Measures of Spread: Variance and Standard Deviation

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Vocabulary Lesso 1-6 rage deviatios populatio variace σ 2 populatio stadard deviatio σ sample variace s 2 sample stadard deviatio s So far i this book you have studied the mea, the mode, the

1 Lesso 1-6 Measures of Spread: Variace ad Stadard Deviatio BIG IDEA Variace ad stadard deviatio deped o the mea of a set of umbers. Calculatig these measures of spread depeds o whether the set is a sample or populatio. Vocabulary Lesso 1-6 rage deviatios populatio variace σ 2 populatio stadard deviatio σ sample variace s 2 sample stadard deviatio s So far i this book you have studied the mea, the mode, the fiveumber summary, ad the IQR. These statistics help to describe the distributio of umbers i a data set. I the Activity below you will use these statistics to compare two give data sets. Activity 1 The two dot plots below display frequecy distributios of the height of the players o two hypothetical wome s basketball teams. Metal Math Ley s average score after 3 tests is 88. What score o the 4th test would brig Ley s average up to exactly 90? Dolphis Sweet Peppers height (i.) Step 1 Describe what you thik is the mai differece betwee the two dot plots from just lookig at the graphs. Step 2 Fid the mea, media, mode, fi ve-umber summary, ad IQR for each data set. Step 3 Determie if ay of the values from Step 2 are appropriate for distiguishig the mai differece betwee the heights of the members of the two teams. Justify your coclusios. I the Activity you may have cocluded that the IQR helps to distiguish betwee the spreads of the two dot plots. It is oe of the measures of spread of a distributio. Yet the IQR for the Sweet Peppers is 0 iches, which implies o spread. This umber is ot sesitive eough to idicate that there are Sweet Pepper players who are ot 5" tall. The simplest measure of the spread of a distributio is its rage. The rage is the differece of the maximum ad miimum values of the variable. Each team has a mea height of 5" ad a rage of 80" 0", or 10". I this way the distributios are quite similar. Yet the heights of the Dolphis seem more spread out tha those of the Sweet Peppers. Measures of Spread: Variace ad Stadard Deviatio 45

Whe the set of data is a populatio, Greek letters are used for mea, variace, ad stadard deviatio. The mea is labeled μ (mu), variace as σ 2 (sigma-squared), ad stadard deviatio as σ (sigma).

2 Chapter 1 So we eed other statistical measures to better describe the spread of the heights of the players of each team. Two measures of spread that are iflueced by every data poit are variace ad stadard deviatio. The Variace ad Stadard Deviatio of a Populatio The Dolphis ca be viewed as a populatio of te wome. Whe the set of data is a populatio, Greek letters are used for mea, variace, ad stadard deviatio. The mea is labeled μ (mu), variace as σ 2 (sigma-squared), ad stadard deviatio as σ (sigma). The variace for a populatio is calculated from the squares of deviatios, or differeces of each data value x i from the mea μ. The shortest Dolphi player is 0" tall. The deviatio x i μ for that player is (0 5) = 5 i. The square of her deviatio is 25 square iches. Aother of the Dolphi players is 8 iches tall. Her squared deviatio is (8 5) 2 = 9 i 2. The populatio variace is the mea of the squared deviatios. That is, where is the umber of objects i a populatio, the variace is the sum of the squared deviatios divided by. The populatio stadard deviatio is the square root of the populatio variace. Example 1 shows how you ca compute populatio variace ad stadard deviatio by had or by usig a statistics utility. Example 1 Fid the variace ad stadard deviatio for the heights of the Dolphis (treatig them as a populatio). Solutio To fi d the variace ad stadard deviatio it helps to orgaize the work step-by-step. Step 1 Write the data x i i a colum. Fid the mea by addig these umbers ad dividig by, the umber of data poits, which i this case is 10. Sice the sum is 50, μ = 5. Step 2 I the ext colum record the result of subtractig the mea from each score, yieldig x i μ, which i this case is x i 5. Deviatios are either positive, zero, or egative. Step 3 Square each deviatio ad record each result i the ext colum. Step 4 Add the squares of the deviatios. Divide the sum of the squared deviatios by, i this case 10, to obtai the variace σ 2. Step 5 Fid the square root of the variace to get the stadard deviatio σ. Results of these steps usig techology are show at the right ad without usig techology o the ext page. 46 Explorig Data

