OCR Statistics 1. Working with data. Section 3: Measures of spread

Size: px
Start display at page:

Download "OCR Statistics 1. Working with data. Section 3: Measures of spread"

Transcription

1 Notes ad Eamples OCR Statistics 1 Workig with data Sectio 3: Measures of spread Just as there are several differet measures of cetral tedec (averages), there are a variet of statistical measures of spread. These otes cotai sub-sectios o: The rage Quartiles ad the iter-quartile rage Bo ad whisker plots Shapes of distributios Idetifig outliers usig quartiles Cumulative frequec tables ad curves Variace ad stadard deviatio The alterative form of the sum of squares Variace ad stadard deviatio usig frequec tables Usig stadard deviatio to idetif outliers Usig codig to calculate stadard deviatio The rage For a set of data, rage = highest item lowest item This is straightforward to calculate, but is highl sesitive to etreme values. For eample, cosider this set of marks for a maths test: {45, 50, 43, 49, 5, 58, 48, 10, 50, 8, 56, 40, 47, 39, 51} The rage of the data is 8 10 = 7 marks, but this does ot give a good measure of the spread, as most of the marks are i the rage Quartiles ad the iter-quartile rage Oe wa of refiig the rage so that it does ot rel completel o the most etreme items of data is to use the iterquartile rage. Iterquartile rage = upper quartile lower quartile. The upper quartile is the media of the upper half of the data, ad the lower quartile is the media of the lower half of the data. MEI, 3/06/09 1/14

2 For a large data set, 5% of the data lie below the lower quartile, ad 75% of the data lie below the upper quartile. The iterquartile rage measures the rage of the middle 50% of the data. For small sets of data, ou use a procedure for placig the lower ad the upper quartile, similar to that used for placig the media. Eample 1 (i) Fid the iterquartile rage of the set of marks below from a test take b 15 studets (ii) Oe studet was abset ad took the test the followig week, scorig 59. Fid the ew iterquartile rage. Solutio (i) First arrage the data i order of size: There are 15 items of data, so the media is the 8 th item, which is 49. Discard this The lower quartile is the media of the lower 7 marks, which is 43. The upper quartile is the media of the upper 7 marks, which is 5. So the iterquartile rage is 5 43 = 9. (ii) The ew set of data has 16 items. For a eve umber of data items, the media falls betwee two items of data, so there is o data item to discard: Media 49.5 The lower quartile is the media of the lower 8 marks, which is 44. The upper quartile is the media of the upper 8 marks, which is 54. The iterquartile rage = = 10 MEI, 07/06/10 /14

3 Bo-ad-whisker plots The media ad quartiles ca be displaed graphicall b meas of a boad-whisker plot, or boplot. This gives a etremel useful summar of the data, ad ca be used to compare sets of data. I this diagram, a bo is draw from the lower to the upper quartile, ad a lie draw i the bo showig the positio of the media. Whiskers eted from the lowest value to the highest: Lowest value Lower quartile Media Upper quartile Highest value Draw to scale Eample Compare the followig sets of data usig their bo ad whisker plots. The represet marks out of 100 for two classes. Class A Class B Solutio The rages of marks are similar, but class A has a lower iter-quartile rage tha class B, which suggests that the majorit of the marks are less spread out for Class A. The media ad quartiles for class A are higher tha those for class B, so o average class A did slightl better o the test. Shapes of distributios The shapes of some histograms for data ca be characterised as follows: Smmetrical Positivel skewed Negativel skewed MEI, 07/06/10 3/14

4 Smmetrical datasets have roughl equal amouts of data either side of a cetral value. Positivel skewed data have greater amouts of data clustered aroud a lower value. Negativel skewed data have greater amouts of data clustered aroud a higher value. Skew ca be see if data is displaed i stem ad leaf diagrams or histograms. Boplots ca also be used to detect skew i the data. The diagram below shows the histogram for a positivel skewed dataset, together with its boplot super-imposed. f.d. You ca see from the boplot that the media is closer to the lower quartile tha the upper quartile, or Upper quartile media > media lower quartile I cotrast, here is a egativel skewed dataset: f.d. MEI, 07/06/10 4/14

5 Here, the boplot shows that the media is closer to the upper quartile tha the lower quartile, so Upper quartile media < media lower quartile. Idetifig outliers usig quartiles Oe defiitio of a outlier uses the quartiles ad iterquartile rage. A outlier ca be idetified as follows (IQR stads for iterquartile rage): a data which are 1.5 IQR below the lower quartile; a data which are 1.5 IQR above the upper quartile. For eample, here is the dataset from Eample 1(ii) Lower quartile 44 Media 49.5 Upper quartile 54 The iterquartile rage is = IQR = = IQR below the lower quartile = = 9, so 10 is a possible outlier. 1.5 IQR above the upper quartile = = 69, so 8 is a possible outlier. The Geogebra resource Boplots ad outliers ca be used to eplore the media ad quartiles, ad ivestigate outliers usig the media ad iterquartile rage. Cumulative frequec tables ad curves Cumulative frequec curves are useful for estimatig the quartiles ad the iter-quartile rage of a large data set. The et eample was also used i sectio to fid the media. Here the iterquartile rage is foud as well. MEI, 07/06/10 5/14

6 Eample 3 Estimate the media ad iterquartile rage of the followig dataset, which gives the mass of 100 eggs: Solutio Mass, m (g) Frequec 40 m < m < m < m < m < m < m < m < 80 0 Mass, m (g) Frequec Mass Cumulative frequec m < m < 45 4 m < m < m < m < m < m < 60 m < m < m < m < m < m < m < The cumulative frequec curve is draw below: c.f mass (kg) 5 of the eggs lie below the lower quartile, show b the ellow lie. 50 of the eggs lie below the media, show b the red lie. 75 of the eggs lie below the upper quartile, show b the blue lie. MEI, 07/06/10 6/14

7 Media = 58 Lower quartile = 53 Upper quartile = 66 Iterquartile rage = = 13. Variace ad stadard deviatio Cosider a small set of data: {0, 1, 1, 3, 5} The mea of this data is give b The deviatio of a item of data from the mea is the differece betwee the data item ad the mea, i.e.. The set of deviatios for this set of data is: {, 1, 1, 1, 3} These deviatios give a measure of spread. However, there is o poit i just addig them up, because their sum is alwas zero! Istead, square each deviatio ad add them up. The sum of their squares is deoted S : For the set of data above: S ( ) ( 1) ( 1) I geeral: ( i ) or i1 S S ( ) Dividig this quatit b, the umber of data, gives the variace. The square root of this quatit is called the stadard deviatio. For the set of data above: 16 Variace S 3. 5 S Stadard deviatio I geeral: ( ) Variace MEI, 07/06/10 7/14

