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

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.

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR. ANTHONY BROWN 8. Statistics 8.1. Measures of Cetre: Mea, Media ad Mode. If we have a series of umbers the

More information

MEASURES OF DISPERSION (VARIABILITY)

MEASURES OF DISPERSION (VARIABILITY) POLI 300 Hadout #7 N. R. Miller MEASURES OF DISPERSION (VARIABILITY) While measures of cetral tedecy idicate what value of a variable is (i oe sese or other, e.g., mode, media, mea), average or cetral

More information

Chapter 2 Descriptive Statistics

Chapter 2 Descriptive Statistics Chapter 2 Descriptive Statistics Statistics Most commoly, statistics refers to umerical data. Statistics may also refer to the process of collectig, orgaizig, presetig, aalyzig ad iterpretig umerical data

More information

CHAPTER 2. Mean This is the usual arithmetic mean or average and is equal to the sum of the measurements divided by number of measurements.

CHAPTER 2. Mean This is the usual arithmetic mean or average and is equal to the sum of the measurements divided by number of measurements. CHAPTER 2 umerical Measures Graphical method may ot always be sufficiet for describig data. You ca use the data to calculate a set of umbers that will covey a good metal picture of the frequecy distributio.

More information

Chapter If n is odd, the median is the exact middle number If n is even, the median is the average of the two middle numbers

Chapter If n is odd, the median is the exact middle number If n is even, the median is the average of the two middle numbers Chapter 4 4-1 orth Seattle Commuity College BUS10 Busiess Statistics Chapter 4 Descriptive Statistics Summary Defiitios Cetral tedecy: The extet to which the data values group aroud a cetral value. Variatio:

More information

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

6.3 Testing Series With Positive Terms

6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

More information

Economics 250 Assignment 1 Suggested Answers. 1. We have the following data set on the lengths (in minutes) of a sample of long-distance phone calls

Economics 250 Assignment 1 Suggested Answers. 1. We have the following data set on the lengths (in minutes) of a sample of long-distance phone calls Ecoomics 250 Assigmet 1 Suggested Aswers 1. We have the followig data set o the legths (i miutes) of a sample of log-distace phoe calls 1 20 10 20 13 23 3 7 18 7 4 5 15 7 29 10 18 10 10 23 4 12 8 6 (1)

More information

ANALYSIS OF EXPERIMENTAL ERRORS

ANALYSIS OF EXPERIMENTAL ERRORS ANALYSIS OF EXPERIMENTAL ERRORS All physical measuremets ecoutered i the verificatio of physics theories ad cocepts are subject to ucertaities that deped o the measurig istrumets used ad the coditios uder

More information

Anna Janicka Mathematical Statistics 2018/2019 Lecture 1, Parts 1 & 2

Anna Janicka Mathematical Statistics 2018/2019 Lecture 1, Parts 1 & 2 Aa Jaicka Mathematical Statistics 18/19 Lecture 1, Parts 1 & 1. Descriptive Statistics By the term descriptive statistics we will mea the tools used for quatitative descriptio of the properties of a sample

More information

Analysis of Experimental Measurements

Analysis of Experimental Measurements Aalysis of Experimetal Measuremets Thik carefully about the process of makig a measuremet. A measuremet is a compariso betwee some ukow physical quatity ad a stadard of that physical quatity. As a example,

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

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

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

Statistics 511 Additional Materials

Statistics 511 Additional Materials Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability

More information

Random Variables, Sampling and Estimation

Random Variables, Sampling and Estimation Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig

More information

Measures of Spread: Standard Deviation

Measures of Spread: Standard Deviation Measures of Spread: Stadard Deviatio So far i our study of umerical measures used to describe data sets, we have focused o the mea ad the media. These measures of ceter tell us the most typical value of

More information

multiplies all measures of center and the standard deviation and range by k, while the variance is multiplied by k 2.

multiplies all measures of center and the standard deviation and range by k, while the variance is multiplied by k 2. Lesso 3- Lesso 3- Scale Chages of Data Vocabulary scale chage of a data set scale factor scale image BIG IDEA Multiplyig every umber i a data set by k multiplies all measures of ceter ad the stadard deviatio

