Chapter Objectves Check off these sklls when you feel that you have mastered them. Construct a hstogram for a small data set. Lst and descrbe two types of dstrbutons for a hstogram. Identfy from a hstogram possble outlers of a data set. Construct a stemplot for a small data set. Calculate the mean of a set of data. Sort a set of data from smallest to largest and then determne ts medan. Determne the upper and lower quartles for a data set. Calculate the fve-number summary for a data set. Construct the dagram of a boxplot from the data set s fve-number summary. Calculate the standard devaton of a small data set. Descrbe a normal curve. Gven the mean and standard devaton of a normally dstrbuted data set, compute the frst and thrd quartles. Explan the 68 95 99.7 rule. Sketch the graph of a normal curve gven ts mean and standard devaton. Gven the mean and standard devaton of a normally dstrbuted data set, compute the ntervals n whch the data set fall nto a gven percentage by applyng the 68 95 99.7 rule. 93
94 Guded Readng Introducton Data, or numercal facts, are essental for makng decsons n almost every area of our lves. But to use them for our purposes, huge collecton of data must be organzed and dstlled nto a few comprehensble summary numbers and vsual mages. Ths wll clarfy the results of our study and allow us to draw reasonable conclusons. The analyss and dsplay of data are thus the groundwork for statstcal nference. In a data set there are ndvduals. These ndvduals may be people, cars, ctes, or anythng to be examned. The characterstc of an ndvdual s a varable. For dfferent ndvduals, a varable can take on dfferent values. Example A Identfy the ndvduals and the varables n the followng data set from a class roster. Name Age Sex Dan 16 Male Edwn 17 Male Adam 16 Male Nada 15 Female Soluton The ndvduals are the names of the people on the class roster. The varables are ther ages and sex. In ths chapter, you wll be dong exploratory data analyss. Ths combnes numercal summares wth graphcal dsplay to see patterns n a set of data. The organzng prncples of data analyss are as follows. 1) Examne ndvdual varables, and then look for relatonshps among varables. ) Draw a graph or graphs and add to t numercal summares. Secton 5.1 Dsplayng Dstrbutons: Hstograms The dstrbuton of a varable tells us what values the varable takes and how often t takes these values. The most common graph of a dstrbuton wth one numercal varable s called a hstogram.
95 Example B Construct a hstogram gven the followng data. How many peces of data are there? Soluton 8 6 4 Value Count 5 10 5 15 7 0 3 5 1 0 5 10 15 0 5 There are + 5+ 7+ 3+ 1= 18 peces of data. When constructng a hstogram, each pece of data must fall nto one class. Each class must be of equal wdth. For any gven data set, there s more than one way to defne the classes. Ether you are nstructed as to how to defne the classes, or you must determne class based on some crtera. Example C Gven the followng exam scores, construct a hstogram wth classes of length 10 ponts. 40 50 50 53 55 55 55 58 60 60 63 65 68 70 70 73 75 75 78 78 83 85 85 88 90 95 96 Soluton It s helpful to frst put the data nto classes and count the ndvdual peces of data n each class. Snce the smallest pece of data s 40, t makes sense to make the frst class 40 to 49, nclusve. Class Count 40 49 1 50 59 7 60 69 5 70 79 7 80 89 4 90 99 3 Notce that the sum of the values n the count column should be 7 (total number of peces of data). Also notce that some of the detals of the scores are lost when raw data are placed n classes.