8 Stadard deviatio ( ) Eample 4 Calculate the variace ad stadard deviatio of the data {0,, 3, 6, 9} Solutio S (0 4) ( 4) (3 4) (6 4) (9 4) Variace 10 5 Stadard deviatio The alterative form of the sum of squares Whe the mea does ot work out eatl, the deviatios will also be difficult to work with. I this case, it is easier to work with a alterative formula for S : S ( ) For the first dataset {0, 1, 1, 3, 5}: S as before. The variace ad stadard deviatio ca ow be writte i the alterative forms: Variace Stadard deviatio Eample 5 Calculate the variace ad stadard deviatio of the data set {1, 1,, 3, 3, 3, 4}. Sice the mea is ot a roud Solutio umber, it is easier to use the secod forms of the formulae Variace Alwas do the whole calculatio at oce. Do ot use a rouded versio of the mea! MEI, 07/06/10 8/14

9 Stadard deviatio For large sets of data, ou are sometimes give a summar of the data: the values of, ad. Eample 6 A set of sample data is summarised as: = Fid (i) (ii) the mea the stadard deviatio 15. Solutio 140 (i) (ii) stadard deviatio Variace ad stadard deviatio usig frequec tables I sectio, ou saw how the formula for the mea ca be adapted for use with data give i a frequec table: f f I the same wa, the formulae for the measures of spread ca be adapted for data give i a frequec table. Be careful: f² meas square, the multipl b f. S f S Variace f Stadard deviatio S f It is ofte coveiet to set out the calculatio i colums, as show i the followig eample: MEI, 07/06/10 9/14

10 Eample 7 The table below shows the umber of occupats of each house i a small village. Number of occupats Frequec Total 194 Fid the mea ad stadard deviatio of the umber of occupats. Solutio Mea f f ² f² f 194 f 690 f 938 f f f ( 194) Stadard deviatio I practice, of course, calculatios like these ca be carried out much more easil usig a spreadsheet, or b eterig the data ito a calculator (most calculators allow ou to eter either raw data or frequecies, ad the will calculate the various statistical measures for ou). You ca also look at the PowerPoit presetatio Variace ad stadard deviatio, which shows fidig the variace ad stadard deviatio of raw data ad data preseted i a frequec table. MEI, 07/06/10 10/14

11 For practice i fidig stadard deviatio, tr the iteractive questios Mea ad stadard deviatio. If the data is grouped, the ou must use mid-iterval values, just as ou did i estimatig the mea. Remember that the results for measures of spread will also be estimates usig this method. Eample 8 Estimate the mea ad stadard deviatio of the data with the followig frequec distributio: Solutio Weight, w, (grams) Frequec, f 0 w < w < w < w < w < 50 4 w Mid-iterval f f ² f² value, 0 w < w < w < w < w < f 30 f 760 f Mea f ( 30 ) Stadard deviatio Usig stadard deviatio to idetif outliers Stadard deviatio ca be used to idetif outliers, usig the followig rule: All data which are over stadard deviatios awa from the mea are idetified as outliers. Eample 9 Use the stadard deviatio to idetif a outliers i the followig set of data: MEI, 07/06/10 11/14

12 Solutio = S S 7714 Stadard deviatio = stadard deviatios below the mea is stadard deviatios above the mea is So a outliers are below 4.0 or above The ol value outside this rage is 99; so this is the ol outlier. The Geogebra resource Histograms, mea ad stadard deviatio ca be used to eplore the shapes of histograms, ad ivestigate outliers usig the mea ad stadard deviatio. Usig codig to calculate stadard deviatio It is sometimes possible to simplif the calculatios of variace ad stadard deviatio b codig the data, i the same wa as for the mea. You ca trasform the data usig a liear codig: a b a You ca udo this codig: b Sice each data item has bee trasformed usig this codig, the mea of the data udergoes the same trasformatio. So the mea of the coded data,, is related to the mea of the origial data,, b the equatio a b. Sice stadard deviatio is a measure of spread, the addig a to all the items of data does ot affect the stadard deviatio. However, multiplig all the data items b b makes the data b times more spread out tha previousl. So the stadard deviatio of the coded data, s, is related to the stadard deviatio of the origial data, s, b the equatio s bs. For eample, the data set {30, 50, 0, 70, 40, 0, 30, 60} could be simplified b dividig all the data b 10. This meas usig the codig. 10 which gives the ew data set {3, 5,, 7, 4,, 3, 6}. MEI, 07/06/10 1/14

13 You ca fid the mea, ad the stadard deviatio, s, of this ew data set. The, sice = 10, ou ca fid the mea of the origial data usig the equatio 10 ad the stadard deviatio of the origial data usig the equatio s 10s. Alterativel, the umbers could be made smaller b subtractig 0 before 0 dividig b 10. This is the codig 10 which gives the ew data set {1, 3, 0, 5,, 0, 1, 4} You ca fid the mea,, ad the stadard deviatio, s, of this ew data set. The, sice = , ou ca fid the mea of the origial data usig the equatio 10 0 ad the stadard deviatio of the origial data usig the equatio s 10s. Codig is especiall useful whe dealig with grouped data, sice i these cases ou are dealig with mid-iterval values which follow a fied patter. For eample, if ou were dealig with heights grouped as , etc., ou would be workig with mid-iterval values of 104.5, 114.5, 14.5 etc B usig the codig, ou would be workig with values of 0, 1, 10, etc. Eample 10 Use liear codig to calculate the mea ad stadard deviatio of the followig data: Weight, w, (grams) Frequec, f 0 w < w < w < w < w < 50 4 Solutio The mid-iterval values (deoted b ) are 5, 15, 5, etc. A coveiet codig is 5 10 The correspodig values become 0, 1,, f f ² f² f 30 f 61 f 169 MEI, 07/06/10 13/14

14 s f s 10s For practice i usig liear codig, tr the iteractive questios Liear codig. MEI, 07/06/10 14/14

