the origial uit of measuremet, we take the positive square root of the variace. The result is kow as the stadard deviatio. STANDARD DEVIATION: S ( x x ) Hece, i this example, our stadard deviatio has come out to be.45 fatalities. I computig the stadard deviatio (or variace) it ca be tedious to first ascertai the arithmetic mea of a series, the subtract it from each value of the variable i the series, ad fially to square each deviatio ad the sum. It is very much more straight-forward to use the short cut formula give below: SHORT CUT FORMULA FOR THE STANDARD DEVIATION: S x x I order to apply the short cut formula, we require oly the aggregate of the series ( x) ad the aggregate of the squares of the idividual values i the series ( x). I other words, oly two colums of figures are called for. The umber of idividual calculatios is also cosiderably reduced, as see below:

7 Total Therefore S ( 16) 6.45 fatalities The formulae that we have just discussed are valid i case of raw data. I case of grouped data i.e. a frequecy distributio, each squared deviatio roud the mea must be multiplied by the appropriate frequecy figure i.e. STANDARD DEVIATION IN CASE OF GROUPED DATA: S f ( x x ) Ad the short cut formula i case of a frequecy distributio is: SHORT CUT FORMULA OF THE STANDARD DEVIATION IN CASE OF GROUPED DATA: fx fx S Which is agai preferred from the computatioal stadpoit? For example, the stadard deviatio life of a batch of electric light bulbs would be calculated as follows: EAMPLE: Life (i Hudreds of Hours) No. of Bulbs f Midpoit x fx fx ad over

8 Therefore, stadard deviatio: S hudredhours 1390 hours As far as the graphical represetatio of the stadard deviatio is cocered, a horizotal lie segmet is draw below the -axis o the graph of the frequecy distributio --- just as i the case of the mea deviatio. f Stadard deviatio The stadard deviatio is a absolute measure of dispersio. Its relative measure called coefficiet of stadard deviatio is defied as: Coefficiet of S.D: Sta dard Deviatio Mea

9 Ad, multiplyig this quatity by 100, we obtai a very importat ad well-kow measure called the coefficiet of variatio. Coefficiet of Variatio: S C.V. 100 As metioed earlier, the stadard deviatio is expressed i absolute terms ad is give i the same uit of measuremet as the variable itself. There are occasios, however, whe this absolute measure of dispersio is iadequate ad a relative form becomes preferable. For example, if a compariso betwee the variability of distributios with differet variables is required, or whe we eed to compare the dispersio of distributios with the same variable but with very differet arithmetic meas. To illustrate the usefuless of the coefficiet of variatio, let us cosider the followig two examples: EAMPLE-1 Suppose that, i a particular year, the mea weekly earigs of skilled factory workers i oe particular coutry were $ with a stadard deviatio of $ 4, while for its eighborig coutry the figures were Rs. 75 ad Rs. 8 respectively. From these figures, it is ot immediately apparet which coutry has the GREATER VARIABILITY i earigs. The coefficiet of variatio quickly provides the aswer: COEFFICIENT OF VARIATION For coutry No. 1: per cet, 19.5 Ad for coutry No. : per cet. 75 From these calculatios, it is immediately obvious that the spread of earigs i coutry No. is greater tha that i coutry No. 1, ad the reasos for this could the be sought. EAMPLE-: The crop yield from 0 acre plots of wheat-lad cultivated by ordiary methods averages 35 bushels with a stadard deviatio of 10 bushels. The yield from similar lad treated with a ew fertilizer averages 58 bushels, also with a stadard deviatio of 10 bushels. At first glace, the yield variability may seem to be the same, but i fact it has improved (i.e. decreased) i view of the higher average to which it relates. Agai, the coefficiet of variatio shows this very clearly: Coefficiet of Variatio:

10 Utreated lad: per cet 35 Treated lad: per cet 58 The coefficiet of variatio for the utreated lad has come out to be 8.57 percet, whereas the coefficiet of variatio for the treated lad is oly 17.4 percet.

Median and IQR The median is the value which divides the ordered data values in half.

