Lesso 3-3 Lesso 3-3 Traslatios of Data Vocabulary ivariat BIG IDEA Addig (or subtractig) the same value to every umber i a data set adds (or subtracts) that value to measures of ceter but does ot affect measures of spread. Traslatig Data The 2007 Uited States Ope Golf Champioship was played at the Oakmot Coutry Club i Pesylvaia. I golf, par is a predetermied umber of strokes that a good golfer should require to complete a hole. For the Oakmot course, pars o the 18 holes totaled 70 strokes, so the course was cosidered to be a par-70 course. The dimples o a golf ball make the ball A golfer s progress through the travel farther. 18 holes of a golf course is tracked by how may strokes the golfer is above or below par. For example, i his first roud of the Ope, Tiger Woods took 71 strokes, which was oe over par, or +1. Agel Cabrera, the evetual champio, took 69 strokes, which was oe uder par, or 1, for the roud. News reports give both the raw scores, 71 ad 69, ad the scores relative to a par, 1 ad 1. Notice that if a player s score is s, the his score i relatio to par is s 70. This is a example of a traslatio of data. Traslatios of data produce distributios that have predictable shapes, ceters, ad spreads. Metal Math Calculate the average score for fi ve basketball games that had scores of 92, 93, 94, 98, ad 98. QY A traslatio of a set of data {x 1,..., x } is a trasformatio that maps each x i to x i + h, where h is some costat. If T is the traslatio, the this trasformatio ca be described as T: x x + h or T(x) = x + h. QY A golf course is rated at par 72. If a player s stroke score is x, what is his score relative to par? The umber x + h, or the poit it represets, is the traslatio image of x. I the U.S. Ope, the trasformatio T mappig each umber of strokes oto its image has the rule T(x) = x - 70. Traslatios of Data 165
Chapter 3 Activity I the Uited States, the passage of the 19th amedmet to the Costitutio i 1920 gave wome the right to vote. This activity compares the year i which wome eared the right to vote i the U.S. to the year wome achieved that right i other coutries. Step 1 Eter the years below ito a statistics utility. Label the colum year. 1893 New Zealad 1920 Uited States 1949 Chia 1974 Jorda 1902 Australia 1921 Swede 1950 Idia 1976 Portugal 1906 Filad 1928 Britai 1954 Colombia 1989 Namibia 1913 Norway 1928 Irelad 1957 Malaysia 1990 Wester Samoa 1915 Demark 1931 Spai 1962 Algeria 1993 Kazakhsta 1917 Caada 1944 Frace 1963 Ira 1993 Moldova 1918 Austria 1945 Italy 1963 Morocco 1994 South Africa 1918 Germay 1947 Argetia 1964 Libya 2005 Kuwait 1918 Polad 1947 Japa 1967 Ecuador 1918 Russia 1947 Mexico 1971 Switzerlad 1919 Netherlads 1947 Pakista 1972 Bagladesh Source: New York Times Step 2 Make a histogram of the data. Use bi size 10. Compute the mea, media, mode, rage, IQR, ad stadard deviatio for the data. Step 3 To compare the year i which wome got the right to vote i differet coutries to the U.S., subtract 1920, the U.S. or baselie value, from all data poits. To do this, fi rst create a slider called baselie that takes o values from 0 to 2000. Title a secod colum ewyear. I the formula lie, eter = year - baselie. Step 4 Adjust your slider so that baselie = 1920. What is the adjusted value for Libya? For Filad? Iterpret each i a setece. Step 5 Compute the mea, media, mode, rage, IQR, ad stadard deviatio for the adjusted data. Which values are the same as the origial data? Which are differet? Step 6 Now make a histogram of the adjusted data. Use bi size 10. How does it compare to the histogram for the origial data? What happes to the histogram bars whe you move the slider? Step 7 Make box plots of both the origial ad the adjusted data. Agai, move the slider aroud. Describe how the box plot chages as you chage the data. Step 8 Which umbers i the fi ve-umber summary are affected by traslatig the data, ad which oes are uaffected by the traslatio? 166 Trasformatios of Graphs ad Data