More information

Discrete Mathematics for CS Spring 2008 David Wagner Note 22

Discrete Mathematics for CS Spring 2008 David Wagner Note 22 CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig

More information

11 Correlation and Regression

11 Correlation and Regression 11 Correlatio Regressio 11.1 Multivariate Data Ofte we look at data where several variables are recorded for the same idividuals or samplig uits. For example, at a coastal weather statio, we might record

More information

BUSINESS STATISTICS (PART-9) AVERAGE OR MEASURES OF CENTRAL TENDENCY: THE GEOMETRIC AND HARMONIC MEANS

BUSINESS STATISTICS (PART-9) AVERAGE OR MEASURES OF CENTRAL TENDENCY: THE GEOMETRIC AND HARMONIC MEANS BUSINESS STATISTICS (PART-9) AVERAGE OR MEASURES OF CENTRAL TENDENCY: THE GEOMETRIC AND HARMONIC MEANS. INTRODUCTION We have so far discussed three measures of cetral tedecy, viz. The Arithmetic Mea, Media

More information

Discrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22

Discrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 22 CS 70 Discrete Mathematics for CS Sprig 2007 Luca Trevisa Lecture 22 Aother Importat Distributio The Geometric Distributio Questio: A biased coi with Heads probability p is tossed repeatedly util the first

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

4.3 Growth Rates of Solutions to Recurrences

4.3 Growth Rates of Solutions to Recurrences 4.3. GROWTH RATES OF SOLUTIONS TO RECURRENCES 81 4.3 Growth Rates of Solutios to Recurreces 4.3.1 Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer.

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

Economics Spring 2015

Economics Spring 2015 1 Ecoomics 400 -- Sprig 015 /17/015 pp. 30-38; Ch. 7.1.4-7. New Stata Assigmet ad ew MyStatlab assigmet, both due Feb 4th Midterm Exam Thursday Feb 6th, Chapters 1-7 of Groeber text ad all relevat lectures

More information

Data Description. Measure of Central Tendency. Data Description. Chapter x i

Data Description. Measure of Central Tendency. Data Description. Chapter x i Data Descriptio Describe Distributio with Numbers Example: Birth weights (i lb) of 5 babies bor from two groups of wome uder differet care programs. Group : 7, 6, 8, 7, 7 Group : 3, 4, 8, 9, Chapter 3

More information

Stat 139 Homework 7 Solutions, Fall 2015

Stat 139 Homework 7 Solutions, Fall 2015 Stat 139 Homework 7 Solutios, Fall 2015 Problem 1. I class we leared that the classical simple liear regressio model assumes the followig distributio of resposes: Y i = β 0 + β 1 X i + ɛ i, i = 1,...,,

More information

Example: Find the SD of the set {x j } = {2, 4, 5, 8, 5, 11, 7}.

Example: Find the SD of the set {x j } = {2, 4, 5, 8, 5, 11, 7}. 1 (*) If a lot of the data is far from the mea, the may of the (x j x) 2 terms will be quite large, so the mea of these terms will be large ad the SD of the data will be large. (*) I particular, outliers

More information

Measures of Variation

Measures of Variation Chapter : Measures of Variatio from Statistical Aalysis i the Behavioral Scieces by James Raymodo Secod Editio 97814669676 01 Copyright Property of Kedall Hut Publishig CHAPTER Measures of Variatio Key

More information

Topic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or

Topic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad

More information

Linear Regression Demystified

Linear Regression Demystified Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to

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

a. For each block, draw a free body diagram. Identify the source of each force in each free body diagram.

a. For each block, draw a free body diagram. Identify the source of each force in each free body diagram. Pre-Lab 4 Tesio & Newto s Third Law Refereces This lab cocers the properties of forces eerted by strigs or cables, called tesio forces, ad the use of Newto s third law to aalyze forces. Physics 2: Tipler

More information

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

Sequences 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 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet

More information

First, note that the LS residuals are orthogonal to the regressors. X Xb X y = 0 ( normal equations ; (k 1) ) So,

