Appedi C. Measures of Cetral Tedecy ad Dispersio A8 Appedi C: Cocepts i Statistics C. Measures of Cetral Tedecy ad Dispersio Mea, Media, ad Mode I may real-life situatios, it is helpful to describe data by a sigle umber that is most represetative of the etire collectio of umbers. Such a umber is called a measure of cetral tedecy. The most commoly used measures are as follows.. The mea, or average, of umbers is the sum of the umbers divided by.. The media of umbers is the middle umber whe the umbers are writte i umerical order. If is eve, the media is the average of the two middle umbers.. The mode of umbers is the umber that occurs most frequetly. If two umbers tie for most frequet occurrece, the collectio has two modes ad is called bimodal. Eample Comparig Measures of Cetral Tedecy O a iterview for a job, the iterviewer tells you that the average aual icome of the compay s employees is $60,89. The actual aual icomes of the employees are show below. What are the mea, media, ad mode of the icomes? $7,0 $78,0 $,678 $8,980 $7,08 $,676 $8,906 $,00 $,0 $,0 $,00 $,8 $7,0 $0, $8,96 $,98 $6,0 $0,9 $6,8 $6,0 $,6 $98, $8,980 $9,0 $,67 What you should lear Fid ad iterpret the mea, media, ad mode of a set of data. Determie the measure of cetral tedecy that best represets a set of data. Fid the stadard deviatio of a set of data. Use bo-ad-whisker plots. Why you should lear it Measures of cetral tedecy ad dispersio provide a coveiet way to describe ad compare sets of data. For istace, i Eercise o page A9, the mea ad stadard deviatio are used to aalyze the prices of gold for the years 98 through 00. Solutio The mea of the icomes is Mea 7,0 78,0,678 8,980...,67,, $60,89. To fid the media, order the icomes as follows. $,00 $,00 $6,0 $7,0 $7,08 $8,980 $0, $,0 $,676 $8,906 $8,96 $,6 $,0 $,8 $,98 $,67 $6,0 $6,8 $7,0 $,678 $8,980 $9,0 $98, $0,9 $78,0 From this list, you ca see that the media icome is $,0. You ca also see that $,00 is the oly icome that occurs more tha oce. So, the mode is $,00. Now try Eercise.
A86 Appedi C Cocepts i Statistics I Eample, was the iterviewer tellig you the truth about the aual icomes? Techically, the perso was tellig the truth because the average is (geerally) defied to be the mea. However, of the three measures of cetral tedecy mea: $60,89, media: $,0, mode: $,00 it seems clear that the media is most represetative. The mea is iflated by the two highest salaries. Choosig a Measure of Cetral Tedecy Which of the three measures of cetral tedecy is most represetative of a particular data set? The aswer is that it depeds o the distributio of the data ad the way i which you pla to use the data. For istace, i Eample, the mea salary of $60,89 does ot seem very represetative to a potetial employee. To a city icome ta collector who wats to estimate % of the total icome of the employees, however, the mea is precisely the right measure. Eample Choosig a Measure of Cetral Tedecy Which measure of cetral tedecy is most represetative of the data give i each frequecy distributio? a. Number 6 7 8 9 b. c. Frequecy 7 0 8 0 Number 6 7 8 9 Frequecy 9 8 7 6 6 7 8 9 Number 6 7 8 9 TECHNOLOGY TIP Calculatig the mea ad media of a large data set ca become time cosumig. Most graphig utilities have mea ad media features that ca be used to fid the meas ad medias of data sets. Eter the data from Eample (a) i the list editor of a graphig utility. The use the mea ad media features to verify the solutio to Eample (a), as show below. Frequecy 6 8 0 Solutio a. For this data set, the mea is., the media is, ad the mode is. Of these, the media or mode is probably the most represetative measure. b. For this data set, the mea ad media are each ad the modes are ad 9 (the distributio is bimodal). Of these, the mea or media is the most represetative measure. c. For this data set, the mea is.9, the media is, ad the mode is 7. Of these, the mea or media is the most represetative measure. Now try Eercise. For istructios o how to use the list feature, the mea feature, ad the media feature, see Appedi A; for specific keystrokes, go to this tetbook s Olie Study Ceter. Variace ad Stadard Deviatio Very differet sets of umbers ca have the same mea. You will ow study two measures of dispersio, which give you a idea of how much the umbers i a set differ from the mea of the set. These two measures are called the variace of the set ad the stadard deviatio of the set.