Median and IQR The median is the value which divides the ordered data values in half. STA 666 Fall 2007 Web-based Course Notes 4: Describig Distributios Numerically Numerical summaries for quatitative variables media ad iterquartile rage (IQR) 5-umber summary mea ad stadard deviatio Media

More information

CURRICULUM INSPIRATIONS: INNOVATIVE CURRICULUM ONLINE EXPERIENCES: TANTON TIDBITS:

CURRICULUM INSPIRATIONS:  INNOVATIVE CURRICULUM ONLINE EXPERIENCES:  TANTON TIDBITS: CURRICULUM INSPIRATIONS: wwwmaaorg/ci MATH FOR AMERICA_DC: wwwmathforamericaorg/dc INNOVATIVE CURRICULUM ONLINE EXPERIENCES: wwwgdaymathcom TANTON TIDBITS: wwwjamestatocom TANTON S TAKE ON MEAN ad VARIATION

More information

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled 1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how

More information

Elementary Statistics

Elementary Statistics Elemetary Statistics M. Ghamsary, Ph.D. Sprig 004 Chap 0 Descriptive Statistics Raw Data: Whe data are collected i origial form, they are called raw data. The followig are the scores o the first test of

More information

Zeros of Polynomials

Zeros of Polynomials Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree

More information

1 Inferential Methods for Correlation and Regression Analysis

1 Inferential Methods for Correlation and Regression Analysis 1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet

More information

A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence

A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as

More information

Measures of Spread: Variance and Standard Deviation

Measures of Spread: Variance and Standard Deviation Lesso 1-6 Measures of Spread: Variace ad Stadard Deviatio BIG IDEA Variace ad stadard deviatio deped o the mea of a set of umbers. Calculatig these measures of spread depeds o whether the set is a sample

More information

Activity 3: Length Measurements with the Four-Sided Meter Stick

Activity 3: Length Measurements with the Four-Sided Meter Stick Activity 3: Legth Measuremets with the Four-Sided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a four-sided meter

More information

1 Lesson 6: Measure of Variation

1 Lesson 6: Measure of Variation 1 Lesso 6: Measure of Variatio 1.1 The rage As we have see, there are several viable coteders for the best measure of the cetral tedecy of data. The mea, the mode ad the media each have certai advatages

More information

Infinite Sequences and Series

Infinite Sequences and Series Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

More information

Section 11.8: Power Series

Section 11.8: Power Series Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i

More information

This is an introductory course in Analysis of Variance and Design of Experiments.

This is an introductory course in Analysis of Variance and Design of Experiments. 1 Notes for M 384E, Wedesday, Jauary 21, 2009 (Please ote: I will ot pass out hard-copy class otes i future classes. If there are writte class otes, they will be posted o the web by the ight before class

More information

II. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation

II. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation II. Descriptive Statistics D. Liear Correlatio ad Regressio I this sectio Liear Correlatio Cause ad Effect Liear Regressio 1. Liear Correlatio Quatifyig Liear Correlatio The Pearso product-momet correlatio

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES 9 SEQUENCES AND SERIES INTRODUCTION Sequeces have may importat applicatios i several spheres of huma activities Whe a collectio of objects is arraged i a defiite order such that it has a idetified first

More information

3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials

3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered

More information

APPENDIX F Complex Numbers

APPENDIX F Complex Numbers APPENDIX F Complex Numbers Operatios with Complex Numbers Complex Solutios of Quadratic Equatios Polar Form of a Complex Number Powers ad Roots of Complex Numbers Operatios with Complex Numbers Some equatios

More information

The Poisson Distribution

The Poisson Distribution MATH 382 The Poisso Distributio Dr. Neal, WKU Oe of the importat distributios i probabilistic modelig is the Poisso Process X t that couts the umber of occurreces over a period of t uits of time. This

More information

The Random Walk For Dummies

The Random Walk For Dummies The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli

More information

Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:

Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman: Math 224 Fall 2017 Homework 4 Drew Armstrog Problems from 9th editio of Probability ad Statistical Iferece by Hogg, Tais ad Zimmerma: Sectio 2.3, Exercises 16(a,d),18. Sectio 2.4, Exercises 13, 14. Sectio

More information

Probability, Expectation Value and Uncertainty

Probability, Expectation Value and Uncertainty Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such

More information

STA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:

STA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to: STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio

More information

Variance of Discrete Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Variance of Discrete Random Variables Class 5, Jeremy Orloff and Jonathan Bloom Variace of Discrete Radom Variables Class 5, 18.05 Jeremy Orloff ad Joatha Bloom 1 Learig Goals 1. Be able to compute the variace ad stadard deviatio of a radom variable.. Uderstad that stadard deviatio

More information

Properties and Hypothesis Testing

Properties and Hypothesis Testing Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.