First, note that the LS residuals are orthogonal to the regressors. X Xb X y = 0 ( normal equations ; (k 1) ) So, 0 2. OLS Part II The OLS residuals are orthogoal to the regressors. If the model icludes a itercept, the orthogoality of the residuals ad regressors gives rise to three results, which have limited practical

More information

Chapter 23: Inferences About Means

Chapter 23: Inferences About Means Chapter 23: Ifereces About Meas Eough Proportios! We ve spet the last two uits workig with proportios (or qualitative variables, at least) ow it s time to tur our attetios to quatitative variables. For

More information

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals

More information

Lecture 24 Floods and flood frequency

Lecture 24 Floods and flood frequency Lecture 4 Floods ad flood frequecy Oe of the thigs we wat to kow most about rivers is what s the probability that a flood of size will happe this year? I 100 years? There are two ways to do this empirically,

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

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

1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable

More information

Chapter 22. Comparing Two Proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc.

Chapter 22. Comparing Two Proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc. Chapter 22 Comparig Two Proportios Copyright 2010, 2007, 2004 Pearso Educatio, Ic. Comparig Two Proportios Read the first two paragraphs of pg 504. Comparisos betwee two percetages are much more commo

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 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations

Section 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?

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

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

An Introduction to Randomized Algorithms

An Introduction to Randomized Algorithms A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis

More information

The Method of Least Squares. To understand least squares fitting of data.

The Method of Least Squares. To understand least squares fitting of data. The Method of Least Squares KEY WORDS Curve fittig, least square GOAL To uderstad least squares fittig of data To uderstad the least squares solutio of icosistet systems of liear equatios 1 Motivatio Curve

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

Simple Linear Regression

Simple Linear Regression Chapter 2 Simple Liear Regressio 2.1 Simple liear model The simple liear regressio model shows how oe kow depedet variable is determied by a sigle explaatory variable (regressor). Is is writte as: Y i

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

Introduction There are two really interesting things to do in statistics.

Introduction There are two really interesting things to do in statistics. ECON 497 Lecture Notes E Page 1 of 1 Metropolita State Uiversity ECON 497: Research ad Forecastig Lecture Notes E: Samplig Distributios Itroductio There are two really iterestig thigs to do i statistics.

More information

STP 226 EXAMPLE EXAM #1

STP 226 EXAMPLE EXAM #1 STP 226 EXAMPLE EXAM #1 Istructor: Hoor Statemet: I have either give or received iformatio regardig this exam, ad I will ot do so util all exams have bee graded ad retured. PRINTED NAME: Siged Date: DIRECTIONS:

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

September 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1

September 2012 C1 Note. C1 Notes (Edexcel) Copyright   - For AS, A2 notes and IGCSE / GCSE worksheets 1 September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright

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

Continuous Data that can take on any real number (time/length) based on sample data. Categorical data can only be named or categorised

Continuous Data that can take on any real number (time/length) based on sample data. Categorical data can only be named or categorised Questio 1. (Topics 1-3) A populatio cosists of all the members of a group about which you wat to draw a coclusio (Greek letters (μ, σ, Ν) are used) A sample is the portio of the populatio selected for

More information

Curve Sketching Handout #5 Topic Interpretation Rational Functions

Curve Sketching Handout #5 Topic Interpretation Rational Functions Curve Sketchig Hadout #5 Topic Iterpretatio Ratioal Fuctios A ratioal fuctio is a fuctio f that is a quotiet of two polyomials. I other words, p ( ) ( ) f is a ratioal fuctio if p ( ) ad q ( ) are polyomials

More information

TOPIC 6 MEASURES OF VARIATION

TOPIC 6 MEASURES OF VARIATION TOPIC 6 MEASURES OF VARIATIO If people s eyes ted to blak out tables if figures, you ca be dar sure that they blak out the small writig that goes aroud them. Ala Graham, 1994 The cocept of variatio sometimes

More information

Response Variable denoted by y it is the variable that is to be predicted measure of the outcome of an experiment also called the dependent variable

Response Variable denoted by y it is the variable that is to be predicted measure of the outcome of an experiment also called the dependent variable Statistics Chapter 4 Correlatio ad Regressio If we have two (or more) variables we are usually iterested i the relatioship betwee the variables. Associatio betwee Variables Two variables are associated