Appedi C. Measures of Cetral Tedecy ad Dispersio A87 Defiitios of Variace ad Stadard Deviatio Cosider a set of umbers,,..., with a mea of. The variace of the set is v... ad the stadard deviatio of the set is letter sigma). v is the lowercase Greek The stadard deviatio of a set is a measure of how much a typical umber i the set differs from the mea. The greater the stadard deviatio, the more the umbers i the set vary from the mea. For istace, each of the followig sets has a mea of.,,,,,, 6, 6, ad,, 7, 7 The stadard deviatios of the sets are 0,, ad. 6 6 7 7 Eample Estimatios of Stadard Deviatio Cosider the three frequecy distributios represeted by the bar graphs i Figure C.. Which set has the smallest stadard deviatio? Which has the largest? 0 Set A Set B Set C Frequecy Frequecy Solutio Of the three sets, the umbers i set A are grouped most closely to the ceter ad the umbers i set C are the most dispersed. So, set A has the smallest stadard deviatio ad set C has the largest stadard deviatio. Now try Eercise 7. 6 7 6 7 6 7 Number Number Number Figure C. Frequecy STUDY TIP I Eample, you may fid it helpful to write each set umerically. For istace, set A is,,,,,,,,,,,,,, 6, 6, 7.
A88 Appedi C Cocepts i Statistics Eample Fidig a Stadard Deviatio Fid the stadard deviatio of each set show i Eample. Solutio Because of the symmetry of each bar graph, you ca coclude that each has a mea of. The stadard deviatio of set A is The stadard deviatio of set B is The stadard deviatio of set C is These values cofirm the results of Eample. That is, set A has the smallest stadard deviatio ad set C has the largest. The followig alterative formula provides a more efficiet way to compute the stadard deviatio. Because of legthy computatios, this formula is difficult to verify. Coceptually, however, the process is straightforward. It cosists of showig that the epressios ad ( 0.. 0. 0.. Now try Eercise....... are equivalet. Try verifyig this equivalece for the set,. 7 6 Alterative Formula for Stadard Deviatio The stadard deviatio of,,..., is give by...., with TECHNOLOGY TIP Calculatig the stadard deviatio of a large data set ca become time-cosumig. Most graphig utilities have statistical features that ca be used to fid differet statistical values of data sets. Eter the data from set A i Eample i the list editor of a graphig utility. The use the oe-variable statistics feature to verify the solutio to Eample, as show below. I the figure above, the stadard deviatio is represeted as, which is about.. For istructios o how to use the oe-variable statistics feature, see Appedi A; for specific keystrokes, go to this tetbook s Olie Study Ceter.
Eample Usig the Alterative Formula Appedi C. Measures of Cetral Tedecy ad Dispersio A89 Use the alterative formula for stadard deviatio to fid the stadard deviatio of the followig set of umbers., 6, 6, 7, 7, 8, 8, 8, 9, 0 Solutio Begi by fidig the mea of the set, which is 7.. So, the stadard deviatio is 6 7 8 9 0 68 0 0.76.0.. You ca use the oe-variable statistics feature of a graphig utility to check this result. Now try Eercise 7. 7. A well-kow theorem i statistics, called Chebychev s Theorem, states that at least k of the umbers i a distributio must lie withi k stadard deviatios of the mea. So, at least 7% of the umbers i a collectio must lie withi two stadard deviatios of the mea, ad at least 88.9% of the umbers must lie withi three stadard deviatios of the mea. For most distributios, these percets are low. For istace, i all three distributios show i Eample, 00% of the umbers lie withi two stadard deviatios of the mea. Eample 6 Describig a Distributio The table at the right shows the umber of outpatiet visits to hospitals (i millios) i each state ad the District of Columbia i 00. Fid the mea ad stadard deviatio of the umbers. What percet of the umbers lie withi two stadard deviatios of the mea? (Source: Health Forum) Solutio Begi by eterig