( n+1 2 ) , position=(7+1)/2 =4,(median is observation #4) Median=10lb

( n+1 2 ) , position=(7+1)/2 =4,(median is observation #4) Median=10lb Chapter 3 Descriptive Measures Measures of Ceter (Cetral Tedecy) These measures will tell us where is the ceter of our data or where most typical value of a data set lies Mode the value that occurs most

More information

Name Date Hr. ALGEBRA 1-2 SPRING FINAL MULTIPLE CHOICE REVIEW #1

Name Date Hr. ALGEBRA 1-2 SPRING FINAL MULTIPLE CHOICE REVIEW #1 Name Date Hr. ALGEBRA - SPRING FINAL MULTIPLE CHOICE REVIEW #. The high temperatures for Phoeix i October of 009 are listed below. Which measure of ceter will provide the most accurate estimatio of the

More information

A Resource for Free-standing Mathematics Qualifications

A Resource for Free-standing Mathematics Qualifications Ope.ls The first sheet is show elow. It is set up to show graphs with equatios of the form = m + c At preset the values of m ad c are oth zero. You ca chage these values usig the scroll ars. Leave the

More information

SAMPLE VERSUS POPULATION. Population - consists of all possible measurements that can be made on a particular item or procedure.

SAMPLE VERSUS POPULATION. Population - consists of all possible measurements that can be made on a particular item or procedure. SAMPLE VERSUS POPULATION Populatio - cosists of all possible measuremets that ca be made o a particular item or procedure. Ofte a populatio has a ifiite umber of data elemets Geerally expese to determie

More information

Intermediate Statistics

Intermediate Statistics Gait Learig Guides Itermediate Statistics Data processig & display, Cetral tedecy Author: Raghu M.D. STATISTICS DATA PROCESSING AND DISPLAY Statistics is the study of data or umerical facts of differet

More information

Descriptive Statistics Summary Lists

Descriptive Statistics Summary Lists Chapter 209 Descriptive Statistics Summary Lists Itroductio This procedure is used to summarize cotiuous data. Large volumes of such data may be easily summarized i statistical lists of meas, couts, stadard

More information

Name Date Hr. ALGEBRA 1-2 SPRING FINAL MULTIPLE CHOICE REVIEW #2

Name Date Hr. ALGEBRA 1-2 SPRING FINAL MULTIPLE CHOICE REVIEW #2 Name Date Hr. ALGEBRA - SPRING FINAL MULTIPLE CHOICE REVIEW # 5. Which measure of ceter is most appropriate for the followig data set? {7, 7, 75, 77,, 9, 9, 90} Mea Media Stadard Deviatio Rage 5. The umber

More information

Normal Distributions

Normal Distributions Normal Distributios Stacey Hacock Look at these three differet data sets Each histogram is overlaid with a curve : A B C A) Weights (g) of ewly bor lab rat pups B) Mea aual temperatures ( F ) i A Arbor,

More information

SD vs. SD + One of the most important uses of sample statistics is to estimate the corresponding population parameters.

SD vs. SD + One of the most important uses of sample statistics is to estimate the corresponding population parameters. SD vs. SD + Oe of the most importat uses of sample statistics is to estimate the correspodig populatio parameters. The mea of a represetative sample is a good estimate of the mea of the populatio that

More information

Arithmetic Sequences

Arithmetic Sequences . Arithmetic Sequeces COMMON CORE Learig Stadards HSF-IF.A. HSF-BF.A.1a HSF-BF.A. HSF-LE.A. Essetial Questio How ca you use a arithmetic sequece to describe a patter? A arithmetic sequece is a ordered

More information

ENGI 4421 Probability and Statistics Faculty of Engineering and Applied Science Problem Set 1 Descriptive Statistics

ENGI 4421 Probability and Statistics Faculty of Engineering and Applied Science Problem Set 1 Descriptive Statistics ENGI 44 Probability ad Statistics Faculty of Egieerig ad Applied Sciece Problem Set Descriptive Statistics. If, i the set of values {,, 3, 4, 5, 6, 7 } a error causes the value 5 to be replaced by 50,

More information

UNIT 4 Section 8 Estimating Population Parameters using Confidence Intervals

UNIT 4 Section 8 Estimating Population Parameters using Confidence Intervals UNIT 4 Sectio 8 Estimatig Populatio Parameters usig Cofidece Itervals To make ifereces about a populatio that caot be surveyed etirely, sample statistics ca be take from a SRS of the populatio ad used

More information

Math 3201 Notes Chapter 4: Rational Expressions & Equations

Math 3201 Notes Chapter 4: Rational Expressions & Equations Learig Goals: See p. tet.. Equivalet Ratioal Epressios ( classes) Read Goal p. 6 tet. Math 0 Notes Chapter : Ratioal Epressios & Equatios. Defie ad give a eample of a ratioal epressio. p. 6. Defie o-permissible

More information

Section 7.2: Direction Fields and Euler s Methods

Section 7.2: Direction Fields and Euler s Methods Sectio 7.: Directio ields ad Euler s Methods Practice HW from Stewart Tetbook ot to had i p. 5 # -3 9-3 odd or a give differetial equatio we wat to look at was to fid its solutio. I this chapter we will

More information

The Closest Line to a Data Set in the Plane. David Gurney Southeastern Louisiana University Hammond, Louisiana

The Closest Line to a Data Set in the Plane. David Gurney Southeastern Louisiana University Hammond, Louisiana The Closest Lie to a Data Set i the Plae David Gurey Southeaster Louisiaa Uiversity Hammod, Louisiaa ABSTRACT This paper looks at three differet measures of distace betwee a lie ad a data set i the plae:

More information

Recursive Procedures. How can you model the relationship between consecutive terms of a sequence?

Recursive Procedures. How can you model the relationship between consecutive terms of a sequence? 6. Recursive Procedures I Sectio 6.1, you used fuctio otatio to write a explicit formula to determie the value of ay term i a Sometimes it is easier to calculate oe term i a sequece usig the previous terms.