96 Secton 5. Interpretng Hstograms An mportant feature of a hstogram s ts overall shape. Although there are many shapes and overall patterns, a dstrbuton may be symmetrc, or t may be skewed to the rght or skewed to the left. If a dstrbuton s skewed to the rght, then the larger values extend out much further to the rght. If a dstrbuton s skewed to the left, then the smaller values extend out much further to the left. The easest way to keep the two terms from beng confused s to thnk of the drecton of the tal. If the tal ponts left, t s skewed to the left. If the tal ponts rght, t s skewed to the rght. Another way to descrbe a dstrbuton s by ts center. For now, we can thnk of the center of a dstrbuton as the mdpont. Another way to descrbe a dstrbuton s by ts spread. The spread of a dstrbuton s statng ts smallest and largest values. In a dstrbuton, we may also observe outlers; that s, a pece or peces of data that fall outsde the overall pattern. Often tmes determnng an outler s a matter of judgment. There are no hard and fast rules for determnng outlers. Example D Gven the followng data regardng exam scores, construct a hstogram. Descrbe ts overall shape and dentfy any outlers. Class Count Class Count Soluton on next page 0 9 1 50 59 6 10 19 0 60 69 8 0 9 0 70 79 7 30 39 0 80 89 5 40 49 3 90 99
97 Soluton The shape s roughly symmetrc. The score n the class 0 9, nclusve, s clearly an outler. Wth a 0 on an exam, the most lkely explanaton s that the student mssed the exam. It s also possble that the student was completely unprepared and performed poorly to obtan a very low score. Example E Gven the followng data regardng exam scores, construct a hstogram. Descrbe ts overall shape and dentfy any outlers. Class Count Class Count Soluton 0 9 0 50 59 6 10 19 1 60 69 8 0 9 70 79 10 30 39 1 80 89 8 40 49 3 90 99 The shape s skewed to the left. There doesn t appear to be any outlers. Queston 1 Gven the followng exam scores, descrbe the overall shape of the dstrbuton and dentfy any outlers. In your soluton, construct a hstogram wth class length of 5 ponts. 1 59 60 61 6 63 64 65 65 66 67 68 68 69 69 70 7 73 74 74 75 76 77 78 80 81 8 85 86 89 91 9 95 Answer The dstrbuton appears to be skewed to the rght. The scores of 1 and appear to be outlers.
98 Secton 5.3 Dsplayng Dstrbutons: Stemplots A stemplot s a good way to represent data for small data sets. Stemplots are qucker to create than hstograms and gve more detaled nformaton. Each value n the data set s represented as a stem and a leaf. The stem conssts of all but the rghtmost dgt, and the leaf s the rghtmost dgt. A stemplot resembles a hstogram turned sdeways. Example F Gven the followng exam scores, construct a stemplot. 40 50 50 53 55 55 55 58 60 60 63 65 68 70 70 73 75 75 78 78 83 85 85 88 90 95 96 Soluton In the stemplot, the tens dgt wll be the stem and the ones dgt wll be the leaf. 4 0 5 0035558 6 00358 7 0035588 8 3558 9 056 Queston The followng are the percentages of salt concentrate taken from lab mxture samples. Descrbe the shape of the dstrbuton and any possble outlers. Ths should be done by frst roundng each pece of data to the nearest percent and then constructng a stemplot. Sample 1 3 4 5 6 7 Percent 39.8 65.7 64.7 0.1 40.8 53.4 70.8 Sample 8 9 10 11 1 13 14 Percent 50.7 68.7 74.3 8.6 58.5 68.0 7. Answer The dstrbuton appears to be roughly symmetrc wth 0 as a possble outler. Secton 5.4 Descrbng Center: Mean and Medan The mean of a data set s obtaned by addng the values of the observatons n the data set and dvdng by the number of data. If the observatons are lsted as values of a varable x (namely x1+ x +... + x x1, x,..., x n ), then the mean s wrtten as x. The formula for the mean s x = n, n where n represents the number of peces of data.
99 Example G Calculate the mean of each data set. a) 13, 111, 105, 115, 11, 113, 117, 119, 114, 118, 111, 150, 147, 19, 138 b) 17, 15, 13,, 14, 15, 10, 1, 16, 16, 17, Soluton 13 + 111+ 105 + 115 + 11 + 113 + 117 + 119 + 114 + 118 + 111+ 150 + 147 + 19 + 138 a) x = 15 18 = 11.5 15 17 + 15 + 13+ + 14 + 15 + 10 + 1+ 16 + 16 + 17 + 158 b) x = = = 13. 1 1 Queston 3 Gven the followng stemplot, determne the mean. Round to the nearest tenth, f necessary. 1 59 3478 3 0334679 4 0159 5 46 6 1 7 3 Answer x = 37 The medan, M, of a dstrbuton s a number n the mddle of the data, so that half of the data are above the medan, and the other half are below t. When determnng the medan, the data should be placed n order, typcally smallest to largest. When there are n peces of data, then the pece of data n+ observatons up from the bottom of the lst s the medan. Ths s farly straghtforward when n s 1 odd. When there are n peces of data and n s even, then you must fnd the average (add together and dvde by two) of the two center peces of data. The smaller of these two peces of data s located n observatons up from the bottom of the lst. The second, larger, of the two peces of data s the next one n order or, n + 1 observatons up from the bottom of the lst. Example H Determne the medan of each data set below. a) 13, 111, 105, 115, 11, 113, 117, 119, 114, 118, 111, 150, 147, 19, 138 b) 17, 15, 13,, 14, 15, 10, 1, 16, 16, 17, Soluton For each of the data sets, the frst step s to place the data n order from smallest to largest. a) 105, 111, 111, 11, 113, 114, 115, 117, 118, 119, 13, 19, 138, 147, 150 15 1 16 th Snce there are 15 peces of data, the + = = 8 pece of data, namely 117, s the medan. b) 1,, 10, 13, 14, 15, 15, 16, 16, 17, 17, 1 th Snce there are 1 peces of data, the mean of the = 6 and 7 th peces of data wll be the 15+ 15 30 n+ 1 medan. Thus, the medan s = = 15. Notce, f you use the general formula, you 1 1 13 would be lookng for a value + = = 6.5 observatons from the bottom. Ths would mply halfway between the actual 6 th observaton and the 7 th observaton.