the umbers i a graphig utility. The use the oe-variable statistics feature to obtai. ad The iterval that cotais all umbers that lie withi two stadard deviatios of the mea is..0,..0 or.08,.. From the table you ca see that all but two of the umbers (96%) lie i this iterval all but the umbers that correspod to the umbers of outpatiet visits to hospitals i Califoria ad New York. Now try Eercise..0. AK AL 9 AR AZ 7 CA 8 CO 7 CT 7 DC DE FL GA HI IA 0 ID IL 7 IN KS 6 KY 9 LA MA 0 MD 7 ME MI 7 MN 9 MO 6 MS MT NC ND NE NH NJ NM NV NY 8 OH 0 OK 6 OR 8 PA RI SC 7 SD TN 0 TX UT VA VT WA 0 WI WV 6 WY
A90 Appedi C Cocepts i Statistics Bo-ad-Whisker Plots Stadard deviatio is the measure of dispersio that is associated with the mea. Quartiles measure dispersio associated with the media. Defiitio of Quartiles Cosider a ordered set of umbers whose media is m. The lower quartile is the media of the umbers that occur o or before m. The upper quartile is the media of the umbers that occur o or after m. Eample 7 Fidig Quartiles of a Set Fid the lower ad upper quartiles of the followig set.,,, 6,, 8, 0,, 6, 6,, 7 Solutio Begi by orderig the set.,,, 6, 6, 8, 0,,, 6, 7, st % d % rd % th % The media of the etire set is 9. The media of the si umbers that are less tha 9 is. So, the lower quartile is. The media of the si umbers that are greater tha 9 is. So, the upper quartile is. Now try Eercise 7(a). Quartiles are represeted graphically by a bo-ad-whisker plot, as show i Figure C.. I the plot, otice that five umbers are listed: the smallest umber, the lower quartile, the media, the upper quartile, ad the largest umber. Also otice that the umbers are spaced proportioally, as though they were o a real umber lie. Figure C. 9 Figure C. TECHNOLOGY TIP You ca use a graphig utility to graph the bo-adwhisker plot i Figure C.. First eter the data i the graphig utility s list editor, as show i Figure C.. The use the statistical plottig feature to set up the bo-ad-whisker plot, as show i Figure C.. Fially, display the bo-ad-whisker plot (usig the ZoomStat feature), as show i Figure C.. For istructios o how to use the list editor ad the statistical plottig features, see Appedi A; for specific keystrokes, go to this tetbook s Olie Study Ceter. Figure C.. 9.8 6. 0. Figure C.
The et eample shows how to fid quartiles whe the umber of elemets i a set is ot divisible by. Appedi C. Measures of Cetral Tedecy ad Dispersio A9 Eample 8 Sketchig Bo-ad-Whisker Plots Sketch a bo-ad-whisker plot for each data set. a. 8, 8, 8, 8, 87, 89, 90, 9, 9, 9, 96, 98, 99 b.,,,, 7, 7, 0,,, 7 Solutio a. This set has umbers. The media is 90 (the seveth umber). The lower quartile is 8 (the media of the first si umbers). The upper quartile is 9. (the media of the last si umbers). See Figure C.6. 8 8 90 9. 99 Figure C.6 b. This set has 0 umbers. The media is 7 (the average of the fifth ad sith umbers). The lower quartile is (the media of the first five umbers). The upper quartile is (the media of the last five umbers). See Figure C.7. 7 7 Figure C.7 Now try Eercise 7(b). C. Eercises See www.calcchat.com for worked-out solutios to odd-umbered eercises. Vocabulary Check Fill i the blaks.. A sigle umber that is the most represetative of a data set is called a of.. If two umbers are tied for the most frequet occurrece, the collectio has two ad is called.. Two measures of dispersio are called the ad the of a data set.. measure dispersio associated with the media. I Eercises 6, fid the mea, media, ad mode of the data set..,, 7,, 8, 9, 7. 0, 7,, 9,,,.,, 7,, 8, 9, 7. 0, 7,, 9,,,.,, 7,, 9, 7 6. 0, 7,, 9,,