Lesso 3-3 Measures of Ceter of Traslated Data Some of the results i the Activity are geeralized i this theorem. Theorem (Ceters of Traslated Data) Addig h to each umber i a data set adds h to each of the mea, media, ad mode. Proof Let {x 1,, x } be a data set. Cosider the traslatio i which x i is mapped to x i + h. To fi d the mea of the image set, you must evaluate ( x i + h) i = 1. By defi itio of Σ, this expressio represets terms (x 1 + h) + (x 2 + h) + (x 3 + h) + + (x + h). Usig the Associative ad Commutative Properties of Additio, we rewrite the expressio as terms (x 1 + x 2 + x 3 + + x ) + (h + h + h + + h) ( x i ) + h = x i = x i + h = + h = _ x + h. Thus uder a traslatio by h, the mea of the image set of data is h uits more tha the mea of the origial set of data. It also ca be show that after a traslatio of h uits, the media ad mode of the image set are also icreased by h uits. Measures of Spread for Traslated Data I the Activity, you saw that the rage of the distributios, the IQR, ad the stadard deviatio before ad after the traslatio remai uchaged. To see that this is true i geeral for a data set {x 1,, x } uder a traslatio by h, first recall that rage = maximum - miimum. Because a traslatio does ot chage the relative positios of the data poits, the miimum value of the traslated data is the image of the origial miimum, ad the maximum value of the traslated data is the image of the origial maximum. Traslatios of Data 167
Chapter 3 Origial Data Traslatio Image miimum maximum miimum maximum 1880 1900 1920 1940 1960 1980 2000 2020 segmet legth = rage -40-20 0 20 40 60 80 100 segmet legth = rage Uder a traslatio of h uits, the miimum m is mapped to m + h ad the maximum M is mapped to M + h. The rage of the traslated data is (M + h) - (m + h) = M - m, which is the rage of the origial data. Therefore the rage remais uchaged after the traslatio. I the calculatio of the variace ad stadard deviatio of image data uder a traslatio by h, the mea _ x becomes _ x + h. So, each ew deviatio equals (x i + h) - ( _ x + h) = x i - _ x, which is the origial deviatio. Because each idividual deviatio stays the same uder a traslatio, the variace ad stadard deviatio also stay the same. Theorem (Spreads of Traslated Data) Addig h to each umber i a data set does ot chage the rage, iterquartile rage, variace, or stadard deviatio of the data. Because the measures of spread of a data set do ot vary uder a traslatio, they are said to be ivariat uder a traslatio. The precedig theorems ca be used to compute measures of ceter or spread whe data are icreased or decreased. GUIDED Example I a local produce store, cataloupes sell for 99 per poud. A clerk weighed 30 melos ad computed the followig statistics: mea = 3.5 lb, stadard deviatio = 8 oz, media = 3.4 lb, ad IQR = 1 lb. After fi ishig his task, the clerk oticed that the scales were ot correctly calibrated. The scale was set at 3 oz as its startig weight, ot 0. Fid the correct values for the mea, stadard deviatio, media, ad iterquartile rage for the 30 melos. Solutio The clerk recorded the weight of each cataloupe as 3 ouces too heavy. The data eed to be traslated 3 ouces smaller. The mea ad the media will therefore be decreased by 3 ouces. Mea = 3.5 lb -? = 3 lb 8 oz -? =? Media = 3.4 lb -? = 3 lb 6.4 oz -? =? But the stadard deviatio ad IQR will be uchaged. Stadard deviatio =? ad IQR =? 168 Trasformatios of Graphs ad Data
Questios COVERING THE IDEAS Lesso 3-3 1. A trasformatio that maps a umber x to x + h is called a()?. 2. The box plots at the right show the distributio of December Salary December salaries for 10 employees i a start-up compay before ad after their year-ed bous was After Bous added. How much was the bous? Before Bous 3. Kayla withdrew the 5 amouts show below from a ATM while visitig her sister i aother tow. 2500 3500 4500 5500 6500 Withdrawal amouts: Salary $50 $100 $80 $120 $100. a. Fid the followig statistics for the five withdrawals. i. rage ii. mode iii. media iv. mea v. variace vi. stadard deviatio b. Sice Kayla was ot usig her ow bak s ATM, she was charged a fee of $1.50 for each trasactio. Use your aswers from Part a to compute the same statistics for the amouts withdraw icludig the trasactio fees. 4. Suppose x 1 = 3 = 5, x 3 = 6.1, x 4 = 2.4, ad x 5 = 3.2. Evaluate the give expressio. 5 5 a. (x i + 8) b. x i + 8 5. For a set of test scores, the mea was x ad the stadard deviatio was s. Later, every score was icreased by b bous poits. a. What is the mea of the traslated scores? b. What is the stadard deviatio of the traslated scores? 6. Name four statistical measures that are ivariat uder a traslatio. 7. A set of data is traslated. Fid the missig values i the table. Origial Data Trasformed Data cases 10? mea? 53 stadard deviatio 8.03? media 70 59.5 rage 23? IQR 12 12 Traslatios of Data 169