100 Queston 4 Gven the followng stemplot, determne the medan. Answer M = 36.5 1 09 3478 3 03345679 4 01359 5 16 6 01 Secton 5.5 Descrbng Spread: The Quartles The quartles Q 1 (the pont below whch 5% of the observatons le) and Q 3 (the pont below whch 75% of the observatons le) gve a better ndcaton of the true spread of the data. More specfcally, Q 1 s the medan of the data to the left of M (the medan of the data set). Q 3 s the medan of the data to the rght of M. Example I Determne the quartles Q 1 and Q 3 of each data set below. a) 13, 111, 105, 115, 11, 113, 117, 119, 114, 118, 111, 150, 147, 19, 138 b) 17, 15, 13,, 14, 15, 10, 1, 16, 16, 17, Soluton For each of the data sets, the frst step s to place the data n order from smallest to largest. a) 105, 111, 111, 11, 113, 114, 115, 117, 118, 119, 13, 19, 138, 147, 150 From Example H we know that the medan s the 8 th pece of data. Thus, there are 7 peces of 7 1 8 th data below M. We therefore can determne Q 1 to be the + = = pece of data. Thus, 4 th Q 1 = 11. Now snce there are 7 peces of data above M, Q 3 wll be the 4 pece of data to the rght of M. Thus, Q 3 = 19. b) 1,, 10, 13, 14, 15, 15, 16, 16, 17, 17, th From Example H we know that the medan s between the 6 and 7 th peces of data. Thus, there are 6 peces of data below M. Snce 6 + 1 7 rd th = = 3.5, Q 1 wll be the mean of 3 and 4 peces of 10+ 13 3 data, namely = = 11.5. Now snce there are 6 peces of data above M, Q 3 wll be the rd th 16+ 17 33 mean of the 3 and 4 peces of data to the rght of M. Thus, Q 3 = = = 16.5. Queston 5 Determne the quartles Q 1 and Q 3 of each data set below. a) 1, 16, 0, 6, 8, 9, 1, 15, 3, 15, 7, 8, 19 b) 14, 1, 11, 1, 4, 8, 6, 4, 8, 10 Answer a) Q 1 = 7.5 and Q 3 = 17.5 b) Q 1 = 8 and Q 3 = 1
101 Secton 5.6 The Fve-Number Summary and Boxplots The fve-number summary conssts of the medan (M), quartles (Q 1 and Q 3 ), and extremes (hgh and low). A boxplot s a graphcal (vsual) representaton of the fve-number summary. A central box spans quartles Q 1 and Q 3. A lne n the mddle of the central box marks the medan, M. Two lnes extend from the box to represent the extreme values. Example J Gven the followng fve-number summary, draw the boxplot. Soluton 00, 50, 300, 450, 700 Queston 6 Gven the followng data, fnd the fve-number summary and draw the boxplot. Answer The fve-number summary s 11, 1, 15.5, 1, 5. The boxplot s as follows. 1, 11, 5, 1, 15, 1, 17, 35, 16, 1
10 Secton 5.7 Descrbng Spread: The Standard Devaton The varance, s, of a set of observatons s an average of the squared dfferences between the ndvdual observatons and ther mean value. In symbols, the varance of n observatons ( ) x1, x,..., xn s s = ( x1 x) + ( x x) +... + ( xn x) n 1. Notce we dvde by n 1. The standard devaton, s, of a set of observatons s the square root of the varance and measures the spread of the data around the mean n the same unts of measurement as the orgnal data set. You should be nstructed as to the method (spreadsheet, calculator wth statstcal capabltes, or by hand) requred for calculatng the varance and n turn the standard devaton. Example K Gven the followng data set, fnd the varance and standard devaton. 8.6, 7., 9., 5.6, 5.5, 4.4 Soluton Placng the data n order (not requred, but helpful) we have the followng hand calculatons. Notce 40.5 that x = 6 = 6.75. Observatons Devatons Squared devatons Thus, x x x 4.4 4.4 6.75.35 5.5 5.5 6.75 1.5 5.6 5.6 6.75 1.15 ( x ) x = (.35) = ( 1.5) = ( 1.15) = 5.55 = 1.565 = 1.35 7. 7. 6.75 =0.45 ( 0.45 ) = 0.05 8.6 8.6 6.75 = 1.85 ( 1.85 ) = 3.45 9. 9. 6.75 =.45 (.45 ) = 6.005 sum = 40.5 sum = 0.00 sum = 18.035 18.035 18.035 s = = = 3.607 6 1 5 and s = 3.607 1.90. Queston 7 Gven the followng data set, fnd the varance and standard devaton. 3.41,.78, 5.6, 6.49, 7.61, 7.9, 8.1, 5.51 Answer s 4.169 and s = 4.169.04.