A9 Appedi C Cocepts i Statistics 7. Reasoig (a) Compare your aswers i Eercises ad with those i Eercises ad. Which of the measures of cetral tedecy is sesitive to etreme measuremets? Eplai your reasoig. (b) Add 6 to each measuremet i Eercise ad calculate the mea, media, ad mode of the revised measuremets. How are the measures of cetral tedecy chaged? (c) If a costat k is added to each measuremet i a set of data, how will the measures of cetral tedecy chage? 8. Cosumer Awareess A perso had the followig mothly bills for electricity. What are the mea ad media of the collectio of bills? Jauary $67.9 February $9.8 March $.00 April $.0 May $7.99 Jue $6. July $8.76 August $7.98 September $87.8 October $8.8 November $6. December $7.00 9. Car Retal A car retal compay kept the followig record of the umbers of miles a retal car was drive. What are the mea, media, ad mode of the data? Moday 0 Tuesday 60 Wedesday 0 Thursday 0 Friday 60 Saturday 0 0. Families A study was doe o families havig si childre. The table shows the umbers of families i the study with the idicated umbers of girls. Determie the mea, media, ad mode of the data. Number of girls 0 6 Frequecy 0 9 7. Bowlig Scores The table shows the bowlig scores for a three-game series of a three-member team. Team member Game Game Game Jay 8 96 Hak 99 9 0 Buck 0 (a) Fid the mea for each team member. (b) Fid the mea for the etire team for the three-game series. (c) Fid the media for the etire team for the three-game series.. Sellig Price The sellig prices of ew homes built i oe subdivisio are listed. $,000 $7,000 $,000 $0,000 $8,000 $00,000 $0,000 $,000 $7,000 $00,000 $0,000 $0,000 (a) Fid the mea, mode, ad media of the sellig prices. (b) Which measure of cetral tedecy best describes the prices? Eplai.. Thik About It Costruct a collectio of umbers that has the followig properties. If this is ot possible, eplai why.. Thik About It Costruct a collectio of umbers that has the followig properties. If this is ot possible, eplai why.. Test Scores A Eglish professor records the followig scores for a 00-poit eam. 99, 6, 80, 77, 9, 7, 87, 79, 9, 88, 90,, 0, 89,, 00, 98, 8, 78, 9 Which measure of cetral tedecy best describes these test scores? 6. Shoe Sales A salesma sold eight pairs of me s brow dress shoes. The sizes of the eight pairs were as follows: 0 8,, 0 0,, ad 0,, 9,. Which measure (or measures) of cetral tedecy best describes (describe) the typical shoe size for this data? I Eercises 7 ad 8, lie plots of data sets are give. Determie the mea ad stadard deviatio of each set. 7. (a) (b) (c) (d) 8. (a) Mea 6, media, mode Mea 6, media 6, mode 8 0 6 6 8 0 8 0 6 6 8 0 6 8
Appedi C. Measures of Cetral Tedecy ad Dispersio A9 (b) (c) (d) 6 8 6 8 6 8 I Eercises 9 6, fid the mea, variace v, ad stadard deviatio of the set. 9., 0, 8, 0.,, 6, 9,. 0,,,,,,,,.,,,,,.,,,,, 6, 7.,,,,,. 9, 6, 0, 9,, 70 6.., 0.,., 0.7, 0.8 I Eercises 7 0, use the alterative formula to fid the stadard deviatio of the set. 7.,, 6, 6,, 8. 6, 6, 7, 67, 9, 9 9. 8., 6.9,.7,., 6. 0. 9.0, 7.,., 7., 6.0. Reasoig Without calculatig the stadard deviatio, eplai why the set,, 0, 0 has a stadard deviatio of 8.. Reasoig If the stadard deviatio of a set of umbers is 0, what does this imply about the set?. Test Scores A istructor adds five poits to each studet s eam score. Will this chage the mea or stadard deviatio of the eam scores? Eplai.. Price of Gold The followig data represets the average prices of gold (i dollars per fie ouce) for the years 98 to 00. Use a computer or graphig utility to fid the mea, variace, ad stadard deviatio of the data. What percet of the data lies withi two stadard deviatios of the mea? (Source: Natioal Miig Associatio) 76,, 6, 7, 68, 7, 7, 8, 8, 6,, 60, 8, 8, 88,, 9, 79, 79, 7, 0, 6, 0,. Test Scores The scores o a mathematics eam give to 600 sciece ad egieerig studets at a college had a mea ad stadard deviatio of ad 8, respectively. Use Chebychev s Theorem to determie the itervals cotaiig at least ad at least 9 of the scores. How would the 8 itervals chage if the stadard deviatio were 6? 6. Thik About It The histograms represet the test scores of two classes of a college course i mathematics. Which histogram has the smaller stadard deviatio? Frequecy 6 86 90 9 98 Score Frequecy I Eercises 7 0, (a) fid the lower ad upper quartiles of the data ad (b) sketch a bo-ad-whisker plot for the data without usig a graphig utility. 