More information

Describing data with graphics and numbers

Describing data with graphics and numbers Describig data with graphics ad umbers Types of Data Categorical Variables also kow as class variables, omial variables Quatitative Variables aka umerical ariables either cotiuous or discrete. Graphig

More information

Performance Plus Software Parameter Definitions

Performance Plus Software Parameter Definitions Performace Plus+ Software Parameter Defiitios/ Performace Plus Software Parameter Defiitios Chapma Techical Note-TG-5 paramete.doc ev-0-03 Performace Plus+ Software Parameter Defiitios/2 Backgroud ad Defiitios

More information

Pattern Recognition Systems Lab 1 Least Mean Squares

Pattern Recognition Systems Lab 1 Least Mean Squares Patter Recogitio Systems Lab 1 Least Mea Squares 1. Objectives This laboratory work itroduces the OpeCV-based framework used throughout the course. I this assigmet a lie is fitted to a set of poits usig

More information

Ones Assignment Method for Solving Traveling Salesman Problem

Ones Assignment Method for Solving Traveling Salesman Problem Joural of mathematics ad computer sciece 0 (0), 58-65 Oes Assigmet Method for Solvig Travelig Salesma Problem Hadi Basirzadeh Departmet of Mathematics, Shahid Chamra Uiversity, Ahvaz, Ira Article history:

More information

MATHEMATICAL METHODS OF ANALYSIS AND EXPERIMENTAL DATA PROCESSING (Or Methods of Curve Fitting)

MATHEMATICAL METHODS OF ANALYSIS AND EXPERIMENTAL DATA PROCESSING (Or Methods of Curve Fitting) MATHEMATICAL METHODS OF ANALYSIS AND EXPERIMENTAL DATA PROCESSING (Or Methods of Curve Fittig) I this chapter, we will eamie some methods of aalysis ad data processig; data obtaied as a result of a give

More information

9.1. Sequences and Series. Sequences. What you should learn. Why you should learn it. Definition of Sequence

9.1. Sequences and Series. Sequences. What you should learn. Why you should learn it. Definition of Sequence _9.qxd // : AM Page Chapter 9 Sequeces, Series, ad Probability 9. Sequeces ad Series What you should lear Use sequece otatio to write the terms of sequeces. Use factorial otatio. Use summatio otatio to

More information

Improving Template Based Spike Detection

Improving Template Based Spike Detection Improvig Template Based Spike Detectio Kirk Smith, Member - IEEE Portlad State Uiversity petra@ee.pdx.edu Abstract Template matchig algorithms like SSE, Covolutio ad Maximum Likelihood are well kow for

More information

Numerical Methods Lecture 6 - Curve Fitting Techniques

Numerical Methods Lecture 6 - Curve Fitting Techniques Numerical Methods Lecture 6 - Curve Fittig Techiques Topics motivatio iterpolatio liear regressio higher order polyomial form expoetial form Curve fittig - motivatio For root fidig, we used a give fuctio

More information

Counting Regions in the Plane and More 1

Counting Regions in the Plane and More 1 Coutig Regios i the Plae ad More 1 by Zvezdelia Stakova Berkeley Math Circle Itermediate I Group September 016 1. Overarchig Problem Problem 1 Regios i a Circle. The vertices of a polygos are arraged o

More information

Alpha Individual Solutions MAΘ National Convention 2013

Alpha Individual Solutions MAΘ National Convention 2013 Alpha Idividual Solutios MAΘ Natioal Covetio 0 Aswers:. D. A. C 4. D 5. C 6. B 7. A 8. C 9. D 0. B. B. A. D 4. C 5. A 6. C 7. B 8. A 9. A 0. C. E. B. D 4. C 5. A 6. D 7. B 8. C 9. D 0. B TB. 570 TB. 5

More information

EVALUATION OF TRIGONOMETRIC FUNCTIONS

EVALUATION OF TRIGONOMETRIC FUNCTIONS EVALUATION OF TRIGONOMETRIC FUNCTIONS Whe first exposed to trigoometric fuctios i high school studets are expected to memorize the values of the trigoometric fuctios of sie cosie taget for the special

More information

Math Section 2.2 Polynomial Functions

Math Section 2.2 Polynomial Functions Math 1330 - Sectio. Polyomial Fuctios Our objectives i workig with polyomial fuctios will be, first, to gather iformatio about the graph of the fuctio ad, secod, to use that iformatio to geerate a reasoably

More information

1.8 What Comes Next? What Comes Later?

1.8 What Comes Next? What Comes Later? 35 1.8 What Comes Next? What Comes Later? A Practice Uderstadig Task For each of the followig tables, CC BY Hiroaki Maeda https://flic.kr/p/6r8odk describe how to fid the ext term i the sequece, write

More information

EM375 STATISTICS AND MEASUREMENT UNCERTAINTY LEAST SQUARES LINEAR REGRESSION ANALYSIS

EM375 STATISTICS AND MEASUREMENT UNCERTAINTY LEAST SQUARES LINEAR REGRESSION ANALYSIS EM375 STATISTICS AND MEASUREMENT UNCERTAINTY LEAST SQUARES LINEAR REGRESSION ANALYSIS I this uit of the course we ivestigate fittig a straight lie to measured (x, y) data pairs. The equatio we wat to fit

More information

Data Analysis. Concepts and Techniques. Chapter 2. Chapter 2: Getting to Know Your Data. Data Objects and Attribute Types

Data Analysis. Concepts and Techniques. Chapter 2. Chapter 2: Getting to Know Your Data. Data Objects and Attribute Types Data Aalysis Cocepts ad Techiques Chapter 2 1 Chapter 2: Gettig to Kow Your Data Data Objects ad Attribute Types Basic Statistical Descriptios of Data Data Visualizatio Measurig Data Similarity ad Dissimilarity

More information

Civil Engineering Computation

Civil Engineering Computation Civil Egieerig Computatio Fidig Roots of No-Liear Equatios March 14, 1945 World War II The R.A.F. first operatioal use of the Grad Slam bomb, Bielefeld, Germay. Cotets 2 Root basics Excel solver Newto-Raphso

More information

FURTHER INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)

FURTHER INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS) Mathematics Revisio Guides More Trigoometric ad Log Itegrals Page of 7 MK HOME TUITION Mathematics Revisio Guides Level: AS / A Level AQA : C Edexcel: C OCR: C OCR MEI: C FURTHER INTEGRATION TECHNIQUES

More information

Polynomial Functions and Models. Learning Objectives. Polynomials. P (x) = a n x n + a n 1 x n a 1 x + a 0, a n 0