103 Secton 5.8 Normal Dstrbutons Samplng dstrbutons, and many other types of probablty dstrbutons, approxmate a bell curve n shape and symmetry. Ths knd of shape s called a normal curve, and can represent a normal dstrbuton, n whch the area of a secton of the curve over an nterval concdes wth the proporton of all values n that nterval. The area under any normal curve s 1. A normal curve s unquely determned by ts mean and standard devaton. The mean of a normal dstrbuton s the center of the curve. The symbol µ wll be used for the mean. The standard devaton of a normal dstrbuton s the dstance from the mean to the pont on the curve where the curvature changes. The symbol σ wll be use for the standard devaton. The frst quartle s located 0.67 standard devaton below the mean, and the thrd quartle s located 0.67 standard devaton above the mean. In other words, we have the followng formulas. Q 1 = µ 0.67σ and Q 3 = µ + 0.67σ Example L The scores on a marketng exam were normally dstrbuted wth a mean of 73 and a standard devaton of 1. a) Fnd the thrd quartle (Q 3 ) for the test scores. b) Fnd a range contanng exactly half of the students scores. Soluton a) Snce Q µ σ ( ) 3 = + 0.67 = 73+ 0.67 1 = 73 + 8.04 = 81.04, we would say the thrd quartle s 81. b) Snce 5% of the data le below the frst quartle and 5% of the data fall above the thrd quartle, 50% of the data would fall between the frst and thrd quartles. Thus, we must fnd the frst quartle. Snce Q 1 = µ 0.67σ = 73 0.67( 1) = 73 8.04 = 64.96, we would say an nterval would be [ 65,81 ].
104 Secton 5.9 The 68 95 99.7 Rule The 68 95 99.7 rule apples to a normal dstrbuton. It s useful n determnng the proporton of a populaton wth values fallng n certan ranges. For a normal curve, the followng rules apply: The proporton of the populaton wthn one standard devaton of the mean s 68%. The proporton of the populaton wthn two standard devatons of the mean s 95%. The proporton of the populaton wthn three standard devatons of the mean s 99.7%. Example M The amount of coffee a certan dspenser flls 16 oz coffee cups wth s normally dstrbuted wth a mean of 14.5 oz and a standard devaton of 0.4 oz. a) Almost all (99.7%) cups dspensed fall wthn what range of ounces? b) What percent of cups dspense less than 13.7 oz? Soluton a) Snce 99.7% of all cups fall wthn 3 standard devatons of the mean, we fnd the followng. µ ± 3σ = 14.5 ± 3( 0.4) = 14.5 ± 1. Thus, the range of ounces s 13.3 to 15.7. b) Make a sketch: 13.7 oz s twoσ below µ ; 95% are wthn σ of µ. 5% le farther than σ. Thus, half of these, or.5%, le below 13.7. Queston 8 Look agan at the marketng exam n whch scores were normally dstrbuted wth a mean of 73 and a standard devaton of 1. a) Fnd a range contanng 34% of the students scores. b) What percentage of the exam scores were between 61 and 97? Answer a) Ether of the ntervals [61, 73] or [73, 85] b) 81.5%
105 Homework Help Exercse 1 Carefully read the Introducton before respondng to ths exercse. Exercses 3 Carefully read Secton 5. before respondng to these exercses. Pay specal attenton to the descrpton of skewed dstrbutons. Exercse 4 Carefully read Sectons 5.1 5.3 before respondng to ths exercse. Frst construct your classes and count ndvduals as descrbed n Example of your text. Include the outler n your hstogram. The followng may be helpful n constructng your hstogram. One possblty s to make the frst class 6 gas mleage < 11 or 11 gas mleage < 16. Class 6 10 11 15 16 0 1 5 6 30 31 35 36 40 41 45 46 50 51 55 56 60 61 65 66 70 Count Exercse 5 Carefully read Sectons 5.1 5. before respondng to ths exercse. Frst construct your classes and count ndvduals as descrbed n Example of your text. Include the outlers n your hstogram. The followng may be helpful n constructng your hstogram. One possblty s to make the frst class 0.0 emmssons <.0. Class 0.0 1.9.0 3.9 4.0 5.9 6.0 7.9 8.0 9.9 10.0 11.9 1.0 13.9 14.0 15.9 16.0 17.9 18.0 19.9 Count Pay specal attenton to the descrpton of skewed dstrbutons and outlers.