Chapter 3 8. Cosider the two frequecy distributios below. Origial Scores Frequecy 2 1 3 2 7 3 8 2 10 6 Trasformed Scores Frequecy 10 1 11 2 15 3 16 2 18 6 a. Make a dot-frequecy diagram showig the two sets of scores. b. Idetify the trasformatio used to get the trasformed scores. c. Fid the iterquartile rage, mode, mea, ad media for the origial scores. d. Use the theorems of the lesso ad your aswers from Parts b ad c to give the IQR, mode, mea, ad media for the trasformed data. APPLYING THE MATHEMATICS I 9 ad 10, cosider the data at the right, which give the scores from the first two tests i a class of 15 studets takig FST. 9. a. Eter these data ito a statistics utility, amig the lists score 1 ad score 2. b. Fid the meas ad stadard deviatios of score 1 ad score 2. c. Draw a scatterplot with score 1 o the horizotal axis ad score 2 o the vertical axis. d. Fid the correlatio coefficiet. e. Fid the lie of best fit for predictig score 2 from score 1. f. Suppose the teacher decided to add 5 bous poits to each score o the first test. Add 5 to each score 1. Draw a ew scatterplot. How is this scatterplot differet from the oe i Part c? g. Compute the correlatio coefficiet ad the least squares regressio equatio for the ew scores o test 1 ad the scores o test 2. Compare your aswers to those i Parts d ad e. What value(s) are ivariat uder this trasformatio? What value(s) have chaged? h. Suppose the teacher made a error whe computig the scores for the secod test, ad to correct his error he subtracted 3 poits from each score. Draw a scatterplot with the score 1 plus bous o the horizotal axis ad score 2 mius 3 o the vertical axis. How is this scatterplot differet from those draw i Parts c ad f? i. Compute the correlatio coefficiet ad the least squares regressio equatio for the trasformed scores. j. Compare the correlatio coefficiets, slopes ad itercepts. What value(s) are ivariat uder these trasformatios? What value(s) have chaged? Test 1 Test 2 84 79 84 81 66 61 98 98 70 67 86 83 74 48 60 61 88 88 93 91 81 81 90 96 82 87 57 53 92 95 Coutig o a calculator 170 Trasformatios of Graphs ad Data
Lesso 3-3 10. Multiple Choice Geeralize the results of Questio 9. If bivariate data are traslated, which (if ay) of the followig are ivariat? A meas of the two variables B stadard deviatios of the two variables C itercept of the lie of best fit D slope of the lie of best fit E correlatio coefficiet betwee the two variables 11. Metally calculate the mea of the heights of the starters o a basketball team: 6, 6 2, 6 3, 6 4, 6 6. 12. Let {x 1, x 3,..., x } be a data set with media m, ad h a costat. a. Suppose that is odd. Explai why the media of the set {x 1 + h + h,..., x + h} is m + h. b. Explai why the media of the traslated set is also m + h whe is eve. REVIEW 1 13. Cosider the equatio y + 3 =. (Lesso 3-2) x + 2 a. Graph the equatio usig paper ad pecil. b. Check your work with a graphig utility. c. Give a rule for the traslatio that maps the graph of y = 1 x oto the graph i Parts a ad b. 14. Suppose the traslatio T: (x, y) (x - 8, y + 13) is applied to the graph of the fuctio with equatio y = 1. (Lesso 3-2) x 2 a. Fid a equatio for the image. b. Sketch graphs of the image ad preimage. 15. Let a real fuctio f be defied by f: x x + 99. What is the domai of f? (Lesso 3-1) 16. Skill Sequece Solve for t. (Previous Course) a. rt = 18 b. rt = 7 + r c. r = 5 d. r + 4 = 1 t t 17. a. Draw ABC with A = (0, 2), B = (4, 4), ad C = (3, 0). b. Let r be the trasformatio that maps each poit (x, y) oto (2x, 3y). Draw A B C = r( ABC). c. What trasformatio is r? (Previous Course) EXPLORATION 18. Give a example of a situatio other tha a athletic evet where data are ofte traslated, ad explai why the traslated data are used. QY ANSWER x 72 Traslatios of Data 171