Techical Assistace Documet 2009 Algebra I Stadard of Learig A.9 Ackowledgemets The Virgiia Departmet of Educatio wishes to express sicere thaks to J. Patrick Liter, Doa Meeks, Dr. Marcia Perry, Amy Siepka, Mike Traylor, ad Dr. Daiel Yates for their assistace i the developmet of this documet. Questios about this documet may be addressed to Michael Bollig at Michael.Bollig@doe.virgiia.gov or (804) 786-648. Virgiia Departmet of Educatio April 200
Stadard of Learig A.9 The studet, give a set of data, will iterpret variatio i real-world cotexts ad calculate ad iterpret mea absolute deviatio, stadard deviatio, ad z-scores. Itroductio The purpose of this documet is to assist teachers as they provide istructio ad assess studets o 2009 Stadard of Learig (SOL) A.9. A.9 is iteded to exted the study of descriptive statistics beyod the measures of ceter studied durig the middle grades. Although calculatio is icluded i SOL A.9, the istructio ad assessmet emphasis should be o uderstadig ad iterpretig statistical values associated with a data set icludig stadard deviatio, mea absolute deviatio, ad z-score. While ot explicitly icluded i this SOL, the arithmetic mea (2009 SOL 5.6), will be itegral to the study of descriptive statistics. I Algebra II, studets will cotiue the study of descriptive statistics as they aalyze the ormal distributio curve ad ormally distributed data ad apply statistical values to determie probabilities associated with areas uder the stadard ormal curve. The otatio ad formulas highlighted i shaded boxes will be used o the Algebra I Ed-of-Course (EOC) SOL assessmet ad icluded o the formula sheet for the Algebra I EOC SOL assessmet begiig i the sprig of 202. Defiitios, formulas, ad otatio The study of statistics icludes gatherig, displayig, aalyzig, iterpretig, ad makig predictios about a larger group of data (populatio) from a sample of those data. Data ca be gathered through scietific experimetatio, surveys, ad/or observatio of groups or pheomea. Numerical data gathered ca be displayed umerically or graphically (examples would iclude lie plots, histograms, ad stem-ad-leaf plots). Sample vs. Populatio Data Sample data ca be collected from a defied statistical populatio. Examples of a statistical populatio might iclude SOL scores of all Algebra I studets i Virgiia, the heights of every U.S. presidet, or the ages of every mathematics teacher i Virgiia. Sample data ca be aalyzed to make ifereces about the populatio. A data set, whether a sample or populatio, is comprised of idividual data poits referred to as elemets of the data set. A elemet of a data set will be represeted as i x, where i represets the i th term of the data set. Whe begiig to teach this stadard, teachers may wat to start with small, defied populatio data sets of approximately 30 items or less to assist i focusig o developmet of uderstadig ad iterpretatio of statistical values ad how they are related to ad affected by the elemets of the data set. Related to the discussio of samples versus populatios of data are discussios about otatio ad variable use. I formal statistics, the arithmetic mea of a populatio is represeted by the Greek letter μ (mu), while the calculated arithmetic mea of a sample is represeted by x, read x bar. Techical Assistace Documet for A.9 April 200
The arithmetic mea of a data set will be represeted by μ. O both brads of approved graphig calculators i Virgiia, the calculated arithmetic mea of a data set is represeted by x. Mea Absolute Deviatio vs. Variace ad Stadard Deviatio Statisticias like to measure ad aalyze the dispersio (spread) of the data set about the mea i order to assist i makig ifereces about the populatio. Oe measure of spread would be to fid the sum of the deviatios betwee each elemet ad the mea; however, this sum is always zero. There are two methods to overcome this mathematical dilemma: ) take the absolute value of the deviatios before fidig the average or 2) square the deviatios before fidig the average. The mea absolute deviatio uses the first method ad the variace ad stadard deviatio uses the secod. If either of these measures is to be computed by had, teachers should ot require studets to use data sets of more tha about 0 elemets. Please ote that studets have ot bee itroduced to summatio otatio prior to Algebra I. A