Statistical Techniques Employed in Atmospheric Sampling

Transcription

1 Appedx A Statstcal Techques Employed Atmospherc Samplg A.1 Itroducto Proper use of statstcs ad statstcal techques s ecessary for assessg the qualty of ambet ar samplg data. For a comprehesve dscusso of the subject of data qualty assessmet (DQA), revew EPA s techcal assstace documet, Gudace for Data Qualty Assessmet, Practcal Methods of Data Aalyss, EPA QA/G-9 (EPA/600/R-96/084), Jauary Ths referece documet provdes practcal demostratos o how to use the data qualty assessmet (DQA) techque evaluatg evrometal data sets ad shows how to apply some graphc ad statstcal tools for performg DQA. Ths chapter s teded as a troducto to statstcs ad statstcal cocepts ad ther use aalyzg ambet ar samplg data. Topcs addressed clude: (a) Data Qualty Objectves (DQO), (b) Data Plottg, (c) Measures of Cetral Tedecy, (d) Measures of Dsperso, ad (e) Dstrbuto Curves. Although these topcs are ot smple, they ca be uderstood ad used by o-statstcas. If a detaled statstcal aalyss of data s requred, t s recommeded that a expereced statstca be cosulted. Studets who could beeft from a revew of basc mathematcs ambet motorg are ecouraged to access the EPA Ar Polluto Trag Isttute course, SI 100: Mathematcs Revew for Ar Polluto Cotrol. Ths selfstructo course ca be foud at: I addto, the Uversty of Illos-Chcago, School of Publc Health- Evrometal ad Occupatoal Health Dvso, has developed a Iteretbased program ettled Itroducto to Evrometal Statstcs. Ths program s preseted as a vdeo seres three modules o topcs whch clude terpretg motorg data, samplg ad aalytcal lmtatos ad sample detecto lmts, ad qualty assurace ad qualty cotrol. Ths program ca be foud at: A-1

2 It s mportat to ote that the statstcal calculatos dscussed ths Appedx are best ad more easly performed by employg oe of may commercally avalable computer-based statstcal software packages. A. The Data Qualty Objectves (DQO) Process Whle the Data Qualty Objectves (DQO) Process s ot a statstcal techque per se, t s mportat because t helps to establsh crtera for data qualty ad the developmet of data collecto desgs. DQOs provde the approprate cotext for uderstadg the purpose of the ambet ar samplg ad aalyss data collecto effort. Also, they establsh the qualtatve ad quattatve crtera for assessg the qualty of the collected data set, based o the predefed teded use of data. Specfc formato o the Data Qualty Objectves Process ca be foud EPA documet, Gudace o Systematc Plag Usg the Data Qualty Objectve Process (EPA QA/G-4), at: DQOs are qualtatve ad quattatve statemets derved from the outputs of the frst sx steps of the DQO Process that ecompass the followg: Clarfy the study objectve. Defe the most approprate type of data to collect. Determe the most approprate codtos from whch to collect the data. Specfy tolerable lmts o decso errors whch wll be used as the bass for establshg the quatty ad qualty of data eeded to support the decso. The DQOs are the used to develop a scetfc ad resource-effectve data collecto desg. The Seve Steps of the DQO Process Step 1: Step : Step 3: Step 4: Step 5: State the Problem. Cocsely descrbe the problem to be studed. Revew pror studes ad exstg formato to ga a suffcet uderstadg to defe the problem. Idetfy the Goal of the Study. Idetfy what questos the study wll attempt to aswer. Idetfy Iformato Iputs. Idetfy the formato that eeds to be obtaed ad the measuremets that eed to be take to resolve the decso statemet. Defe Boudares of the Study. Specfy the tme perods ad spatal area to whch decsos wll apply. Determe whe ad where data should be collected. Develop the Aalytcal Approach. Defe the statstcal parameters of terest, specfy the acto level, ad tegrate the prevous DQO outputs to a sgle statemet that descrbes the logcal bass for choosg amog alteratve actos. A-