106 Exercse 6 Carefully read Secton 5. before respondng to ths exercse. Pay specal attenton to the descrpton of symmetrc and skewed dstrbutons. Thnk about how gender and rght/left-handedness are dstrbuted n real lfe. Exercses 7 10 Carefully read Secton 5.3 before respondng to these exercses. Carefully read the descrpton of how to descrbe each pece of data n Exercse 8. You may choose to use the followng stems n the exercses. Exercse 8 0 1 3 Exercse 9 10 11 1 13 14 15 16 17 18 19 0 Exercse 10 48 49 50 51 5 53 54 55 56 57 58 Exercse 11 Carefully read Secton 5.4 before respondng to ths exercse. Make sure to show all steps n your calculatons, unless otherwse nstructed. Exercse 1 (a) Make the stemplot, wth the outler. 1 3 4 5 6 7 (b) Calculate the mean. Use the stemplot to put the data n order from smallest to largest n order to fnd the medan. Snce there s an even number of peces of data, you wll need to examne two peces of data to determne the medan. Remove the outler and recalculate the mean and determne the medan of the 17 peces of data. Compare the results wth and wthout the outler. Exercse 13 14 Carefully read Secton 5. before respondng to these exercses. The followng drawngs may be helpful to show the relatve locatons of the medan and the mean. Exercses 15 16 Examples wll vary. Exercses 17 0 Carefully read Secton 5.5 5.6 before respondng to these exercses. Make sure to frst put data n order from smallest to largest. Double check that you have accounted for all peces of data. Pay specal attenton when you are dealng wth an even number of peces of data n determnng the medan. When determnng quartles, remember f there s an even number of peces to the left of the mean, there wll also be an even number of peces to the rght of the mean.
107 Exercse 1 Carefully read Secton 5.5 5.6 before respondng to ths exercse. It would be helpful to create a stem plot to organze your data from smallest to largest. 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 Exercse Carefully read Sectons 5.5 5.6 before respondng to ths exercse. The data are already organzed from smallest to largest. Exercse 3 Look carefully at the referenced fgure and compare as many features as possble. Exercse 4 Your values of the medan and quartles may dffer slghtly from another student. Try roundng to the nearest thousand. Exercse 5 Carefully read Sectons 5.1 5.6 before respondng to ths exercse. Make sure to frst put data n order from smallest to largest and round to the nearest whole number. Double check that you have accounted for all peces of data. In part a, you can ether create a hstogram or a stemplot. Pay specal attenton when you are dealng wth an even number of peces of data n determnng the medan. When determnng quartles, remember f there s an even number of peces to the left of the mean, there wll also be an even number of peces to the rght of the mean. Exercse 6 Approxmate the bar heghts. You wll need to determne n whch bar the 5%, 50%, and 75% marks occur for Q, M, and Q. 1 3 Exercses 7 8 Both of these exercses rely on the descrpton of nterquartle range gven n Exercse 7.