itroductory lesso o how to iterpret the otatio will be ecessary. Examples of summatio otatio 5 4 i = + 2+ 3+ 4+ 5 xi = x + x + x + x i= i= 2 3 4 Mea Absolute Deviatio Mea absolute deviatio is oe measure of spread about the mea of a data set, as it is a way to address the dilemma of the sum of the deviatios of elemets from the mea beig equal to zero. The mea absolute deviatio is the arithmetic mea of the absolute values of the deviatios of elemets from the mea of a data set. xi μ i= Mea absolute deviatio =, where μ represets the mea of the data set, represets the umber of elemets i the data set, ad x i represets the i th elemet of the data set. The mea absolute deviatio is less affected by outlier data tha the variace ad stadard deviatio. Outliers are elemets that fall at least.5 times the iterquartile rage (IQR) below the first quartile ( ) or above the third quartile ( ). Graphig calculators idetify ad i the list of computed -varible statistics. Mea absolute deviatio caot be directly computed o the graphig calculator as ca the stadard deviatio. The mea absolute deviatio must be computed by had or by a series of keystrokes usig computatio with lists of data. More iformatio (keystrokes ad screeshots) o usig graphig calculators to compute this ca be foud later i this documet. Variace The secod way to address the dilemma of the sum of the deviatios of elemets from the mea beig equal to zero is to square the deviatios prior to fidig the arithmetic mea. The average Techical Assistace Documet for A.9 2 April 200
of the squared deviatios from the mea is kow as the variace, ad is aother measure of the spread of the elemets i a data set. Variace ( ) 2 i= σ = Stadard deviatio ( ) ( x μ ) 2 i = i= σ ( x μ ) 2 Techical Assistace Documet for A.9 3 April 200, where μ represets the mea of the data set, represets the umber of elemets i the data set, ad x i represets the i th elemet of the data set. The differeces betwee the elemets ad the arithmetic mea are squared so that the differeces do ot cacel each other out whe fidig the sum. Whe squarig the differeces, the uits of measure are squared ad larger differeces are weighted more heavily tha smaller differeces. I order to provide a measure of variatio i terms of the origial uits of the data, the square root of the variace is take, yieldig the stadard deviatio. Stadard Deviatio The stadard deviatio is the positive square root of the variace of the data set. The greater the value of the stadard deviatio, the more spread out the data are about the mea. The lesser (closer to 0) the value of the stadard deviatio, the closer the data are clustered about the mea. i, where μ represets the mea of the data set, represets the umber of elemets i the data set, ad x i represets the i th elemet of the data set. Ofte, textbooks will use two distict formulas for stadard deviatio. I these formulas, the Greek letter σ, writte ad read sigma, represets the stadard deviatio of a populatio, ad s represets the sample stadard deviatio. The populatio stadard deviatio ca be estimated by calculatig the sample stadard deviatio. The formulas for sample ad populatio stadard deviatio look very similar except that i the sample stadard deviatio formula, - is used istead of i the deomiator. The reaso for this is to accout for the possibility of greater variability of data i the populatio tha what is see i the sample. Whe - is used i the deomiator, the result is a larger umber. So, the calculated value of the sample stadard deviatio will be larger tha the populatio stadard deviatio. As sample sizes get larger ( gets larger), the differece betwee the sample stadard deviatio ad the populatio stadard deviatio gets smaller. The use of - to calculate the sample stadard deviatio is kow as Bessel s correctio. As a remider, i A.9 studets ad teachers should use the formula for stadard deviatio with i the deomiator as oted i the shaded box above. Whe usig Casio or Texas Istrumets (TI) graphig calculators to compute the stadard deviatio for a data set, two computatios for the stadard deviatio are give, oe for a populatio (usig i the deomiator) ad oe for a sample (usig i the deomiator). Studets should be asked to use the computatio of stadard deviatio for populatio data i istructio ad assessmets. O a Casio calculator, it is idicated with xσ ad o a TI