More information

MA131 - Analysis 1. Workbook 2 Sequences I

MA131 - Analysis 1. Workbook 2 Sequences I MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................

More information

Dotting The Dot Map, Revisited. A. Jon Kimerling Dept. of Geosciences Oregon State University

Dotting The Dot Map, Revisited. A. Jon Kimerling Dept. of Geosciences Oregon State University Dottig The Dot Map, Revisited A. Jo Kimerlig Dept. of Geoscieces Orego State Uiversity Dot maps show the geographic distributio of features i a area by placig dots represetig a certai quatity of features

More information

Sequences I. Chapter Introduction

Sequences I. Chapter Introduction Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which

More information

Example 3.3: Rainfall reported at a group of five stations (see Fig. 3.7) is as follows. Kundla. Sabli

Example 3.3: Rainfall reported at a group of five stations (see Fig. 3.7) is as follows. Kundla. Sabli 3.4.4 Spatial Cosistecy Check Raifall data exhibit some spatial cosistecy ad this forms the basis of ivestigatig the observed raifall values. A estimate of the iterpolated raifall value at a statio is

More information

Practice Test Problems for Test IV, with Solutions

Practice Test Problems for Test IV, with Solutions Practice Test Problems for Test IV, with Solutios Dr. Holmes May, 2008 The exam will cover sectios 8.2 (revisited) to 8.8. The Taylor remaider formula from 8.9 will ot be o this test. The fact that sums,

More information

Complex Numbers Primer

Complex Numbers Primer Before I get started o this let me first make it clear that this documet is ot iteded to teach you everythig there is to kow about complex umbers. That is a subject that ca (ad does) take a whole course

More information

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting

Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would

More information

Analysis of Experimental Data

Analysis of Experimental Data Aalysis of Experimetal Data 6544597.0479 ± 0.000005 g Quatitative Ucertaity Accuracy vs. Precisio Whe we make a measuremet i the laboratory, we eed to kow how good it is. We wat our measuremets to be both

More information

Series III. Chapter Alternating Series

Series III. Chapter Alternating Series Chapter 9 Series III With the exceptio of the Null Sequece Test, all the tests for series covergece ad divergece that we have cosidered so far have dealt oly with series of oegative terms. Series with

More information

Final Examination Solutions 17/6/2010

Final Examination Solutions 17/6/2010 The Islamic Uiversity of Gaza Faculty of Commerce epartmet of Ecoomics ad Political Scieces A Itroductio to Statistics Course (ECOE 30) Sprig Semester 009-00 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:

More information

7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals

7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals 7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses

More information

ECE 901 Lecture 12: Complexity Regularization and the Squared Loss

ECE 901 Lecture 12: Complexity Regularization and the Squared Loss ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality

More information

Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.

Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1. Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio

More information

Binomial Distribution

Binomial Distribution 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible

More information

Algebra II Notes Unit Seven: Powers, Roots, and Radicals

Algebra II Notes Unit Seven: Powers, Roots, and Radicals Syllabus Objectives: 7. The studets will use properties of ratioal epoets to simplify ad evaluate epressios. 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. b a, the b is the radical.

More information

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 9 Multicolliearity Dr Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Multicolliearity diagostics A importat questio that

More information

Introducing Sample Proportions

Introducing Sample Proportions Itroducig Sample Proportios Probability ad statistics Aswers & Notes TI-Nspire Ivestigatio Studet 60 mi 7 8 9 0 Itroductio A 00 survey of attitudes to climate chage, coducted i Australia by the CSIRO,

More information

TRACEABILITY SYSTEM OF ROCKWELL HARDNESS C SCALE IN JAPAN

TRACEABILITY SYSTEM OF ROCKWELL HARDNESS C SCALE IN JAPAN HARDMEKO 004 Hardess Measuremets Theory ad Applicatio i Laboratories ad Idustries - November, 004, Washigto, D.C., USA TRACEABILITY SYSTEM OF ROCKWELL HARDNESS C SCALE IN JAPAN Koichiro HATTORI, Satoshi

More information

11.6 Absolute Convergence and the Ratio and Root Tests

11.6 Absolute Convergence and the Ratio and Root Tests .6 Absolute Covergece ad the Ratio ad Root Tests The most commo way to test for covergece is to igore ay positive or egative sigs i a series, ad simply test the correspodig series of positive terms. Does