Lesso 1-6 Height (i.) x i Deviatio (i.) x i μ Square of Deviatio (i 2 ) (x i μ) 2 0 0-5 = 5 25 1 1-5 = 4 16 1 4 16 3 2 4 4 1 1 5 0 0 8 3 9 9 4 16 9 4 16 Sum 50 0 128 The mea μ is _ 50 10 = 5 i.

Also, otice that whe the deviatios are squared, values farther from the mea cotribute more to the variace tha values close to the mea.

3 Lesso 1-6 Height (i.) x i Deviatio (i.) x i μ Square of Deviatio (i 2 ) (x i μ) = = Sum The mea μ is _ = 5 i., the variace σ2 = _ = 64_ 5 = 12.8 i 2, ad the stadard deviatio σ = _ = _ = i. Notice that the sum of the deviatios equals 0. This is a great way to check your work. Also, otice that whe the deviatios are squared, values farther from the mea cotribute more to the variace tha values close to the mea. For istace, a height of 80 cotributes (80-5) 2 = 5 2 = 25 to the sum of squared deviatios, but 4 cotributes oly (4-5) 2 = ( 1) 2 = 1. Because of this, groups with more data close to the mea geerally have smaller stadard deviatios tha groups with more data far from the mea. Below is a picture of the Dolphis data showig how the stadard deviatio of 3.58 relates to the distributio. Dolphis μ σ σ height (i.) Basketball Begiigs Basketball was itroduced to wome at Smith College i 1892, just oe year after the game was iveted. You are asked to calculate the variace ad stadard deviatio for the Sweet Peppers i Questio 3. Formulas for the variace ad stadard deviatio are usually writte usig -otatio. For a set of umbers x 1, x 2, x 3,..., x each deviatio from the mea ca be writte as x i - μ, ad the square of the deviatio as (x i - μ) 2. Because the defiitio of the variace ad stadard deviatio are based o the mea, they ca be used oly whe it makes sese to calculate a mea. Measures of Spread: Variace ad Stadard Deviatio 4

4 Chapter 1 Defiitio of Variace ad Stadard Deviatio of a Populatio Let μ be the mea of the populatio data set x 1, x 2,..., x. The the variace σ 2 ad stadard deviatio σ of the populatio are ( x σ 2 sum of squared deviatios i μ ) 2 = i=1 = ad σ = _ variace = ( x i μ ) 2 i=1. Variace ad Stadard Deviatio of a Sample For may years statisticias oly used the populatio formulas. Over time, mathematicias ad statisticias established that dividig by for a sample variace did ot produce the best estimate of the populatio variace. They showed that whe usig a sample, dividig by 1 rather tha by provided a better estimate of populatio variace. Whe the data is a sample, Roma letters are used for mea, variace, ad stadard deviatio. The mea is labeled x (read x-bar ), variace is s 2, ad stadard deviatio is s. The oly differece i the formula is that 1 is used i place of. Defiitio of Variace ad Stadard Deviatio of a Sample Let x be the mea of the sample data set x 1, x 2, x. The the variace s 2 ad stadard deviatio s of the sample are ( x s 2 sum of squared deviatios i _ x ) 2 = i=1 = 1 1 ad s = _ variace = ( x i _ x ) 2 i=1. 1 CAUTION: I this book most of the data come from samples, so uless directed otherwise, use the variace ad stadard deviatio formulas for samples. GUIDED Example 2 Accordig to the U.S. Departmet of Agriculture, te to twety earthworms per cubic foot is a sig of healthy soil. Mr. Gree checked the soil i his garde by diggig oe-cubic-foot holes ad coutig the earthworms. He foud the followig couts: 4, 23, 15, 10, 8, 12, Explorig Data