graphig calculator as σ x. More iformatio (keystrokes ad screeshots) o usig graphig calculators as a istructioal tool for 2009 SOL A.9 ca be foud later i this documet. z-scores A z-score, also called a stadard score, is a measure of positio derived from the mea ad stadard deviatio of the data set. I Algebra I, the z-score will be used to determie how may stadard deviatios a elemet is above or below the mea of the data set. It ca also be used to determie the value of the elemet, give the z-score of a ukow elemet ad the mea ad stadard deviatio of a data set. The z-score has a positive value if the elemet lies above the mea ad a egative value if the elemet lies below the mea. A z-score associated with a elemet of a data set is calculated by subtractig the mea of the data set from the elemet ad dividig the result by the stadard deviatio of the data set. x μ z-score( z) =, where x represets a elemet of the σ data set, μ represets the mea of the data set, ad σ represets the stadard deviatio of the data set. A z-score ca be computed for ay elemet of a data set; however, they are most useful i the aalysis of data sets that are ormally distributed. I Algebra II, z-scores will be used to determie the relative positio of elemets withi a ormally distributed data set, to compare two or more distict data sets that are distributed ormally, ad to determie percetiles ad probabilities associated with occurrece of data values withi a ormally distributed data set. Computatio of descriptive statistics usig graphig calculators Example Maya s compay produces a special product o 4 days oly each year. Her job requires that she report o productio at the ed of the 4 days. She recorded the umber of products produced each day (below) ad decided to use descriptive statistics to report o product productio. Computatio of Variace ad Stadard Deviatio Texas Istrumets (TI-83/84) Casio (9750/9850/9860) Eter the data ito L Eter the data ito List From the home scree click o TS. To eter data ito lists choose optio :Edit or e. I L, eter each elemet of the data set. Eter the first elemet ad press e to move to the ext lie ad cotiue util all the elemets have bee etered ito L. From the meu scree select W ad press l. I List, eter the data set. Eter the first elemet ad press l to move to the ext lie ad cotiue util all the elemets have bee etered ito List. Techical Assistace Documet for A.9 4 April 200
Calculate variace ad stadard deviatio by computig -Variable Statistics for L Press TS > Choose optio : -Var Stats or e Press e to compute the -variable statistics (defaults to L, eter list ame after -Var Stats if data are i aother list). Note: Screeshot from 9860/9750. The 9850 would ot show the SUB row. Calculate variace ad stadard deviatio by computig -Variable Statistics for List Press w(calc) Press q(var) Ÿ = arithmetic mea of the data set Σx = sum of the x values Σx = sum of the x values Sx = sample stadard deviatio σ x = populatio stadard deviatio = umber of data poits (elemets) x = arithmetic mea of the data set Σx = sum of the x values Σx = sum of the x values σ x = populatio stadard deviatio sx = sample stadard deviatio = umber of data poits (elemets) NOTE: σ x will represet the stadard deviatio (σ ). Squarig σ will yield the 2 variace ( σ ). NOTE: σ x will represet the stadard deviatio (σ ). Squarig σ will yield the 2 variace ( σ ). Techical Assistace Documet for A.9 5 April 200
Computatio of Mea Absolute Deviatio Texas Istrumets (TI-83/84) Casio (9750/9850/9860) Compute the mea absolute deviatio usig the data i L From the home scree click o TS. Choose optio :Edit or e. Press > to move to L2. Press : to highlight L2. Press e (to get L2 = at the bottom of the scree) m > Choose optio : abs ( or e Press ` (to get L )- v choose optio 5: Statistics... ad the choose optio 2: Ÿ (to get Ÿ or type i value) Press ) e (L2 will automatically fill with data) Compute the mea absolute deviatio usig the data i List Cotiuig from the calculatio of stadard deviatio, pressdutil the fuctio butto meus read GRAPH CALC TEST INTR DIST. Press $ to move the cursor to List 2. Press B util List 2 is highlighted. Press i [9750/9860]u(more-arrow right) q(num)q(abs) [9850]r(NUM)q(Abs) Press (iqq(to get List ) -oeyw(to get x or type i value))l (List 2 will automatically fill with data) L2 ow cotais