7.,,,,,,, 0, 8., 0,,, 7, 6,,, 8,, 0 9. 6, 8, 8, 0,, 7,, 7, 9, 0., 0,, 8,, 8,, 9, 7, 9, 8, 8 88 9 96 Score I Eercises, use a graphig utility to create a bo-ad-whisker plot for the data.. 9,,, 9,,, 7,, 9,, 0, 9. 9,,,, 6,,,, 7, 0, 7,, 8, 9, 9. 0.,.,.9,.9,.,.,.,.,.7, 7.,.8,., 7., 6.,.8. 78., 76., 07., 78., 9., 90., 77.8, 7., 97., 7., 8.8, 6.6. Product Lifetime A compay has redesiged a product i a attempt to icrease the lifetime of the product. The two sets of data list the lifetimes (i moths) of 0 uits with the origial desig ad 0 uits with the ew desig. Create a bo-ad-whisker plot for each set of data, ad the commet o the differeces betwee the plots. Origial Desig. 78. 6. 68.9 0.6 7...7 7.7 0..0..0 8. 8. 0.8 8. 8. 0.0.6 New Desig.8 7..6 9.0. 7. 60.0. 8.9 80. 6.7. 67.9. 99..0...8 87.8 6
A9 Appedi C Cocepts i Statistics C. Least Squares Regressio I may of the eamples ad eercises i this tet, you have bee asked to use the regressio feature of a graphig utility to fid mathematical models for sets of data. The regressio feature of a graphig utility uses the method of least squares to fid a mathematical model for a set of data. As a measure of how well a model fits a set of data poits, y,, y,, y,...,, y you ca add the squares of the differeces betwee the actual y-values ad the values give by the model to obtai the sum of the squared differeces. For istace, the table shows the heights (i feet) ad the diameters y (i iches) of eight trees. The table also shows the values of a liear model y* 0. 9. for each -value. The sum of squared differeces for the model is.7. What you should lear Use the sum of squared differeces to determie a least squares regressio lie. Fid a least squares regressio lie for a set of data. Fid a least squares regressio parabola for a set of data. Why you should lear it The method of least squares provides a way of creatig a mathematical model for a set of data, which ca the be aalyzed. 70 7 7 76 8 78 77 80 y 8. 0..0..9.0 6. 8.0 y* 8. 9.8.0. 6..6.08.7 y y* 0. 0 0.096..90 7.808 8.9 The model that has the least sum of squared differeces is the least squares regressio lie for the data. The least squares regressio lie for the data i the table is y 0. 0.. The sum of squared differeces is.. To fid the least squares regressio lie y a b for the poits, y,, y,, y,...,, y algebraically, you eed to solve the followig system for a ad b. i b i a i y i I the system, b i a y i i... y i y y... y i... i y i y y... y. TECHNOLOGY SUPPORT For istructios o how to use the regressio feature, see Appedi A; for specific keystrokes, go to this tetbook s Olie Study Ceter. TECHNOLOGY TIP Recall from Sectio.7 that whe you use the regressio feature of a graphig utility, the program may output a correlatio coefficiet, r. Whe is close to, the model is a good fit for the data. r
Appedi C. Least Squares Regressio A9 Eample Fidig a Least Squares Regressio Lie Fid the least squares regressio lie for, 0,,, 0,, ad,. Solutio Begi by costructig a table, as show below. y y 0 0 9 0 0 0 6 i y i 6 i y i i Applyig the system for the least squares regressio lie with produces y i b a 6 b a. i y i i b i a b i a Solvig this system of equatios produces a 8 ad b 7 So, the least squares regressio lie is y 8 7 6, 6. as show i Figure C.8. Now try Eercise. Figure C.8 8 y = + 7 6 The least squares regressio parabola y a b c for the poits, y,, y,, y,...,, y is obtaied i a similar maer by solvig the followig system of three equatios i three ukows for a, b, ad c. c i b i a i c i b i a y i i y i i c i b i a i y i C. Eercises See www.calcchat.com for worked-out solutios to odd-umbered eercises. I Eercises, fid the least squares regressio lie for the poits. Verify your aswer with a graphig utility..,,,,,,, 6. 0,,, 0,,, 6,.,,,,,,,. 0,,,,,,,