3 Step 6: Step 7: Specfy the Performace or Acceptace Crtera. Defe the decso maker s tolerable decso error rates based o a cosderato of the cosequeces of makg a correct decso. Develop the Pla for Obtag Data. Evaluate formato from the prevous steps ad geerate alteratve data collecto desgs. Choose the most resource-effectve desg that meets all DQOs. Outputs of the DQO Process The DQO Process leads to the developmet of a quattatve ad qualtatve framework for a study. Each step of the Process derves valuable crtera that wll be used to establsh the fal data collecto desg. The frst fve steps of the DQO Process detfy mostly qualtatve crtera, such as what problem has tated the study ad what decso t attempts to resolve. These steps also defe the type of data that wll be collected, where ad whe the data wll be collected, ad a decso rule that specfes how the decso wll be made. The sxth step defes quattatve crtera expressed as lmts o decso errors that the decso maker ca tolerate. The fal step s used to develop a data collecto desg based o the crtera developed the frst sx steps. The fal product of the DQO Process s a data collecto desg that meets the quattatve ad qualtatve eeds of the study. A.3 Data Collecto Desg A data collecto desg specfes the fal cofgurato of the evrometal motorg or measuremet effort requred to satsfy the DQOs. It desgates: the types ad quattes of samples or motorg formato to be collected; where, whe, ad uder what codtos they should be collected; what varables are to be measured; ad QA/QC procedures to esure that samplg desg ad measuremet errors are cotrolled suffcetly to meet the tolerable decso error rates specfed the DQOs. Data Plottg Data s usually uterpretable the form whch t s collected. I ths secto, we shall cosder the graphcal techques of summarzg such data so that the meagful formato ca be extracted from t. There are two kds of varables to whch we assg data: cotuous varables ad dscrete varables. A cotuous varable s oe that ca assume ay value some terval of values. Examples of cotuous varables are weght, volume, legth, tme, ad temperature. Most ar polluto data are take from cotuous varables. Dscrete varables, o the other had, are those varables whose possble values are tegers. Therefore, they volve coutg rather tha measurg. Examples of dscrete varables are the umber of sample statos, umber of people a room, ad umber of tmes a cotrol stadard s volated. Sce ay measurg A-3

4 devce s of lmted accuracy, measuremets real lfe are actually dscrete ature rather tha cotuous, but ths should ot keep us from regardg such varables as cotuous. Whe a weght s recorded as 165 pouds, t s assumed that the actual weght s somewhere betwee ad pouds. A.4 Graphcal Aalyss Frequecy Tables Let us cosder the set of data Table A-1, whch represets SO levels for a gve hour for 5 days. The frst step summarzg the data s to form a frequecy table. A frequecy table s a table prepared by dvdg a data set to selected uts or class tervals, the coutg ad sertg the umber of pots (frequecy of occurreces) wth the uts or class tervals. Table A- s a frequecy table prepared from the SO data set gve Table A-1. I costructg the frequecy table, we have dvded the 5 pots the data set to 11 class tervals wth each terval beg 15 uts legth. The choce of dvdg the data to 11 tervals was purely arbtrary. However, dealg wth data t s best to choose the legth of the class terval such that 8 to 15 tervals wll clude all of the data uder cosderato. Dervg the frequecy of occurrece colum volves othg more tha coutg the umber of values each terval. The relatve frequecy colum s obtaed by dvdg the umber of pots or frequecy of occurreces wth a ut by the total umber of evets wth the data set, whch ths example s 5. From observato of the frequecy table, we ca ow see the data takg form. The values appear to be clustered betwee 5 ad 85 ppb. I fact, early 80% are ths terval. A-4

5 Table A-1. SO levels. Days SO Cocetrato (ppb)* *ppb = parts per bllo collected SO levels. Class Iterval (ppb) Table A-. Frequecy table. Frequecy of Occurrece (total 5) Relatve Frequecy /5 = /5 = /5= /5= /5 = /5= /5= /5 = A-5