the xi μ part of the mea absolute deviatio formula. I order to complete the calculatio of the mea absolute deviatio, fid the arithmetic mea of L2 by calculatig the -variable statistics of L2. Press S > Choose optio : -Var Stats or e Press `2 (to get L2 ) e to compute the -variable statistics for L2. List 2 ow cotais the xi μ part of the mea absolute deviatio formula. I order to complete the calculatio of the mea absolute deviatio, fid the arithmetic mea of List 2 by calculatig the -variable statistics of List 2. Cotiue to pressdutil the fuctio butto meus read GRAPH CALC TEST INTR DIST Press w(calc) u(set) [9750/9860]q2(to set it to List 2) [9850]w(List 2) Press ld. Techical Assistace Documet for A.9 6 April 200
Press [9750/9860]q(VAR) [9850]w(CALC)q(VAR) Ÿ = 8.5742857 from the -variable statistics of L2 represets the mea absolute deviatio of the origial data set recorded i L. x = 8.5742857 from the -variable statistics of List 2 represets the mea absolute deviatio of the origial data set recorded i List. Iterpretatio of descriptive statistics Example 2 Data set Data set 2 Number of Basketball Players Recorded Oce Each Day from April -4 Number of Basketball Players Recorded Oce Each Day from April 5-28 7 7 6 6 Frequecy 5 4 3 2 Frequecy 5 4 3 2 0 0-0 -20 2-30 3-40 4-50 5-60 6-70 -0-20 2-30 3-40 4-50 5-60 6-70 Number of Players Number of Players Mea = 45.0 Variace = 06. Stadard Deviatio = 0.3 Mea Absolute Deviatio = 9. Mea = 45.0 Variace = 420.3 Stadard Deviatio = 20.5 Mea Absolute Deviatio = 6 What types of ifereces ca be made about the data set with the give iformatio? The stadard deviatio of data set is less tha the stadard deviatio of data set 2. That tells us that there was less variatio (more cosistecy) i the umber of people playig basketball durig April -4 (data set ). Whe comparig the variace of the two data sets, the differece betwee the two idicates that data set 2 has much more dispersio of data tha data set. The stadard deviatio of data set 2 is almost twice the stadard deviatio of data set, idicatig that the elemets of data set 2 are more spread out with respect to the mea. Techical Assistace Documet for A.9 7 April 200
Give a stadard deviatio ad graphical represetatios of differet data sets, the stadard deviatio could be matched to the appropriate graph by comparig the spread of data i each graph. Whe a data set cotais clear outliers (the elemets with values of ad 2 i data set 2), the outlyig elemets have a lesser affect o the calculatio of the mea absolute deviatio tha o the stadard deviatio. How ca z-scores be used to make ifereces about data sets? A z-score ca be calculated for a specific elemet s value withi the set of data. The z-score for a elemet with value of 30 ca be computed for data set 2. z = 30 45 = 0.73 20.5 The value of 0.73 idicates that the elemet falls just uder oe stadard deviatio below (egative) the mea of the data set. If the mea, stadard deviatio, ad z-score are kow, the value of the elemet associated with the z-score ca be determied. For istace, give a stadard deviatio of 2.0 ad a mea of 8.0, what would be the value of the elemet associated with a z-score of.5? Sice the z-score is positive, the associated elemet lies above the mea. A z-score of.5 meas that the elemet falls.5 stadard deviatios above the mea. So, the elemet falls.5(2.0) = 3.0 poits above the mea of 8. Therefore, the z-score of.5 is associated with the elemet with a value of.0. Iterpretatio of descriptive statistics Example 3 Maya represeted the heights of boys i Mrs. Costatie s ad Mr. Kluge s classes o a lie plot ad calculated the mea ad stadard deviatio. Heights of Boys i Mrs. Costatie s ad Mr. Kluge s Classes (i iches) x x x x x x x x x x x x x x x x x x x x x 64 65 66 67 68 69 70 7 72 73 Mea = 68.4 Stadard Deviatio = 2.3 Note: I this problem, a small, defied populatio of the boys i Mrs. Costatie s ad Mr. Kluge s classes is assumed. How may elemets are above the mea? There are 9 elemets above the mea value of 68.4. Techical Assistace Documet for A.9 8 April 200