More information

Chapter 10: Power Series

Chapter 10: Power Series Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because

More information

Chapter 12 Correlation

Chapter 12 Correlation Chapter Correlatio Correlatio is very similar to regressio with oe very importat differece. Regressio is used to explore the relatioship betwee a idepedet variable ad a depedet variable, whereas correlatio

More information

Chapter 4. Fourier Series

Chapter 4. Fourier Series Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,

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

n outcome is (+1,+1, 1,..., 1). Let the r.v. X denote our position (relative to our starting point 0) after n moves. Thus X = X 1 + X 2 + +X n,

n outcome is (+1,+1, 1,..., 1). Let the r.v. X denote our position (relative to our starting point 0) after n moves. Thus X = X 1 + X 2 + +X n, CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 9 Variace Questio: At each time step, I flip a fair coi. If it comes up Heads, I walk oe step to the right; if it comes up Tails, I walk oe

More information

Alternating Series. 1 n 0 2 n n THEOREM 9.14 Alternating Series Test Let a n > 0. The alternating series. 1 n a n.

Alternating Series. 1 n 0 2 n n THEOREM 9.14 Alternating Series Test Let a n > 0. The alternating series. 1 n a n. 0_0905.qxd //0 :7 PM Page SECTION 9.5 Alteratig Series Sectio 9.5 Alteratig Series Use the Alteratig Series Test to determie whether a ifiite series coverges. Use the Alteratig Series Remaider to approximate

More information

Average-Case Analysis of QuickSort

Average-Case Analysis of QuickSort Average-Case Aalysis of QuickSort Comp 363 Fall Semester 003 October 3, 003 The purpose of this documet is to itroduce the idea of usig recurrece relatios to do average-case aalysis. The average-case ruig

More information

µ and π p i.e. Point Estimation x And, more generally, the population proportion is approximately equal to a sample proportion

µ and π p i.e. Point Estimation x And, more generally, the population proportion is approximately equal to a sample proportion Poit Estimatio Poit estimatio is the rather simplistic (ad obvious) process of usig the kow value of a sample statistic as a approximatio to the ukow value of a populatio parameter. So we could for example

More information

Chapter 6 Part 5. Confidence Intervals t distribution chi square distribution. October 23, 2008

Chapter 6 Part 5. Confidence Intervals t distribution chi square distribution. October 23, 2008 Chapter 6 Part 5 Cofidece Itervals t distributio chi square distributio October 23, 2008 The will be o help sessio o Moday, October 27. Goal: To clearly uderstad the lik betwee probability ad cofidece

More information

Parameter, Statistic and Random Samples

Parameter, Statistic and Random Samples Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,

More information

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

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

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

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

CHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS. 8.1 Random Sampling. 8.2 Some Important Statistics

CHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS. 8.1 Random Sampling. 8.2 Some Important Statistics CHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS 8.1 Radom Samplig The basic idea of the statistical iferece is that we are allowed to draw ifereces or coclusios about a populatio based

More information

Chapter 6 Overview: Sequences and Numerical Series. For the purposes of AP, this topic is broken into four basic subtopics:

Chapter 6 Overview: Sequences and Numerical Series. For the purposes of AP, this topic is broken into four basic subtopics: Chapter 6 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals (which is what most studets

More information

Calculus with Analytic Geometry 2

Calculus with Analytic Geometry 2 Calculus with Aalytic Geometry Fial Eam Study Guide ad Sample Problems Solutios The date for the fial eam is December, 7, 4-6:3p.m. BU Note. The fial eam will cosist of eercises, ad some theoretical questios,

More information

x a x a Lecture 2 Series (See Chapter 1 in Boas)

x a x a Lecture 2 Series (See Chapter 1 in Boas) Lecture Series (See Chapter i Boas) A basic ad very powerful (if pedestria, recall we are lazy AD smart) way to solve ay differetial (or itegral) equatio is via a series expasio of the correspodig solutio

More information

Chapter 22. Comparing Two Proportions. Copyright 2010 Pearson Education, Inc.