6 The Frequecy Polygo The ext step s to graph the formato the frequecy table. Oe way of dog ths would be to plot the frequecy for the mdpot of each class terval. The sold le coectg the pots of Fgure A-1 forms a frequecy polygo. Fgure A-1. Polluto cocetrato (mdpot of class terval) frequecy polygo. The Hstogram Aother method of graphg the formato would be by costructg a hstogram as show Fgure A-. The hstogram s a two-dmesoal graph whch the legth of the class terval s take to cosderato. The hstogram ca be a very useful tool statstcs, especally f we covert the gve frequecy scale to a relatve scale so that the sum of all the ordates equals oe. Ths s show Fgure A-3. Thus, each ordate value s derved by dvdg the orgal value by the umber of observatos the sample, ths case, 5. The advatage costructg a hstogram lke ths oe s that we ca read probabltes from t, f we ca assume a scale o the abscssa such that a gve value wll fall ay oe terval the area uder the curve that terval. For example, the probablty that a value wll fall betwee 55 ad 70 s equal to ts assocated terval's porto of the total area of tervals, whch s A-6

7 Fgure A-. Pollutat cocetrato hstogram of frequecy dstrbuto curve. Fgure A-3. Hstogram of percet frequecy dstrbuto curve. A-7

8 The Cumulatve Frequecy Dstrbuto Usg the frequecy table ad hstogram dscussed above, we ca costruct a cumulatve frequecy table ad curve as show Table A-3 ad Fgure A-4. Table A-3. Cumulatve frequecy table. SO level Cumulatve frequecy Relatve cumulatve frequecy Uder Fgure A-4. Cumulatve frequecy dstrbuto curve. A-8

9 The cumulatve frequecy table gves the umber of observatos less tha a gve value. Probabltes ca be read from the cumulatve frequecy curve or cumulatve frequecy table. For example, to fd the probablty that a value wll be less tha 85, we read up to the curve at the pot x = 85 ad across to the value 0.76 o the y-axs. A alteratve way to use the table s to go to the row where the SO level shows uder 85, the go across to the relatve cumulatve frequecy value of Dstrbuto of Data Whe we draw a hstogram for a set of data, we are represetg the dstrbuto of the data. Dfferet sets of data wll vary relato to oe aother ad, cosequetly, ther hstograms wll look dfferet. I ths chapter, we detfy three characterstcs that wll dstgush the dstrbutos of dfferet sets of data. These are cetral locato, dsperso, ad skewess. These are characterzed Fgure A-5. Curves A ad B have the same cetral locato, but B s more dspersed. However, both A ad B are symmetrcal ad are, therefore, sad ot to be skewed. Curve C s skewed to the rght ad has a dfferet cetral locato tha A ad B. Mathematcal measures of cetral locato ad dsperso wll be dscussed later. Fgure A-5. Relatve frequecy dstrbuto showg: Curve A ad B both cetrally located, Curve B beg more dspersed tha Curve A, ad the skewess of Curve C. Trasformato of Data I most statstcal work, data that closely approxmate a partcular symmetrcal curve, called the ormal curve, are requred. Both curves A ad B Fgure A-5 are examples of ormal curves. I dealg wth skewed curves, such as C the same fgure, t s desrable to trasform the data some way so that a symmetrcal curve resemblg the ormal curve s obtaed. Referrg to the frequecy table (Table A-) ad hstogram (Fgure A-) of the data used earler, t A-9

10 ca be see that for ths set of data, the dstrbuto s skewed ( the opposte drecto as Curve C above), hece the data are ot ormally dstrbuted. The Logarthmc Trasformato Oe of the most successful ways of dervg a symmetrcal dstrbuto from a skewed dstrbuto s by expressg the orgal data terms of logarthms. The logarthms of the orgal data are gve Table A-4. Arbtrarly dvdg the logarthmc data to e class tervals, each of 0.1 ut legth, we ca prepare the logarthmc frequecy table Table A-5. As ca be see Fgure A-6, a frequecy plot of the log trasformed data more closely approxmates a symmetrcal curve tha the arthmetc plot of the orgal data. Table A-4. Logarthmc trasformato. Day Pollutat coc. Log A-10