5 Lesso 1-6 Calculate the sample variace ad sample stadard deviatio of the umbers of earthworms per cubic foot. Solutio Follow the same steps used i Example 1, but sice this data represets a sample, use the variace ad stadard deviatio formulas for samples. The symbols x i, x i _ x, ad (x i _ x ) 2 represet the earthworm cout, deviatio from the mea, ad squared deviatio, respectively. Cout Deviatio Square of Deviatio (worms) (worms) x i x i - x _ (worms squared) (x i - x _ ) ? ? ? 23.62? ? 5.14? sum x i =?? 0 i=1 i=1 ( xi - x ) 2 =? Wiggle, Squiggle, ad Squirm I oe acre of lad, you ca ofte fi d more tha oe millio worms. xi The mea x i=1 = _ = _ worms. ( xi _ x ) 2 The variace s 2 i=1 = =?_ =? worms squared.?? The stadard deviatio s = variace =?? worms, to two decimal places. Because of these differet formulas, some statistics utilities have two sets of symbols: s 2 ad s, ad σ ad σ 2. Other calculators ad programs use oly oe set of formulas for variace ad stadard deviatio. Activity 2 Use your calculator or statistics software to fi d (to the earest teth) the stadard deviatio of the followig data set. 89, 9, 4, 6, 99, 91, 84, 81 If more tha oe stadard deviatio is give, record both. (You should fi d that the mea is 83 ad both the sample ad populatio stadard deviatio are betwee 9 ad 11.) Measures of Spread: Variace ad Stadard Deviatio 49

6 Chapter 1 Questios COVERING THE IDEAS 1. State whether the statistic is a measure of ceter or a measure of spread. a. mea b. rage c. variace d. iterquartile rage e. stadard deviatio f. media 2. Multiple Choice The stadard deviatio of a set of scores is A the sum of the deviatios. B the differece betwee the highest ad lowest scores. C the score that occurs with the greatest frequecy. D oe of the above. 3. Use the heights of the Dolphis ad Sweet Peppers. a. Calculate the variace ad stadard deviatio for the heights of the Sweet Peppers usig the formulas for populatios. b. Fid the differece of the meas. c. Fid the differece of the rages. d. Fid the differece of the stadard deviatios. e. Explai i your ow words what the differeces i Parts b d tell you about the two data sets. 4. What statistics chage if the Dolphis ad Sweet Peppers are cosidered samples of all wome basketball players? 5. If the stadard deviatio s = 4.5 cm, fid the sample variace. I 6 ad, the measuremets refer to pulse rates of two studets while joggig, i beats per miute. Use a calculator or statistical software to fid the mea ad sample stadard deviatio for each situatio. 6. studet A with four measuremets: 100, 120, 115, 133. studet B with five measuremets: 110, 120, 124, 116, Suppose you used the formula for the sample stadard deviatio i Example 1. Would your aswer be greater tha, equal to, or less tha the populatio stadard deviatio show there? 9. Each of the followig situatios produces data that ca be summarized with mea ad stadard deviatio. Which would require populatio formulas for variace ad stadard deviatio rather tha sample formulas? A The Evirometal Protectio Agecy measures carbo mooxide cotet of air at 15 locatios of a metropolita area. B A algebra teacher has scores from his studets fial exams. C A cosumer magazie tests four cars from each of three brads of hybrid vehicles to evaluate operatig cost per mile. D Number of home rus per game for the Bosto Red Sox i the 2008 seaso. Los Ageles, CA 50 Explorig Data