Polynomial Functions and Models. Learning Objectives. Polynomials. P (x) = a n x n + a n 1 x n a 1 x + a 0, a n 0 Polyomial Fuctios ad Models 1 Learig Objectives 1. Idetify polyomial fuctios ad their degree 2. Graph polyomial fuctios usig trasformatios 3. Idetify the real zeros of a polyomial fuctio ad their multiplicity

More information

. Written in factored form it is easy to see that the roots are 2, 2, i,

. Written in factored form it is easy to see that the roots are 2, 2, i, CMPS A Itroductio to Programmig Programmig Assigmet 4 I this assigmet you will write a java program that determies the real roots of a polyomial that lie withi a specified rage. Recall that the roots (or

More information

The isoperimetric problem on the hypercube

The isoperimetric problem on the hypercube The isoperimetric problem o the hypercube Prepared by: Steve Butler November 2, 2005 1 The isoperimetric problem We will cosider the -dimesioal hypercube Q Recall that the hypercube Q is a graph whose

More information

Parabolic Path to a Best Best-Fit Line:

Parabolic Path to a Best Best-Fit Line: Studet Activity : Fidig the Least Squares Regressio Lie By Explorig the Relatioship betwee Slope ad Residuals Objective: How does oe determie a best best-fit lie for a set of data? Eyeballig it may be

More information

condition w i B i S maximum u i

condition w i B i S maximum u i ecture 10 Dyamic Programmig 10.1 Kapsack Problem November 1, 2004 ecturer: Kamal Jai Notes: Tobias Holgers We are give a set of items U = {a 1, a 2,..., a }. Each item has a weight w i Z + ad a utility

More information

A New Morphological 3D Shape Decomposition: Grayscale Interframe Interpolation Method

A New Morphological 3D Shape Decomposition: Grayscale Interframe Interpolation Method A ew Morphological 3D Shape Decompositio: Grayscale Iterframe Iterpolatio Method D.. Vizireau Politehica Uiversity Bucharest, Romaia ae@comm.pub.ro R. M. Udrea Politehica Uiversity Bucharest, Romaia mihea@comm.pub.ro

More information

IMP: Superposer Integrated Morphometrics Package Superposition Tool

IMP: Superposer Integrated Morphometrics Package Superposition Tool IMP: Superposer Itegrated Morphometrics Package Superpositio Tool Programmig by: David Lieber ( 03) Caisius College 200 Mai St. Buffalo, NY 4208 Cocept by: H. David Sheets, Dept. of Physics, Caisius College

More information

Chapter 3 Descriptive Statistics Numerical Summaries

Chapter 3 Descriptive Statistics Numerical Summaries Secto 3.1 Chapter 3 Descrptve Statstcs umercal Summares Measures of Cetral Tedecy 1. Mea (Also called the Arthmetc Mea) The mea of a data set s the sum of the observatos dvded by the umber of observatos.

More information

The number n of subintervals times the length h of subintervals gives length of interval (b-a).

The number n of subintervals times the length h of subintervals gives length of interval (b-a). Simulator with MadMath Kit: Riema Sums (Teacher s pages) I your kit: 1. GeoGebra file: Ready-to-use projector sized simulator: RiemaSumMM.ggb 2. RiemaSumMM.pdf (this file) ad RiemaSumMMEd.pdf (educator's

More information

Learning to Shoot a Goal Lecture 8: Learning Models and Skills

Learning to Shoot a Goal Lecture 8: Learning Models and Skills Learig to Shoot a Goal Lecture 8: Learig Models ad Skills How do we acquire skill at shootig goals? CS 344R/393R: Robotics Bejami Kuipers Learig to Shoot a Goal The robot eeds to shoot the ball i the goal.

More information

Here are the coefficients of the terms listed above: 3,5,2,1,1 respectively.

Here are the coefficients of the terms listed above: 3,5,2,1,1 respectively. *. Operatios with Poloials: Let s start b defiig soe words. Ter: A ter is a uber, variable or the product of a uber ad variable(s). For eaple:,, z, a Coefficiet: A coefficiet is the ueric factor of the

More information

Module 8-7: Pascal s Triangle and the Binomial Theorem

Module 8-7: Pascal s Triangle and the Binomial Theorem Module 8-7: Pascal s Triagle ad the Biomial Theorem Gregory V. Bard April 5, 017 A Note about Notatio Just to recall, all of the followig mea the same thig: ( 7 7C 4 C4 7 7C4 5 4 ad they are (all proouced

More information

Ch 9.3 Geometric Sequences and Series Lessons

Ch 9.3 Geometric Sequences and Series Lessons Ch 9.3 Geometric Sequeces ad Series Lessos SKILLS OBJECTIVES Recogize a geometric sequece. Fid the geeral, th term of a geometric sequece. Evaluate a fiite geometric series. Evaluate a ifiite geometric

More information

Consider the following population data for the state of California. Year Population

Consider the following population data for the state of California. Year Population Assigmets for Bradie Fall 2016 for Chapter 5 Assigmet sheet for Sectios 5.1, 5.3, 5.5, 5.6, 5.7, 5.8 Read Pages 341-349 Exercises for Sectio 5.1 Lagrage Iterpolatio #1, #4, #7, #13, #14 For #1 use MATLAB

More information

ECE4050 Data Structures and Algorithms. Lecture 6: Searching

ECE4050 Data Structures and Algorithms. Lecture 6: Searching ECE4050 Data Structures ad Algorithms Lecture 6: Searchig 1 Search Give: Distict keys k 1, k 2,, k ad collectio L of records of the form (k 1, I 1 ), (k 2, I 2 ),, (k, I ) where I j is the iformatio associated

More information

Learning Log Title: CHAPTER 7: PROPORTIONS AND PERCENTS. Date: Lesson: Chapter 7: Proportions and Percents

Learning Log Title: CHAPTER 7: PROPORTIONS AND PERCENTS. Date: Lesson: Chapter 7: Proportions and Percents Chapter 7: Proportions and Percents CHAPTER 7: PROPORTIONS AND PERCENTS Date: Lesson: Learning Log Title: Date: Lesson: Learning Log Title: Chapter 7: Proportions and Percents Date: Lesson: Learning Log

More information

Linear Time-Invariant Systems

Linear Time-Invariant Systems 9/9/00 LIEAR TIE-IVARIAT SYSTES Uit, d Part Liear Time-Ivariat Sstems A importat class of discrete-time sstem cosists of those that are Liear Priciple of superpositio Time-ivariat dela of the iput sequece