11 Class terval Table A-5. Logarthmc frequecy table. Frequecy of occurrece Cumulatve frequecy Relatve cumulatve frequecy Probablty Graph Paper Probablty graph paper s used the aalyss of cumulatve frequecy curves; for example, the graph paper ca be used as a rough test of whether the arthmetc or the logarthmc scale best approxmates a ormal dstrbuto. The scale, arthmetc or logarthmc, o whch the cumulatve frequecy dstrbuto of the data s more early a straght le, s the oe provdg the better approxmato to a ormal dstrbuto. Plottg the cumulatve dstrbuto curve of the data above o the two scales shows that the logarthmc scale yelds the better ft (Fgure A-6). Fgure A-6. Normalzed data plot vs. o-trasformed data. A-11

12 These probablty plots ca be used, f the data are ormally dstrbuted, to estmate the mea ad stadard devato of the data. The estmate of the mea, as wll be show later, s the 50 th percetle pot, ad the estmato of the stadard devato s the dstace from the 50 th percetle to the 16 th percetle. A percetle s a measure of the relatve posto of oe of several observatos relato to all of the observatos, ad provdes a measure of relatve stadg that s useful for summarzg data. Least-Square Lear Regresso If the lear relatoshp betwee two varables s sgfcat, a least-square lear regresso le, or le of best ft, may be draw to represet the data. Ths relatoshp ca the be used to determe the value of a ukow varable. For example, f the ambet ar cocetrato s ukow, but learly related to the respose of a ambet ar motor, we ca estmate the ambet ar cocetrato based o a observed respose from the ar motor. Algebracally, a straght le has the followg form: (Eq. A-1) y mx b Where: y = depedet varable plotted o the ordate (y-axs) x = explaatory varable (depedet varable) plotted o the abscssa (x-axs) b = the pot at whch the le tercepts the y-axs at x = 0 m = slope, whch shows how much of a chage of 1 ut of x affects y Lear regresso mmzes the vertcal dstace betwee all data pots ad the straght le (Fgure A-7). Fgure A-7. Lear regresso curve. The costats m ad b for the least-square le ca be determed usg the followg equatos: A-1

13 (Eq. A-) x y xy m x x (Eq. A-3) b y mx Where: = umber of observatos y = y ; x x Example Problem Calbrato of a ambet ar aalyzer s requred before t ca be used to provde relable ambet ar cocetrato measuremets. A typcal calbrato cossts of the troducto of kow ad certfed stadard cocetratos, typcally parts per mllo (ppm) over the lear operatoal rage of the strumet, ad the recordg of the correspodg respose of the strumet uts such as volts. Based o the recorded resposes ad the kow cocetratos, a least-square lear relatoshp betwee the varables ca be calculated ad subsequetly used to determe ambet cocetratos based o the respose of the aalyzer. The followg data were collected durg a calbrato of a chemlumescet NO aalyzer. x = Cocetrato NO x (ppm) y = Istrumet respose (volts) Values for m ad b for the least-square or best ft le ca be calculated from: x, y, x, x y,, y, ad x. Soluto: x x y xy 5 x x y y A-13

14 7.33 m b The equato for ths calbrato curve would be y = 1.6x 0.10, where y (the strumet respose volts) s equal to the ambet cocetrato ppm tmes the slope of the le whch s 1.6, plus the y-tercept of x, whch s To calculate ambet cocetratos ppm, we solve the equato for x : x x ppm ppm y b m y A.5 Measures of Cetral Tedecy Arthmetc Average, or Mea A basc way of summarzg data s by the computato of a cetral value. The most commoly used cetral value statstc s the arthmetc average, or the mea. Ths statstc s partcularly useful whe appled to a set of data havg a farly symmetrcal dstrbuto. The mea s a effcet statstc that t summarzes all the data the set, ad because each pece of data s take to accout ts computato. The formula for computg the mea s: (Eq. A-4) Where: = arthmetc mea = th measuremet = total umber of observatos The arthmetc mea s ot a perfect measure of the true cetral value of a gve data set. Arthmetc meas overemphasze the mportace of oe or two extreme data pots. May measuremets of a ormally dstrbuted data set wll have a arthmetc mea that closely approxmates the true cetral value. A-14