How may elemets are below the mea? There are 2 elemets below the mea value of 68.4. How may elemets fall withi oe stadard deviatio of the mea? There are 2 elemets that fall withi oe stadard deviatio of the mea. The values of the mea plus oe stadard deviatio ad the mea mius oe stadard deviatio ( μ σ =66. ad μ + σ =70.7) determie how may elemets fall withi oe stadard deviatio of the mea. I other words, all 2 elemets betwee μ σ ad μ + σ (boys that measure 67, 68, 69, or 70 ) are withi oe stadard deviatio of the mea. Applicatio scearios. Diae oversees productio of ball bearigs with a diameter of 0.5 iches at three locatios i the Uited States. She collects the stadard deviatio of a sample of ball bearigs each moth from each locatio to compare ad moitor productio. Stadard deviatio of 0.5 ich diameter ball bearig productio (i iches) July August September Plat locatio # 0.0 0.0 0.02 Plat locatio #2 0.02 0.04 0.05 Plat locatio #3 0.02 0.0 0.0 Compare ad cotrast the stadard deviatios from each plat locatio, lookig for treds or potetial issues with productio. What coclusios or questios might be raised from the statistical data provided? What other statistical iformatio ad/or other data might eed to be gathered i order for Diae to determie ext steps? Sample respose: Plat # ad Plat #3 had stadard deviatios that seemed steady, but oe would be wise to keep a eye o Plat # i the comig moths, because it had icreases i August ad September. A potetial cocer with the stadard deviatio of Plat #2 exists. The growig stadard deviatio idicates that there might be a issue with growig variability i the size of the ball bearigs. Plat #2 should be asked to take more frequet samples ad cotiue to moitor, to check the calibratio of the equipmet, ad/or to check for a equipmet problem. 2. Jim eeds to purchase a large umber of 20-watt florescet light bulbs for his compay. He has arrowed his search to two compaies offerig the 20-watt bulbs for the same price. The Bulb Emporium ad Lights-R-Us claim that their 20-watt bulbs last for 0,000 hours. Which descriptive statistic might assist Jim i makig the best purchase? Explai why it would assist him. Sample respose: The stadard deviatio of the lifespa of each compay s 20- watt bulbs should be compared. The bulbs with the lowest lifespa stadard deviatio will have the slightest variatio i umber of hours that the bulbs last. The bulbs with the slightest variatio i the umber of hours that they last meas that they are more likely to last close to 0,000 hours. Techical Assistace Documet for A.9 9 April 200
3. I a school district, Mr. Mills is i charge of SAT testig. I a meetig, the superitedet asks him how may studets scored less tha oe stadard deviatio below the mea o the mathematics portio of the SAT i 2009. He looks through his papers ad fids that the mea of the scores is 525 ad 653 studets took the SAT i 2009. He also foud a chart with percetages of z-scores o the SAT i 2009 as follows: z-score (mathematics) Percet of studets z < - 3 0. -3 < z < -2 2. -2 < z < - 3.6 - < z < 0 34.0 0 < z < 34.0 < z < 2 3.6 2 < z < 3 2. z > 3 0. How ca Mr. Mills determie the umber of studets that scored less tha oe stadard deviatio below the mea o the mathematics portio of the SAT? Sample respose: The z-score tells you how may stadard deviatios a elemet (i this case a score) is from the mea. If the z-score of a score is -, the that score is stadard deviatio below the mea. There are 5.8% (3.6% + 2.% + 0.%) of the scores that have a z-score < -, so there are about 26 (0.58 x 653) studets that scored less tha oe stadard deviatio below the mea. Questios to explore with studets. Give a frequecy graph, a stadard deviatio of, ad a mea of, how may elemets fall withi stadard deviatio(s) from the mea? Why? 2. Give the stadard deviatio ad mea or mea absolute deviatio ad mea, which frequecy graph would most likely represet the situatio ad why? 3. Give two data sets with the same mea ad differet spreads, which oe would best match a data set with a stadard deviatio or mea absolute deviatio of? How do you kow? 4. Give two frequecy graphs, explai why oe might have a larger stadard deviatio. 5. Give a data set with a mea of, a stadard deviatio of, ad a z-score of, what is the value of the elemet associated with the z-score? 6. What do z-scores tell you about positio of elemets with respect to the mea? How do z-scores relate to their associated elemet s value? 7. Give the stadard deviatio, the mea, ad the value of a elemet of the data set, explai how you would fid the associated z-score. Techical Assistace Documet for A.9 0 April 200