More information

Project 2.5 Improved Euler Implementation

Project 2.5 Improved Euler Implementation Project 2.5 Improved Euler Implemetatio Figure 2.5.10 i the text lists TI-85 ad BASIC programs implemetig the improved Euler method to approximate the solutio of the iitial value problem dy dx = x+ y,

More information

n n B. How many subsets of C are there of cardinality n. We are selecting elements for such a

n n B. How many subsets of C are there of cardinality n. We are selecting elements for such a 4. [10] Usig a combiatorial argumet, prove that for 1: = 0 = Let A ad B be disjoit sets of cardiality each ad C = A B. How may subsets of C are there of cardiality. We are selectig elemets for such a subset

More information

MAXIMUM MATCHINGS IN COMPLETE MULTIPARTITE GRAPHS

MAXIMUM MATCHINGS IN COMPLETE MULTIPARTITE GRAPHS Fura Uiversity Electroic Joural of Udergraduate Matheatics Volue 00, 1996 6-16 MAXIMUM MATCHINGS IN COMPLETE MULTIPARTITE GRAPHS DAVID SITTON Abstract. How ay edges ca there be i a axiu atchig i a coplete

More information

Image Segmentation EEE 508

Image Segmentation EEE 508 Image Segmetatio Objective: to determie (etract) object boudaries. It is a process of partitioig a image ito distict regios by groupig together eighborig piels based o some predefied similarity criterio.

More information

The Graphs of Polynomial Functions

The Graphs of Polynomial Functions Sectio 4.3 The Graphs of Polyomial Fuctios Objective 1: Uderstadig the Defiitio of a Polyomial Fuctio Defiitio Polyomial Fuctio 1 2 The fuctio ax a 1x a 2x a1x a0 is a polyomial fuctio of degree where

More information

The Nature of Light. Chapter 22. Geometric Optics Using a Ray Approximation. Ray Approximation

The Nature of Light. Chapter 22. Geometric Optics Using a Ray Approximation. Ray Approximation The Nature of Light Chapter Reflectio ad Refractio of Light Sectios: 5, 8 Problems: 6, 7, 4, 30, 34, 38 Particles of light are called photos Each photo has a particular eergy E = h ƒ h is Plack s costat

More information

CIS 121 Data Structures and Algorithms with Java Fall Big-Oh Notation Tuesday, September 5 (Make-up Friday, September 8)

CIS 121 Data Structures and Algorithms with Java Fall Big-Oh Notation Tuesday, September 5 (Make-up Friday, September 8) CIS 11 Data Structures ad Algorithms with Java Fall 017 Big-Oh Notatio Tuesday, September 5 (Make-up Friday, September 8) Learig Goals Review Big-Oh ad lear big/small omega/theta otatios Practice solvig

More information

Homework 1 Solutions MA 522 Fall 2017

Homework 1 Solutions MA 522 Fall 2017 Homework 1 Solutios MA 5 Fall 017 1. Cosider the searchig problem: Iput A sequece of umbers A = [a 1,..., a ] ad a value v. Output A idex i such that v = A[i] or the special value NIL if v does ot appear

More information

Counting II 3, 7 3, 2 3, 9 7, 2 7, 9 2, 9

Counting II 3, 7 3, 2 3, 9 7, 2 7, 9 2, 9 Coutig II Sometimes we will wat to choose objects from a set of objects, ad we wo t be iterested i orderig them For example, if you are leavig for vacatio ad you wat to pac your suitcase with three of

More information

Lecture Notes 6 Introduction to algorithm analysis CSS 501 Data Structures and Object-Oriented Programming

Lecture Notes 6 Introduction to algorithm analysis CSS 501 Data Structures and Object-Oriented Programming Lecture Notes 6 Itroductio to algorithm aalysis CSS 501 Data Structures ad Object-Orieted Programmig Readig for this lecture: Carrao, Chapter 10 To be covered i this lecture: Itroductio to algorithm aalysis

More information

ENGR Spring Exam 1

ENGR Spring Exam 1 ENGR 300 Sprig 03 Exam INSTRUCTIONS: Duratio: 60 miutes Keep your eyes o your ow work! Keep your work covered at all times!. Each studet is resposible for followig directios. Read carefully.. MATLAB ad

More information

Xbar/R Chart for x1-x3

Xbar/R Chart for x1-x3 Chapter 6 Selected roblem Solutios Sectio 6-5 6- a) X-bar ad Rage - Iitial Study Chartig roblem 6- X-bar Rage ----- ----- UCL:. sigma 7.4 UCL:. sigma 5.79 Ceterlie 5.9 Ceterlie.5 LCL: -. sigma.79 LCL:

More information

Chapter 3 Classification of FFT Processor Algorithms

Chapter 3 Classification of FFT Processor Algorithms Chapter Classificatio of FFT Processor Algorithms The computatioal complexity of the Discrete Fourier trasform (DFT) is very high. It requires () 2 complex multiplicatios ad () complex additios [5]. As

More information

Package popkorn. R topics documented: February 20, Type Package

Package popkorn. R topics documented: February 20, Type Package Type Pacage Pacage popkor February 20, 2015 Title For iterval estimatio of mea of selected populatios Versio 0.3-0 Date 2014-07-04 Author Vi Gopal, Claudio Fuetes Maitaier Vi Gopal Depeds

More information

1.2 Binomial Coefficients and Subsets

1.2 Binomial Coefficients and Subsets 1.2. BINOMIAL COEFFICIENTS AND SUBSETS 13 1.2 Biomial Coefficiets ad Subsets 1.2-1 The loop below is part of a program to determie the umber of triagles formed by poits i the plae. for i =1 to for j =

More information

3D Model Retrieval Method Based on Sample Prediction

3D Model Retrieval Method Based on Sample Prediction 20 Iteratioal Coferece o Computer Commuicatio ad Maagemet Proc.of CSIT vol.5 (20) (20) IACSIT Press, Sigapore 3D Model Retrieval Method Based o Sample Predictio Qigche Zhag, Ya Tag* School of Computer

More information

9 x and g(x) = 4. x. Find (x) 3.6. I. Combining Functions. A. From Equations. Example: Let f(x) = and its domain. Example: Let f(x) = and g(x) = x x 4