15 Example Problem Calculate the mea of 3.0,.5,., 3.4, 3.. Soluto: Meda Whe a dstrbuto of data s asymmetrcal, such as that of Fgure A-8, t s sometmes desrable to compute a dfferet measure of cetral value. Ths secod measure, kow as the meda, s smply the mddle value of a dstrbuto, or the quatty above whch half the data le ad below whch the other half of the data le. If data are lsted ther order of magtude (from lowest to hghest), the meda s the [(+1)/] value. If the umber of data s eve, the the umercal data of the meda s the value mdway betwee the two data earest the mddle. The meda, beg a postoal value, s less flueced by extreme values a dstrbuto tha the mea. Fgure A-8. Example of a asymmetrcal dstrbuto of data (meda vs. mea). A-15

16 Example Problem Fd the meda of, 10, 15, 8, 13, 18. Soluto: The data must frst be arraged order of magtude, such as: 8, 10, 13, 15, 18, Sce = 6, the meda s the 7/ = 3.5 value, thus the meda s 14, or the value halfway betwee 13 ad 15, sce ths data set has a eve umber of measuremets. Geometrc Mea Aother measure of cetral tedecy used more specalzed applcatos s the geometrc mea ( z ). The geometrc mea s defed by usg the followg equato: (Eq. A-5)... z 1 If scetfc calculators are ot avalable, a formula that more readly leds tself to a four-fucto calculator s: 1 Log 10 z Log 10 The formula s derved as follows. Log... Log 10 z Log Where: log s to base 10 1 but 1 1 Log Log ad Log Y Log LogY Therefore: Log Log Log 1 Log... 1 Log Log A-16

17 The geometrc mea s most ofte used for data whose causes behave expoetally rather tha learly, such as the growth of bactera, measuremets that are ratos, or logormal dstrbutos. I a dstrbuto shaped lke that of Fgure A-8, the geometrc mea, lke the meda, wll yeld a value closer to the ma cluster of values tha wll the mea. The arthmetc mea s always hgher tha the geometrc mea. Example Problem Calculate the geometrc mea of 3.0,.5,., 3.4, 3.. Soluto: Log z or Log 10 z z z A.6 Measures of Dsperso Measures of cetral tedecy are more meagful f accompaed by formato o measures of dsperso. Measures of dsperso descrbe how the data spread out from the ceter. Examples of measures of dsperso a data set clude the rage, sample stadard devato, coeffcet of varato, ad the stadard geometrc devato. The Rage The easest measure of dsperso of a set of data s the dfferece betwee the maxmum ad the mmum values the set, termed the rage. The rage does ot make full use of the formato cotaed the data, sce oly two of the data pots are take to accout. Thus the rage s a useful measure of varablty for data sets of 10 or less. A-17

18 Fgure A-9. Dsperso characterstc curves. Stadard Devato The most commoly used measure of dsperso, or varablty, of sets of data s the stadard devato. Its defg formula s gve by the expresso: (Eq. A-6) s 1 Where: s = the stadard devato (always postve) = th measuremet = the mea of the data sample = the umber of observatos The expresso shows how the devato of each measuremet from the overall mea s corporated to the stadard devato. A algebracally equvalet formula that makes computato much easer s: s 1 where the varables are defed as above. A-18

19 Example Problem: Stadard Devato Usg the data provded the followg table, calculate the stadard devato: Soluto: s s s s s s s Coeffcet of Varato The coeffcet of varato (CV) s a utless measure that allows the comparso of dsperso across several sets of data. It s the stadard devato dvded by the sample mea. The CV s ofte used evrometal applcatos because varablty (expressed as stadard devato) s ofte proportoal to the mea. A-19