7 Lesso 1-6 APPLYING THE MATHEMATICS 10. Suppose you kow the distace i miles each studet i a class lives from school. For this data set, state the uit for each statistic. a. mea b. rage c. variace d. stadard deviatio 11. Beth foud the variace of a data set to be 11. Why must her aswer be wrog? 12. Suppose two samples have the same mea, but differet stadard deviatios s 1 ad s 2, with s 1 < s 2. Which sample shows more variability? 13. a. Cosider the weights (i kilograms) of a group of deer. If the stadard deviatio is.8 kg, what is the variace? b. If the variace is 19 kg 2 6, what is the stadard 4 deviatio? Use the hypothetical frequecy distributios of 0 ACT scores for groups X, Y, ad Z at the right. a. Match each group with its best descriptio. i. cosistetly ear the mea ii. very widely spread iii. evely distributed. b. Without calculatig, tell which group s ACT scores have the greatest stadard deviatio ad which have the smallest. c. Verify your aswer to Part b with calculatios. 15. More tha 1.3 millio studets i the class of 200 took the ACT. O the mathematics sectio, μ = 21.0 ad σ = 5.1. Studets receive scores rouded to the earest whole umber. What is the iterval of studet scores that lie withi oe stadard deviatio of the mea? For 16 ad 1, use the followig data that represet the 0 times (rouded to the earest 5 secods) for 20 sixthgraders to ru 400 meters Fid a sample of five studets out of the 20 whose stadard deviatio for ruig time is as small as possible. 1. Fid a sample of four ruig times whose stadard deviatio is larger tha 25 secods. Compute the stadard deviatio. Frequecy Frequecy Frequecy Group X ACT Score Group Y ACT Score Group Z ACT Score Measures of Spread: Variace ad Stadard Deviatio 51

8 Chapter Multiple Choice A class of studets is said to be homogeeous if the studets i the class are very much alike o some measure. Here are four classes of studets who were tested o a 20-poit chemistry test. Which class is the most homogeeous with respect to scores o the test? Explai your aswer. A = 20 x = 15.3 s = 2.5 B = 25 x = 12.1 s = 5.4 C = 18 x = 11.3 s = 3.2 D = 30 x = 10.4 s = 3.2 REVIEW 19. Use the table below o seasoally adjusted U.S. domestic imports for 200. U.S. Total Imports i Goods ad Services ($ billios) TOTAL JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Total (per moth) Cumulative Total???????????? Source: U.S. Cesus Bureau a. Complete the table ad make both a histogram ad a cumulative data lie graph for imports. b. What was the total cost of U.S. domestic imports for 200? (Lessos 1-5, 1-3) 20. Two data sets of heights of people each have miimum = 50", media = 6", ad maximum = 80". Oe data set has IQR = 15"; the other has IQR = 10". a. Draw possible box plots for each data set. b. Which data set shows more spread? (Lessos 1-4) 21. The histogram below shows the umber of states receivig the umber of legal permaet residets specified i each iterval i Write a paragraph describig immigratio i Iclude both specific iformatio such as maximum, miimum, mea, or media values (whe possible), ad geeral treds such as skewess. (Lessos 1-3, 1-2, 1-1) Number of States Number of Legal Permaet Residets (thousads) Source: U.S. Departmet of Homelad Security 52 Explorig Data

9 Lesso 1-6 I 22 ad 23, use these data o the percet of Advaced Placemet Examiatios i Mathematics or Computer Sciece take by female high school studets. Year Percet Source: The College Board 22. a. Multiple Choice Which of the followig would be a appropriate graph for represetig these data? (There may be more tha oe correct choice.) A box plot B cumulative frequecy graph C histogram D lie graph b. Draw such a graph, ad describe tred(s) i the data. (Lessos 1-3, 1-1) 23. The total umber of studets takig AP Exams i Mathematics or Computer Sciece was about 121,000 i 1994 ad about 311,520 i a. What was the average aual chage i the umber of wome takig AP Exams i these areas durig this period? b. What was the average aual icrease i the umber of me takig these exams i this period? (Lesso 1-2, Previous Course) 24. Multiple Choice xi equals (Lesso 1-2) i=1 x A x. B _. C _ x. D oe of these. EXPLORATION 25. The Russia mathematicia Pafuti L. Chebychev ( ) proved a remarkable theorem called Chebychev s Iequality: I ay data set, if p is the fractio of the data that lies withi k stadard deviatios to either side of the mea, the p 1 1_ k. 2 a. Accordig to Chebychev s Theorem, what percet of a data set must lie withi 2 stadard deviatios of the mea? b. What percet must lie withi 3 stadard deviatios? c. Test Chebychev s Theorem o a data set of your choice. Measures of Spread: Variace ad Stadard Deviatio 53

Elementary Statistics

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