9 x and g(x) = 4. x. Find (x) 3.6. I. Combining Functions. A. From Equations. Example: Let f(x) = and its domain. Example: Let f(x) = and g(x) = x x 4 1 3.6 I. Combiig Fuctios A. From Equatios Example: Let f(x) = 9 x ad g(x) = 4 f x. Fid (x) g ad its domai. 4 Example: Let f(x) = ad g(x) = x x 4. Fid (f-g)(x) B. From Graphs: Graphical Additio. Example:

More information

Lecture 1: Introduction and Strassen s Algorithm

Lecture 1: Introduction and Strassen s Algorithm 5-750: Graduate Algorithms Jauary 7, 08 Lecture : Itroductio ad Strasse s Algorithm Lecturer: Gary Miller Scribe: Robert Parker Itroductio Machie models I this class, we will primarily use the Radom Access

More information

Creating Exact Bezier Representations of CST Shapes. David D. Marshall. California Polytechnic State University, San Luis Obispo, CA , USA

Creating Exact Bezier Representations of CST Shapes. David D. Marshall. California Polytechnic State University, San Luis Obispo, CA , USA Creatig Exact Bezier Represetatios of CST Shapes David D. Marshall Califoria Polytechic State Uiversity, Sa Luis Obispo, CA 93407-035, USA The paper presets a method of expressig CST shapes pioeered by

More information

ENGR 132. Fall Exam 1

ENGR 132. Fall Exam 1 ENGR 3 Fall 03 Exam INSTRUCTIONS: Duratio: 60 miutes Keep your eyes o your ow work. Keep your work covered at all times.. Each studet is resposible for followig directios. Read carefully.. MATLAB ad Excel

More information

Intro to Scientific Computing: Solutions

Intro to Scientific Computing: Solutions Itro to Scietific Computig: Solutios Dr. David M. Goulet. How may steps does it take to separate 3 objects ito groups of 4? We start with 5 objects ad apply 3 steps of the algorithm to reduce the pile

More information

Chapter 5. Functions for All Subtasks. Copyright 2015 Pearson Education, Ltd.. All rights reserved.

Chapter 5. Functions for All Subtasks. Copyright 2015 Pearson Education, Ltd.. All rights reserved. Chapter 5 Fuctios for All Subtasks Copyright 2015 Pearso Educatio, Ltd.. All rights reserved. Overview 5.1 void Fuctios 5.2 Call-By-Referece Parameters 5.3 Usig Procedural Abstractio 5.4 Testig ad Debuggig

More information

An (or ) is a sequence in which each term after the first differs from the preceding term by a fixed constant, called the.

An (or ) is a sequence in which each term after the first differs from the preceding term by a fixed constant, called the. Sectio.2 Arithmetic Sequeces ad Series -.2 Arithmetic Sequeces ad Series Arithmetic Sequeces Arithmetic Series Key Terms: arithmetic sequece (arithmetic progressio), commo differece, arithmetic series

More information

Designing a learning system

Designing a learning system CS 75 Itro to Machie Learig Lecture Desigig a learig system Milos Hauskrecht milos@pitt.edu 539 Seott Square, -5 people.cs.pitt.edu/~milos/courses/cs75/ Admiistrivia No homework assigmet this week Please

More information

CHAPTER IV: GRAPH THEORY. Section 1: Introduction to Graphs

CHAPTER IV: GRAPH THEORY. Section 1: Introduction to Graphs CHAPTER IV: GRAPH THEORY Sectio : Itroductio to Graphs Sice this class is called Number-Theoretic ad Discrete Structures, it would be a crime to oly focus o umber theory regardless how woderful those topics

More information

Designing a learning system

Designing a learning system CS 75 Machie Learig Lecture Desigig a learig system Milos Hauskrecht milos@cs.pitt.edu 539 Seott Square, x-5 people.cs.pitt.edu/~milos/courses/cs75/ Admiistrivia No homework assigmet this week Please try

More information

Lecture 28: Data Link Layer

Lecture 28: Data Link Layer Automatic Repeat Request (ARQ) 2. Go ack N ARQ Although the Stop ad Wait ARQ is very simple, you ca easily show that it has very the low efficiecy. The low efficiecy comes from the fact that the trasmittig

More information

LU Decomposition Method

LU Decomposition Method SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS LU Decompositio Method Jamie Traha, Autar Kaw, Kevi Marti Uiversity of South Florida Uited States of America kaw@eg.usf.edu http://umericalmethods.eg.usf.edu Itroductio

More information

Python Programming: An Introduction to Computer Science

Python Programming: An Introduction to Computer Science Pytho Programmig: A Itroductio to Computer Sciece Chapter 6 Defiig Fuctios Pytho Programmig, 2/e 1 Objectives To uderstad why programmers divide programs up ito sets of cooperatig fuctios. To be able to

More information

15-859E: Advanced Algorithms CMU, Spring 2015 Lecture #2: Randomized MST and MST Verification January 14, 2015

15-859E: Advanced Algorithms CMU, Spring 2015 Lecture #2: Randomized MST and MST Verification January 14, 2015 15-859E: Advaced Algorithms CMU, Sprig 2015 Lecture #2: Radomized MST ad MST Verificatio Jauary 14, 2015 Lecturer: Aupam Gupta Scribe: Yu Zhao 1 Prelimiaries I this lecture we are talkig about two cotets:

More information

Optimum Solution of Quadratic Programming Problem: By Wolfe s Modified Simplex Method

Optimum Solution of Quadratic Programming Problem: By Wolfe s Modified Simplex Method Volume VI, Issue III, March 7 ISSN 78-5 Optimum Solutio of Quadratic Programmig Problem: By Wolfe s Modified Simple Method Kalpaa Lokhade, P. G. Khot & N. W. Khobragade, Departmet of Mathematics, MJP Educatioal

More information

The Magma Database file formats

The Magma Database file formats The Magma Database file formats Adrew Gaylard, Bret Pikey, ad Mart-Mari Breedt Johaesburg, South Africa 15th May 2006 1 Summary Magma is a ope-source object database created by Chris Muller, of Kasas City,

More information

EE123 Digital Signal Processing

EE123 Digital Signal Processing Last Time EE Digital Sigal Processig Lecture 7 Block Covolutio, Overlap ad Add, FFT Discrete Fourier Trasform Properties of the Liear covolutio through circular Today Liear covolutio with Overlap ad add