20 (Eq. A-7) Where: s = stadard = sample mea CV s Example Problem: Coeffcet of Varato Use the data preseted the prevous example problem to solve for the CV. CV s CV CV Stadard Geometrc Devato Dsperso of skewed data such as logormal dstrbutos s measured by the stadard geometrc devato. The stadard geometrc devato s very smlar to the stadard devato. The dsperso the log of the measuremets s measured by the geometrc stadard devato stead of the dsperso of the measuremets whch would provde a arthmetc stadard devato. The log calculato ormalzes the data to better approxmate a ormal dstrbuto. The formula for calculatg the stadard geometrc devato s: (Eq. A-8) s z atlog Where: log s to the base 10 s z = stadard geometrc devato = th measuremet = the mea of the sample log 1 log The followg formula s mathematcally detcal, yet t s much easer to use calculato: s z atlog log 1 log 1 1 A-0

21 Example Problem: Stadard Geometrc Devato Usg the data provded the followg table, calculate the stadard geometrc devato: log log log. 541 log log [.e. (.541) ] s z atlog log 1 log 1 s z atlog s z atlog 4 1 s z atlog sz atlog sz atlog 0786 s z or 1.0 A-1

22 A.7 Dstrbuto Curves Dstrbuto curves are graphcal dsplays of the dvdual data pots a data set ad are mportat because they ca detfy patters ad treds data that mght go uotced f the data were ot plotted. May types of dstrbuto curves exst: bomal, t, ch, F, ormal, ad logormal are just a few of the exstg dstrbutos. However, ar polluto measuremets, the ormal ad logormal are the most commoly occurrg oes. Thus, oly these two wll be dscussed. The Normal Dstrbuto Oe reaso the ormal (Gaussa) dstrbuto s so mportat s that a umber of atural pheomea are ormally dstrbuted or closely approxmate t. I fact, may expermets whe repeated a large umber of tmes wll approach the ormal dstrbuto curve. I ts pure form, the ormal curve s a cotuous symmetrcal, smooth curve shaped lke the oe show Fgure A-10. Naturally, a fte dstrbuto of dscrete data ca oly approxmate ths curve. The ormal curve has the followg defte relatos to the descrptve measures of a dstrbuto. Fgure A-10. Normal dstrbuto curve. A-

23 The Mea ad Meda The ormal dstrbuto curve s symmetrcal; therefore, the mea ad the meda are equal ad are foud at the ceter of the curve. Recall that, geeral, the mea ad meda of a asymmetrcal dstrbuto do ot cocde. The Rage The ormal curve rages alog the x-axs from mus fty to plus fty. Therefore, the rage of a ormal dstrbuto s fte. The Stadard Devato The stadard devato, s, becomes a most meagful measure whe related to the ormal curve. A total of 68.% of the area lyg uder a ormal curve s cluded by the part ragg from 1 stadard devato below to 1 stadard devato above the mea. A total of 95.4% les + stadard devatos from the mea ad 99.7% les wth 3 stadard devatos (Fgure A-11). By usg tables foud statstcs texts ad hadbooks, oe ca determe the area lyg uder ay part of the ormal curve. Fgure A-11. Characterstcs of the ormal dstrbuto. These areas uder the ormal dstrbuto curve ca be gve probablty terpretatos. For example, f a expermet yelds a early ormal dstrbuto wth a mea equal to 30 ad a stadard devato of 10, we ca expect about 68% of a large umber of expermetal results to rage from 0 to 40, so that the probablty of ay partcular expermetal result's havg a value betwee 0 ad 40 s about I applyg the propertes of the ormal curve to the testg of data readgs, oe ca determe whether a chage the codtos beg measured s show A-3