108 Exercse 9 (a) Placng the data n order s helpful, but not requred. If you are performng the calculatons by hand, the followng table may be helpful. Observatons x Devatons x x Squared devatons ( x ) x sum = sum = 0.00 sum = (b) If your data are n order above, the medan can easly be determned. If you have already worked Exercse 10, then the mean has already been calculated for comparson n ths exercse.
109 Exercse 30 Placng the data n order s helpful, but not requred. When performng the calculatons by hand, the followng table may be helpful. Observatons x Devatons x x Squared devatons ( x ) x sum = sum = 0.00 sum = Exercse 31 Snce the square root of the varance s the standard devaton, the square of the standard devaton s the varance. Exercse 3 Thnk about what each of the measures represents before answerng each part of ths exercse. Exercse 33 Round the data to the nearest tenth to create the stemplots. 3 4 5 6 7 8 9 Data A 5 6 7 8 9 10 11 1 Data B Exercse 34 Look carefully at each fgure before answerng each part of ths exercse. Exercse 35 Answers wll vary on ths exercse dependng on what s used for calculatons. Exercse 36 The followng may be helpful n creatng stemplots. Group 1 Group Group 3 0 0 0 1 1 1 3 3 3 Contnued on next page
110 Exercse 36 contnued The followng may be helpful n creatng hstograms usng class wdths of 10 trees per 0.1 hectare n area. Placng the data n order s helpful, but not requred. When performng the calculatons by hand, the followng table may be helpful. Group 1: Observatons Devatons Squared devatons x x x ( x ) x Contnued on next page sum = sum = sum =
111 Exercse 36 contnued In Group and Group 3, you may choose to use 6 decmal place accuracy for x n order to calculate x x. places and s to three. Group : You may also choose to round ( ) Observatons x x x to fve decmal places. Round s to four decmal Devatons x x Squared devatons ( x ) x sum = sum = sum = Group 3: Observatons x Devatons x x Squared devatons ( x ) x sum = sum = sum = Exercse 37 (a) Snce the standard devaton cannot be negatve, thnk about what the smallest value t can be and ts mplcaton. (b) Thnk about a way to make the data set as spread out as possble. Try wth two peces of data and see f that helps to determne four peces of data. You may choose to try dfferent data sets to to convnce yourself of your answer.
11 Exercse 38 Carefully read Secton 5.8 before respondng to ths exercse. Sketch a normal curve, mark the axs wth the mean as the center of the curve and one standard devaton to the rght and left wll mark the change-of-curvature ponts. These three ponts set the proper scale. Exercse 39 Refer to exercses 13 14. Exercse 40 Thnk of the letter M. Exercses 41 4 and 44 45 and 48 49 Carefully read Secton 5.9 before respondng to these exercses. Make a sketch for each exercse by drawng a normal curve, placng the mean and 3 standard devatons to the rght and left. The followng may be helpful. Make sure to use symmetry when determnng percentages above or below a value. Exercse 41 Exercse 4 Exercse 44 Exercse 45 Exercse 48 Exercse 49
113 Exercses 43 and 46 Carefully read Secton 5.8 before respondng to these exercses regardng quartles of a normal dstrbuton. Exercse 47 Carefully read Sectons 5.8 5.9 before respondng to ths exercse. Exercse 50 Apply the formula gven n part a for parts a and b and compare the results n part c. Exercse 51 The followng may be helpful n creatng the stemplots. Lengths of red flowers 37 38 39 40 41 4 43 Lengths of yellow flowers 34 35 36 37 38 Exercse 5 Arrange the data (separately) n order from smallest to largest n order to determne the fve-number summary for each varety. Draw the boxplots and compare the skewness and the varabltes. Exercses 53 Placng the data n order s helpful, but not requred. When performng the calculatons by hand, the followng tables may be helpful. Note the order of red and yellow were swtched for room consderatons. Yellow: Observatons Devatons Squared devatons x x x ( x ) x Contnued on next page sum = sum = sum =
114 Exercses 53 contnued Red: In the red data, you may choose to use 6 decmal place accuracy for x n order to calculate x x. You may also choose to round ( x x) and s to three. Observatons x to fve decmal places. Round s to four decmal places Devatons x x Squared devatons ( x ) x sum = sum = sum = Exercses 54 55 Carefully read Sectons 5.8 5.9 before respondng to these exercses.