More information

Linear Regression Models

Linear Regression Models Liear Regressio Models Dr. Joh Mellor-Crummey Departmet of Computer Sciece Rice Uiversity johmc@cs.rice.edu COMP 528 Lecture 9 15 February 2005 Goals for Today Uderstad how to Use scatter diagrams to ispect

More information

Statisticians use the word population to refer the total number of (potential) observations under consideration

Statisticians use the word population to refer the total number of (potential) observations under consideration 6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space

More information

Statistical Intervals for a Single Sample

Statistical Intervals for a Single Sample 3/5/06 Applied Statistics ad Probability for Egieers Sixth Editio Douglas C. Motgomery George C. Ruger Chapter 8 Statistical Itervals for a Sigle Sample 8 CHAPTER OUTLINE 8- Cofidece Iterval o the Mea

More information

Sequences of Definite Integrals, Factorials and Double Factorials

Sequences of Definite Integrals, Factorials and Double Factorials 47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology

More information

Lesson 10: Limits and Continuity

Lesson 10: Limits and Continuity www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals

More information

The standard deviation of the mean

The standard deviation of the mean Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider

More information

Chapter 13, Part A Analysis of Variance and Experimental Design

Chapter 13, Part A Analysis of Variance and Experimental Design Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chapter 13, Part A Aalysis of Variace ad Eperimetal Desig Itroductio to Aalysis of Variace Aalysis of Variace: Testig for the Equality of

More information

Summary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram.

Summary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram. Key Cocepts: 1) Sketchig of scatter diagram The scatter diagram of bivariate (i.e. cotaiig two variables) data ca be easily obtaied usig GC. Studets are advised to refer to lecture otes for the GC operatios

More information

The Sample Variance Formula: A Detailed Study of an Old Controversy

The Sample Variance Formula: A Detailed Study of an Old Controversy The Sample Variace Formula: A Detailed Study of a Old Cotroversy Ky M. Vu PhD. AuLac Techologies Ic. c 00 Email: kymvu@aulactechologies.com Abstract The two biased ad ubiased formulae for the sample variace

More information

Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence

Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i

More information

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. XI-1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI-2 (1075) STATISTICAL DECISION MAKING Advaced

More information

UNIT #5. Lesson #2 Arithmetic and Geometric Sequences. Lesson #3 Summation Notation. Lesson #4 Arithmetic Series. Lesson #5 Geometric Series

UNIT #5. Lesson #2 Arithmetic and Geometric Sequences. Lesson #3 Summation Notation. Lesson #4 Arithmetic Series. Lesson #5 Geometric Series UNIT #5 SEQUENCES AND SERIES Lesso # Sequeces Lesso # Arithmetic ad Geometric Sequeces Lesso #3 Summatio Notatio Lesso #4 Arithmetic Series Lesso #5 Geometric Series Lesso #6 Mortgage Paymets COMMON CORE

More information

MAT1026 Calculus II Basic Convergence Tests for Series

MAT1026 Calculus II Basic Convergence Tests for Series MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real

More information

(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?

(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m? MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle

More information

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A)

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A) REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data

More information

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series

More information

How to Maximize a Function without Really Trying

How to Maximize a Function without Really Trying How to Maximize a Fuctio without Really Tryig MARK FLANAGAN School of Electrical, Electroic ad Commuicatios Egieerig Uiversity College Dubli We will prove a famous elemetary iequality called The Rearragemet

More information

THE SYSTEMATIC AND THE RANDOM. ERRORS - DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS

THE SYSTEMATIC AND THE RANDOM. ERRORS - DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS R775 Philips Res. Repts 26,414-423, 1971' THE SYSTEMATIC AND THE RANDOM. ERRORS - DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS by H. W. HANNEMAN Abstract Usig the law of propagatio of errors, approximated

More information

Paired Data and Linear Correlation

Paired Data and Linear Correlation Paired Data ad Liear Correlatio Example. A group of calculus studets has take two quizzes. These are their scores: Studet st Quiz Score ( data) d Quiz Score ( data) 7 5 5 0 3 0 3 4 0 5 5 5 5 6 0 8 7 0

More information

INFINITE SEQUENCES AND SERIES

INFINITE SEQUENCES AND SERIES 11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS

More information

REVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.

REVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains. the Further Mathematics etwork wwwfmetworkorguk V 07 The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values

More information

EDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 4 - CALCULUS

EDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 4 - CALCULUS EDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 4 - CALCULUS TUTORIAL 1 - DIFFERENTIATION Use the elemetary rules of calculus arithmetic to solve problems that ivolve differetiatio

More information

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

More information

Tennessee Department of Education

Tennessee Department of Education Teessee Departmet of Educatio Task: Comparig Shapes Geometry O a piece of graph paper with a coordiate plae, draw three o-colliear poits ad label them A, B, C. (Do ot use the origi as oe of your poits.)

More information

Chapter 6 Sampling Distributions

Chapter 6 Sampling Distributions Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to

More information

Confidence Intervals for the Population Proportion p

Confidence Intervals for the Population Proportion p Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:

More information

P.3 Polynomials and Special products

P.3 Polynomials and Special products Precalc Fall 2016 Sectios P.3, 1.2, 1.3, P.4, 1.4, P.2 (radicals/ratioal expoets), 1.5, 1.6, 1.7, 1.8, 1.1, 2.1, 2.2 I Polyomial defiitio (p. 28) a x + a x +... + a x + a x 1 1 0 1 1 0 a x + a x +... +

More information

Complex Analysis Spring 2001 Homework I Solution

Complex Analysis Spring 2001 Homework I Solution Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle

More information

Sampling Error. Chapter 6 Student Lecture Notes 6-1. Business Statistics: A Decision-Making Approach, 6e. Chapter Goals

Sampling Error. Chapter 6 Student Lecture Notes 6-1. Business Statistics: A Decision-Making Approach, 6e. Chapter Goals Chapter 6 Studet Lecture Notes 6-1 Busiess Statistics: A Decisio-Makig Approach 6 th Editio Chapter 6 Itroductio to Samplig Distributios Chap 6-1 Chapter Goals After completig this chapter, you should

More information

Rademacher Complexity

Rademacher Complexity EECS 598: Statistical Learig Theory, Witer 204 Topic 0 Rademacher Complexity Lecturer: Clayto Scott Scribe: Ya Deg, Kevi Moo Disclaimer: These otes have ot bee subjected to the usual scrutiy reserved for

More information

Appendix F: Complex Numbers

Appendix F: Complex Numbers Appedix F Complex Numbers F1 Appedix F: Complex Numbers Use the imagiary uit i to write complex umbers, ad to add, subtract, ad multiply complex umbers. Fid complex solutios of quadratic equatios. Write

More information

Confidence Intervals QMET103

Confidence Intervals QMET103 Cofidece Itervals QMET103 Library, Teachig ad Learig CONFIDENCE INTERVALS provide a iterval estimate of the ukow populatio parameter. What is a cofidece iterval? Statisticias have a habit of hedgig their

More information

Lecture 1 Probability and Statistics

Lecture 1 Probability and Statistics Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark

More information

4.1 Sigma Notation and Riemann Sums

4.1 Sigma Notation and Riemann Sums 0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

More information

BHW #13 1/ Cooper. ENGR 323 Probabilistic Analysis Beautiful Homework # 13

BHW #13 1/ Cooper. ENGR 323 Probabilistic Analysis Beautiful Homework # 13 BHW # /5 ENGR Probabilistic Aalysis Beautiful Homework # Three differet roads feed ito a particular freeway etrace. Suppose that durig a fixed time period, the umber of cars comig from each road oto the

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

Chapter 4 - Summarizing Numerical Data

Chapter 4 - Summarizing Numerical Data Chapter 4 - Summarizig Numerical Data 15.075 Cythia Rudi Here are some ways we ca summarize data umerically. Sample Mea: i=1 x i x :=. Note: i this class we will work with both the populatio mea µ ad the

More information

WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? ABSTRACT

WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? ABSTRACT WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? Harold G. Loomis Hoolulu, HI ABSTRACT Most coastal locatios have few if ay records of tsuami wave heights obtaied over various time periods. Still

More information

Addition: Property Name Property Description Examples. a+b = b+a. a+(b+c) = (a+b)+c

Addition: Property Name Property Description Examples. a+b = b+a. a+(b+c) = (a+b)+c Notes for March 31 Fields: A field is a set of umbers with two (biary) operatios (usually called additio [+] ad multiplicatio [ ]) such that the followig properties hold: Additio: Name Descriptio Commutativity

More information

Once we have a sequence of numbers, the next thing to do is to sum them up. Given a sequence (a n ) n=1

Once we have a sequence of numbers, the next thing to do is to sum them up. Given a sequence (a n ) n=1 . Ifiite Series Oce we have a sequece of umbers, the ext thig to do is to sum them up. Give a sequece a be a sequece: ca we give a sesible meaig to the followig expressio? a = a a a a While summig ifiitely

More information

DISTRIBUTION LAW Okunev I.V.