24 or whether oly chace fluctuatos the readgs are represeted. For a wellestablshed set of crtero data, a frequetly used set of cotrol lmts s ± 3 stadard devatos. That s, a specal vestgato of data readgs tryg these lmts ca be used to determe whether the codtos uder whch the orgal data were take have chaged. Sce the lmts of 3 stadard devatos o ether sde of the mea clude 99.7% of the area uder the ormal curve, t s very ulkely that a readg outsde these lmts s due to the codtos producg the crtero set of data. The purpose of ths techque s to separate the purely chace fluctuatos from the other causes of varato. For example, f a log seres of observatos of a evrometal measuremet yeld a mea of 50 ad a stadard devato of 10, the cotrol lmts wll be set up as 50 ± 30 - other words, ± 3 stadard devatos, or from 0 to 80. So, a value of 81 would suggest that the uderlyg codtos have chaged, ad that a large umber of smlar observatos at ths tme would yeld a dstrbuto of results wth a mea dfferet (larger) tha 50. Ths process of determg whether a value represets a sgfcat chage s closely related to the use of cotrol charts. I settg up cotrol lmts, t s ofte ecessary to dvde the avalable data to subgroups ad calculate the mea ad stadard devatos of each of these groups, makg careful ote of the codtos prevalg uder each subgroup. I collectg data to establsh cotrol lmts, as much formato as possble should be gathered about the causes ad codtos effect durg the perod of obtag a crtero set of data. Geerally, the codtos durg ths perod should be ormal, or as much cotrol as possble. I the stuato where oe takes readgs of some evrometal quatty, the appearace of data beyod the cotrol lmts mght suggest the startg of a ew data groupg to further ascerta whether the uderlyg evrometal varable has chaged. It should be kept md that the lmts of ± 3 stadard devatos are tradtoal rather tha absolute. They have bee foud through experece to be very useful may cotrol stuatos, but each expermeter must decde what lmts would be most sutable for a gve purpose by determg what levels of probablty would be eeded to quatfy acceptace ad rejecto bouds. Logormal Dstrbutos Logormal dstrbutos ca best be demostrated by meas of a example: If hourly sulfur doxde cocetratos are plotted agast frequecy of occurrece as the Data Plottg Secto, a skewed dstrbuto would exst smlar to the oe Fgure A-1. Such a curve dcates that may cocetratos are close to zero ad that few are very hgh. Ulke temperature, sulfur doxde cocetratos are blocked o the left because values less tha zero do ot exst. Because umerous ads exst for ormal dstrbutos, t s desrable to ormalze ths type of dstrbuto. By plottg the log of hourly SO cocetratos agast the frequecy of occurrece, a bell-shaped curve smlar to Fgure A-10 s obtaed. By makg ths ample ormalzg feature, all exstg ormal dstrbuto tables ca be used to make probablty terpretatos. A-4

25 Fgure A-1. Frequecy vs. cocetrato of SO. A-5

26 Refereces Ambet Motorg Techology Iformato Ceter [Iteret]. Avalable at: The Clea Ar Act. 4 USC 85, School of Publc Health, Evrometal ad Occuapatoal Health Dvso, Uversty of Illos-Chcago. Course: Itroducto to Evrometal Statstcs [Iteret]. Avalable at: U.S. Evrometal Protecto Agecy July CFR Pt. 50, Appedx A. U.S. Evrometal Protecto Agecy. 40 CFR Pt. 53. U.S. Evrometal Protecto Agecy. 40 CFR Pt. 58. U.S. Evrometal Protecto Agecy Dec Fed. Reg U.S. Evrometal Protecto Agecy Oct Fed. Reg., o. 194, pp U.S. Evrometal Protecto Agecy, Ar Polluto Trag Isttute. Course SI 100: Mathematcs Revew for Ar Polluto Cotrol. U.S. Evrometal Protecto Agecy. Gudace o systematc plag usg the Data Qualty Objectves Process. EPA QA/G-4 (EPA/40/B-06/001). U.S. Evrometal Protecto Agecy. Qualty assurace hadbook for ar polluto measuremet systems. EPA 454/R A-6