More information

Sorting in Linear Time. Data Structures and Algorithms Andrei Bulatov

Sorting in Linear Time. Data Structures and Algorithms Andrei Bulatov Sortig i Liear Time Data Structures ad Algorithms Adrei Bulatov Algorithms Sortig i Liear Time 7-2 Compariso Sorts The oly test that all the algorithms we have cosidered so far is compariso The oly iformatio

More information

Investigation Monitoring Inventory

Investigation Monitoring Inventory Ivestigatio Moitorig Ivetory Name Period Date Art Smith has bee providig the prits of a egravig to FieArt Gallery. He plas to make just 2000 more prits. FieArt has already received 70 of Art s prits. The

More information

27 Refraction, Dispersion, Internal Reflection

27 Refraction, Dispersion, Internal Reflection Chapter 7 Refractio, Dispersio, Iteral Reflectio 7 Refractio, Dispersio, Iteral Reflectio Whe we talked about thi film iterferece, we said that whe light ecouters a smooth iterface betwee two trasparet

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES SEQUENCES AND SERIES U N I The umber of gifts set i the popular Christmas Carol days of Christmas form a sequece. A part of the sog goes this way O the th day of Christmas my true love gave to me drummers

More information

Lecture 7 7 Refraction and Snell s Law Reading Assignment: Read Kipnis Chapter 4 Refraction of Light, Section III, IV

Lecture 7 7 Refraction and Snell s Law Reading Assignment: Read Kipnis Chapter 4 Refraction of Light, Section III, IV Lecture 7 7 Refractio ad Sell s Law Readig Assigmet: Read Kipis Chapter 4 Refractio of Light, Sectio III, IV 7. History I Eglish-speakig coutries, the law of refractio is kow as Sell s Law, after the Dutch

More information

EE University of Minnesota. Midterm Exam #1. Prof. Matthew O'Keefe TA: Eric Seppanen. Department of Electrical and Computer Engineering

EE University of Minnesota. Midterm Exam #1. Prof. Matthew O'Keefe TA: Eric Seppanen. Department of Electrical and Computer Engineering EE 4363 1 Uiversity of Miesota Midterm Exam #1 Prof. Matthew O'Keefe TA: Eric Seppae Departmet of Electrical ad Computer Egieerig Uiversity of Miesota Twi Cities Campus EE 4363 Itroductio to Microprocessors

More information

Octahedral Graph Scaling

Octahedral Graph Scaling Octahedral Graph Scalig Peter Russell Jauary 1, 2015 Abstract There is presetly o strog iterpretatio for the otio of -vertex graph scalig. This paper presets a ew defiitio for the term i the cotext of

More information

Measures of Dispersion

Measures of Dispersion Measures of Dispersion 6-3 I Will... Find measures of dispersion of sets of data. Find standard deviation and analyze normal distribution. Day 1: Dispersion Vocabulary Measures of Variation (Dispersion

More information

Computational Geometry

Computational Geometry Computatioal Geometry Chapter 4 Liear programmig Duality Smallest eclosig disk O the Ageda Liear Programmig Slides courtesy of Craig Gotsma 4. 4. Liear Programmig - Example Defie: (amout amout cosumed

More information

On Infinite Groups that are Isomorphic to its Proper Infinite Subgroup. Jaymar Talledo Balihon. Abstract

On Infinite Groups that are Isomorphic to its Proper Infinite Subgroup. Jaymar Talledo Balihon. Abstract O Ifiite Groups that are Isomorphic to its Proper Ifiite Subgroup Jaymar Talledo Baliho Abstract Two groups are isomorphic if there exists a isomorphism betwee them Lagrage Theorem states that the order

More information

1. The lines intersect. There is one solution, the point where they intersect. The system is called a consistent system.

1. The lines intersect. There is one solution, the point where they intersect. The system is called a consistent system. Commo Core Math 3 Notes Uit Day Systems I. Systems of Liear Equatios A system of two liear equatios i two variables is two equatios cosidered together. To solve a system is to fid all the ordered pairs

More information

Chapter 18: Ray Optics Questions & Problems

Chapter 18: Ray Optics Questions & Problems Chapter 18: Ray Optics Questios & Problems c -1 2 1 1 1 h s θr= θi 1siθ 1 = 2si θ 2 = θ c = si ( ) + = m = = v s s f h s 1 Example 18.1 At high oo, the su is almost directly above (about 2.0 o from the

More information

Apparent Depth. B' l'

Apparent Depth. B' l' REFRACTION by PLANE SURFACES Apparet Depth Suppose we have a object B i a medium of idex which is viewed from a medium of idex '. If '

More information

ENGR 132. Fall Exam 1 SOLUTIONS

ENGR 132. Fall Exam 1 SOLUTIONS ENGR 3 Fall 03 Exam SOLUTIONS INSTRUCTIONS: Duratio: 60 miutes Keep your eyes o your ow work! Keep your work covered at all times!. Each studet is resposible for followig directios. Read carefully.. MATLAB

More information

CSC165H1 Worksheet: Tutorial 8 Algorithm analysis (SOLUTIONS)

CSC165H1 Worksheet: Tutorial 8 Algorithm analysis (SOLUTIONS) CSC165H1, Witer 018 Learig Objectives By the ed of this worksheet, you will: Aalyse the ruig time of fuctios cotaiig ested loops. 1. Nested loop variatios. Each of the followig fuctios takes as iput a

More information

PLEASURE TEST SERIES (XI) - 04 By O.P. Gupta (For stuffs on Math, click at theopgupta.com)

PLEASURE TEST SERIES (XI) - 04 By O.P. Gupta (For stuffs on Math, click at theopgupta.com) wwwtheopguptacom wwwimathematiciacom For all the Math-Gya Buy books by OP Gupta A Compilatio By : OP Gupta (WhatsApp @ +9-9650 350 0) For more stuffs o Maths, please visit : wwwtheopguptacom Time Allowed

More information

What is the difference between a statistician and a mortician? Nobody's dying to see the statistician! Chapter 8 Interval Estimation

What is the difference between a statistician and a mortician? Nobody's dying to see the statistician! Chapter 8 Interval Estimation Slides Preared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 What is the differece betwee a statisticia ad a morticia? Nobody's dyig to see the statisticia! Slide Chater 8 Iterval Estimatio Poulatio

More information