Chapter 22. Comparing Two Proportions. Copyright 2010 Pearson Education, Inc. Chapter 22 Comparig Two Proportios Copyright 2010 Pearso Educatio, Ic. Comparig Two Proportios Comparisos betwee two percetages are much more commo tha questios about isolated percetages. Ad they are more

More information

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense, 3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [

More information

Lecture 15: Learning Theory: Concentration Inequalities

Lecture 15: Learning Theory: Concentration Inequalities STAT 425: Itroductio to Noparametric Statistics Witer 208 Lecture 5: Learig Theory: Cocetratio Iequalities Istructor: Ye-Chi Che 5. Itroductio Recall that i the lecture o classificatio, we have see that

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

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More information

3 Resampling Methods: The Jackknife

3 Resampling Methods: The Jackknife 3 Resamplig Methods: The Jackkife 3.1 Itroductio I this sectio, much of the cotet is a summary of material from Efro ad Tibshirai (1993) ad Maly (2007). Here are several useful referece texts o resamplig

More information

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

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

ENGI 4421 Probability and Statistics Faculty of Engineering and Applied Science Problem Set 1 Solutions Descriptive Statistics. None at all!

ENGI 4421 Probability and Statistics Faculty of Engineering and Applied Science Problem Set 1 Solutions Descriptive Statistics. None at all! ENGI 44 Probability ad Statistics Faculty of Egieerig ad Applied Sciece Problem Set Solutios Descriptive Statistics. If, i the set of values {,, 3, 4, 5, 6, 7 } a error causes the value 5 to be replaced

More information

Expectation and Variance of a random variable

Expectation and Variance of a random variable Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio

More information

The axial dispersion model for tubular reactors at steady state can be described by the following equations: dc dz R n cn = 0 (1) (2) 1 d 2 c.

The axial dispersion model for tubular reactors at steady state can be described by the following equations: dc dz R n cn = 0 (1) (2) 1 d 2 c. 5.4 Applicatio of Perturbatio Methods to the Dispersio Model for Tubular Reactors The axial dispersio model for tubular reactors at steady state ca be described by the followig equatios: d c Pe dz z =

More information

Error & Uncertainty. Error. More on errors. Uncertainty. Page # The error is the difference between a TRUE value, x, and a MEASURED value, x i :

Error & Uncertainty. Error. More on errors. Uncertainty. Page # The error is the difference between a TRUE value, x, and a MEASURED value, x i : Error Error & Ucertaity The error is the differece betwee a TRUE value,, ad a MEASURED value, i : E = i There is o error-free measuremet. The sigificace of a measuremet caot be judged uless the associate

More information

MATH/STAT 352: Lecture 15

MATH/STAT 352: Lecture 15 MATH/STAT 352: Lecture 15 Sectios 5.2 ad 5.3. Large sample CI for a proportio ad small sample CI for a mea. 1 5.2: Cofidece Iterval for a Proportio Estimatig proportio of successes i a biomial experimet

More information

NAME: ALGEBRA 350 BLOCK 7. Simplifying Radicals Packet PART 1: ROOTS

NAME: ALGEBRA 350 BLOCK 7. Simplifying Radicals Packet PART 1: ROOTS NAME: ALGEBRA 50 BLOCK 7 DATE: Simplifyig Radicals Packet PART 1: ROOTS READ: A square root of a umber b is a solutio of the equatio x = b. Every positive umber b has two square roots, deoted b ad b or

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

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

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

U8L1: Sec Equations of Lines in R 2

U8L1: Sec Equations of Lines in R 2 MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (-D) Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie

More information

SOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1.

SOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1. SOLUTIONS TO EXAM 3 Problem Fid the sum of the followig series 2 + ( ) 5 5 2 5 3 25 2 2 This series diverges Solutio: Note that this defies two coverget geometric series with respective radii r 2/5 < ad

More information

Big Picture. 5. Data, Estimates, and Models: quantifying the accuracy of estimates.

Big Picture. 5. Data, Estimates, and Models: quantifying the accuracy of estimates. 5. Data, Estimates, ad Models: quatifyig the accuracy of estimates. 5. Estimatig a Normal Mea 5.2 The Distributio of the Normal Sample Mea 5.3 Normal data, cofidece iterval for, kow 5.4 Normal data, cofidece

More information