DISTRIBUTION LAW Okunev I.V. 1 DISTRIBUTION LAW Okuev I.V. Distributio law belogs to a umber of the most complicated theoretical laws of mathematics. But it is also a very importat practical law. Nothig ca help uderstad complicated

More information

Central Limit Theorem the Meaning and the Usage

Central Limit Theorem the Meaning and the Usage Cetral Limit Theorem the Meaig ad the Usage Covetio about otatio. N, We are usig otatio X is variable with mea ad stadard deviatio. i lieu of sayig that X is a ormal radom Assume a sample of measuremets

More information

Math 116 Practice for Exam 3

Math 116 Practice for Exam 3 Math 6 Practice for Eam 3 Geerated April 4, 26 Name: SOLUTIONS Istructor: Sectio Number:. This eam has questios. Note that the problems are ot of equal difficulty, so you may wat to skip over ad retur

More information

Teaching Mathematics Concepts via Computer Algebra Systems

Teaching Mathematics Concepts via Computer Algebra Systems Iteratioal Joural of Mathematics ad Statistics Ivetio (IJMSI) E-ISSN: 4767 P-ISSN: - 4759 Volume 4 Issue 7 September. 6 PP-- Teachig Mathematics Cocepts via Computer Algebra Systems Osama Ajami Rashaw,

More information

NUMERICAL METHODS FOR SOLVING EQUATIONS

NUMERICAL METHODS FOR SOLVING EQUATIONS Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:

More information

Discrete probability distributions

Discrete probability distributions Discrete probability distributios I the chapter o probability we used the classical method to calculate the probability of various values of a radom variable. I some cases, however, we may be able to develop

More information

Lecture 9: September 19

Lecture 9: September 19 36-700: Probability ad Mathematical Statistics I Fall 206 Lecturer: Siva Balakrisha Lecture 9: September 9 9. Review ad Outlie Last class we discussed: Statistical estimatio broadly Pot estimatio Bias-Variace

More information

Notes on iteration and Newton s method. Iteration

Notes on iteration and Newton s method. Iteration Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f

More information

Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 19

Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 19 CS 70 Discrete Mathematics ad Probability Theory Sprig 2016 Rao ad Walrad Note 19 Some Importat Distributios Recall our basic probabilistic experimet of tossig a biased coi times. This is a very simple

More information

Statistical Fundamentals and Control Charts

Statistical Fundamentals and Control Charts Statistical Fudametals ad Cotrol Charts 1. Statistical Process Cotrol Basics Chace causes of variatio uavoidable causes of variatios Assigable causes of variatio large variatios related to machies, materials,

More information

Lecture 1: Simple descriptive statistics I

Lecture 1: Simple descriptive statistics I Lecture : Simple descriptive statistics I L. Example: Number of egg clutches received by male sticklebacks The three-spie stickleback, Gasterosteus aculeatus, reproduces as follows. Male sticklebacks build

More information

On an Application of Bayesian Estimation

On an Application of Bayesian Estimation O a Applicatio of ayesia Estimatio KIYOHARU TANAKA School of Sciece ad Egieerig, Kiki Uiversity, Kowakae, Higashi-Osaka, JAPAN Email: ktaaka@ifokidaiacjp EVGENIY GRECHNIKOV Departmet of Mathematics, auma

More information

7.1 Finding Rational Solutions of Polynomial Equations

7.1 Finding Rational Solutions of Polynomial Equations Name Class Date 7.1 Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio? Resource Locker Explore Relatig Zeros ad Coefficiets of Polyomial

More information

Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t

Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said

More information

Singular Continuous Measures by Michael Pejic 5/14/10

Singular Continuous Measures by Michael Pejic 5/14/10 Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable

More information

Polynomial Multiplication and Fast Fourier Transform

Polynomial Multiplication and Fast Fourier Transform Polyomial Multiplicatio ad Fast Fourier Trasform Com S 477/577 Notes Ya-Bi Jia Sep 19, 2017 I this lecture we will describe the famous algorithm of fast Fourier trasform FFT, which has revolutioized digital

More information

KLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions

KLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give

More information