115 Do You Know the Terms? Cut out the followng 19 flashcards to test yourself on Revew Vocabulary. You can also fnd these flashcards at http://www.whfreeman.com/fapp7e. Boxplot Dstrbuton Exploratory data analyss Fve-number summary Hstogram Indvduals Mean Medan Normal dstrbutons Outler
116 The pattern of outcomes of a varable. The dstrbuton descrbes what values the varable takes and how often each value occurs. A graph of the fve-number summary. A box spans the quartles, wth an nteror lne markng the medan. Lnes extend out from ths box to the extreme hgh and low observatons. A summary of a dstrbuton that gves the medan, the frst and thrd quartles, and the largest and smallest observatons. The practce of examnng data for overall patterns and specal features, wthout necessarly seekng answers to specfc questons. The people, anmals, or thngs descrbed by a data set. A graph of the dstrbuton of outcomes (often dvded nto classes) for a sngle varable. The heght of each bar s the number of observatons n the class of outcomes covered by the base of the bar. All classes should have the same wdth. The mdpont of a set of observatons. Half the observatons fall below the medan and half fall above. The ordnary arthmetc average of a set of observatons. To fnd the mean, add all the observatons and dvde the sum by the number of observatons summed. A data pont that falls clearly outsde the overall pattern of a set of data. A famly of dstrbutons that descrbe how often a varable takes ts values by areas under a curve. The normal curves are symmetrc and bell-shaped. A specfc normal curve s completely descrbed by gvng ts mean and ts standard devaton.
117 Quartles 68 95-99.7 rule Skewed dstrbuton Standard devaton Standard devaton of a normal curve Stemplot Symmetrc dstrbuton Varable Varance
118 In any normal dstrbuton, 68% of the observatons le wthn 1 standard devaton on ether sde of the mean; 95% le wthn standard devatons of the mean; and 99.7% le wthn 3 standard devatons of the mean. The frst quartle of a dstrbuton s the pont wth 5% of the observatons fallng below t; the thrd quartle s the pont wth 75% below t. A measure of the spread of a dstrbuton about ts mean as center. It s the square root of the average squared devaton of the observatons from ther mean. A dstrbuton n whch observatons on one sde of the medan extend notably farther from the medan than do observatons on the other sde. In a rght-skewed dstrbuton, the larger observatons extend farther to the rght of the medan than the smaller observatons extend to the left. A dsplay of the dstrbuton of a varable that attaches the fnal dgts of the observatons as leaves on stems made up of all but the fnal dgt. The standard devaton of a normal curve s the dstance from the mean to the change-of-curvature ponts on ether sde. Any characterstc of an ndvdual. A dstrbuton wth a hstogram or stemplot n whch the part to the left of the medan s roughly a mrror mage of the part to the rght of the medan. A measure of the spread of a dstrbuton about ts mean. It s the average squared devaton of the observatons from ther mean. The square root of the varance s the standard devaton.
119 Learnng the Calculator Example 1 Construct a hstogram gven the followng. Value Count 1 13 4 15 6 16 8 0 3 Soluton Frst enter the data by pressng the button. The followng screen wll appear. If there s data already stored, you may wsh the clear t out. For example, f you wsh to remove the data n L1, toggle to the top of the data and press then. Repeat for any other data sets you wsh to clear. Enter the new data beng sure to press after each pece of data s dsplayed. In order to dsplay a hstogram, you press then. Ths s equvalent to. The followng screen (or smlar) wll appear. You wll need to turn a stat plot On and choose the hstogram opton ( ). You wll also need to make sure Xlst and Freq reference the correct data. In ths case L1 and L, respectvely.
10 Next, you wll need to make sure that no other graphs appear on your hstogram. Press another relaton s present, ether toggle to = and press enter to deselect or delete the relaton. and f You wll next need to choose an approprate wndow. By pressng you need to enter an approprate wndow that ncludes your smallest and largest peces of data. These values dctate your choces of Xmn and Xmax. Your choce of Xscl s determned by the knd of data you are gven. In ths case, the approprate choce s 1. If you are gven data such as 10, 1, 14, 16, and values such as 11, 13, and 15 are not consdered then the approprate choce would be n order to make the vertcal bars touch. In terms of choces for frequency, Ymn should be set at zero. Ymax should be at least as large as the hghest frequency value. Your choce of Yscl s determned by how large the maxmum frequency value s from your table. Next, we dsplay the hstogram by pressng the button. Notce that the hstogram dffers slghtly from how a hand drawng should be. Ideally, the base of each rectangle should be shfted left by half of a unt. Example Gven the followng data, construct a hstogram. Class Count 0 9 10 19 1 0 9 3 30 39 6 40 49
11 Soluton Follow the nstructons n Example 1 n order to nput data and set up the wndow to dsplay the hstogram. The wdth of the classes should be the Xscl n order to make the vertcal bars touch. Also, n a case lke ths where you are gven classes, use the left endpont of the class as data peces. Example 3 Consder the followng data. 1, 34, 55, 6, 54, 3, 34, 5, 50, 55, 5, 50 Arrange the data n order from Fnd the standard devaton. smallest to largest. Fnd the fve number summary. Fnd the mean. Dsplay the boxplot. Enter the data, notng that there are 1 peces of data. Make sure the locaton of the last entry corresponds to the total number of peces of data. To arrange the data n order from smallest to largest, press the button and choose the SortA( opton whch sorts the data n ascendng order. Choose the approprate data set (n ths case L1) and then press. The calculator wll dsplay Done ndcatng the data s sorted. By pressng the opton. button, you can then vew the data arranged n order by choosng the Edt
1 The data arranged from smallest to largest s as follows. 1, 3, 5, 34, 34, 50, 50, 5, 54, 55, 55, 6 To fnd the mean and standard devaton, press the button. Toggle over to CALC and choose the 1-Var Stats opton and then press the. You wll get your home screen. Press agan and you wll then be able to determne the mean and standard devaton. The mean s (approxmately 4.917) and the standard devaton s Sx (approxmately 14.519). To determne the fve number summary, from the last screen press the down arrow ( ) fve tmes. The fve number summary s 1, 9.5, 50, 54.5, 6. To dsplay the box plot, press then. Ths s equvalent to. You wll need to choose for boxplot. Make sure the proper data are chosen for Xlst and Freq should be set at 1. Choose an approprate wndow for Xmn and Xmax based on the mnmum and maxmum values. The values you choose for Ymn and Ymax do not have an effect on the boxplot. You may choose values for Xscl and Yscl based on appearance of the axes. Dsplay boxplot by pressng the button.
13 Practce Quz 1. The weghts (n pounds) of your cousns are: 10, 89, 108, 76, 1. Whch are the outlers? a. 1 only. b. 10 only c. both 10 and 1. Below s a stemplot of the ages of adults on your block. Whch statement s true? 115 3 05788 4 15 5 57 6 5 7 8 8 1 a. The stemplot s roughly symmetrc. b. The stemplot s skewed to the hgher ages. c. The two oldest people are outlers. 3. Here are 7 measured lengths (n nches): 13, 8, 5, 3, 8, 9, 1. Fnd ther medan. a. 3 b. 8 c. 8.3 4. Here are 7 measured lengths (n nches): 13, 8, 5, 3, 8, 9, 1. Fnd ther mean. a. 3 b. 8 c. 8.3 5. The boxplot graph always ncludes the a. mean and medan. b. quartles and the standard devaton. c. quartles and the medan. 6. The percentage of scores on a standardzed exam that le between the frst and thrd quartles s a. 5%. b. 50%. c. 75%. 7. If the mean of the data, 4, 6, 3, 5, 8, 7 s 5, what s ts standard devaton? a. 1 7 b. 4 c.
14 8. The scores on a marketng exam were normally dstrbuted wth a mean of 67 and a standard devaton of 9. Fnd the frst quartle (Q 1 ) for the test scores. a. 58 b. 61 c. 5 9. Gven the followng data, fnd the fve-number summary. 5, 8, 1, 15, 11, 1, 9, 1 a. 5, 8.5, 11.5, 13.5, 1 b. 5, 8.5, 1, 13, 1 c. 5, 1, 13, 1, 1 10. The amount of coffee a certan dspenser flls 1 oz coffee cups wth s normally dstrbuted wth a mean of 10.9 oz and a standard devaton of 0. oz. What percent of cups dspense more than 11.1 oz? a. 68% b. 5% c. 16%
15 Word Search Refer to pages 07 08 of your text to obtan the Revew Vocabulary. There are 17 hdden vocabulary words/expressons n the word search below. Standard devaton of a normal curve and 68-95-99.7 rule were both omtted from the word search. It should be noted that spaces and hyphens are removed. 1.. 3. 4. 5. 6. 7. 8. 10. 11. 1. 13. 14. 15. 16. 17. 9.