UNIT 1 MEASURES OF CENTRAL TENDENCY

Transcription

1 UIT MEASURES OF CETRAL TEDECY Measures o Cetral Tedecy Structure Itroducto Objectves Measures o Cetral Tedecy 3 Armetc Mea 4 Weghted Mea 5 Meda 6 Mode 7 Geometrc Mea 8 Harmoc Mea 9 Partto Values Quartles Decles Percetles 0 Summary Solutos /Aswers ITRODUCTIO As we ow at ater e classcato ad tabulato o data oe ote ds too much detal or may uses at may be made o ormato avalable We, ereore, eed urer aalyss o e tabulated data to draw erece I s ut we are gog to dscuss about measures o cetral tedeces For e purpose o aalyss, very mportat ad powerul tool s a sgle average value at represets e etre mass o data The term average Statstcs reers to a oe gure summary o a dstrbuto It gves a value aroud whch e dstrbuto s cocetrated For s reaso at average s also called e measure o cetral tedecy For eample, suppose Mr X drves hs car at a average speed o 60 m/hr We get a dea at he drves ast (o Ida roads o course!) To compare e perormace o two classes, we ca compare e average scores e same test gve to ese two classes Thus, calculato o average codeses a dstrbuto to a sgle value at s supposed to represet e dstrbuto Ths helps bo dvdual assessmets o a dstrbuto as well as comparso w aoer dstrbuto Ths ut comprses some sectos as e Secto gves e deto o measures o cetral tedecy The sgcace ad propertes o a good measure o cetral tedecy are also descrbed Sub-sectos ad I sub sequet Sectos 3, 4, 5 ad 6, drect ad drect meods or calculatg Armetc mea, Weghted mea, Meda ad mode, respectvely are eplaed w er merts ad demerts, whereas Sectos 7 ad 8 meods or calculatg Geometrc mea ad Harmoc mea or ugrouped 7

2 Aalyss o Quattatve Data ad grouped data, respectvely are eplaed w er merts ad demerts The cocepts ad meods o calculatg e partto values are descrbed Secto 9 Objectves Ater studyg s ut, you would be able to dee a average; epla e sgcace o a measure o cetral tedecy; epla e propertes o a good average; calculate e deret types o measures o cetral tedecy; descrbe e merts ad demerts deret types o measures o cetral tedecy; ad descrbe e meods o calculato o partto values MEASURES OF CETRAL TEDECY Accordg to Proessor Bowley, averages are statstcal costats whch eable us to comprehed a sgle eort e sgcace o e whole They row lght as to how e values are cocetrated e cetral part o e dstrbuto For s reaso as o last page at ey are also called e measures o cetral tedecy, a average s a sgle value whch s cosdered as e most represetatve or a gve set o data Measures o cetral tedecy show e tedecy o some cetral value aroud whch data ted to cluster Sgcace o e Measure o Cetral Tedecy The ollowg are two ma reasos or studyg a average: To get a sgle represetatve Measure o cetral tedecy eables us to get a sgle value rom e mass o data ad also provde a dea about e etre data For eample t s mpossble to remember e heghts measuremet o all studets a class But e average heght s obtaed, we get a sgle value at represets e etre class To acltate comparso Measures o cetral tedecy eable us to compare two or more a two populatos by reducg e mass o data oe sgle gure The comparso ca be made eer at e same tme or over a perod o tme For eample, a subject has bee taught more a two classes so by obtag e average mars o ose classes, comparso ca be made Propertes o a Good Average The ollowg are e propertes o a good measure o average: It should be smple to uderstad 8 Sce we use e measures o cetral tedecy to smply e complety o a data, so a average should be uderstadable easly oerwse ts use s boud to be very lmted

3 It should be easy to calculate A average ot oly should be easy to uderstad but also should be smple to compute, so at t ca be used as wdely as possble 3 It should be rgdly deed A measure o cetral tedecy should be deed properly so at t has a approprate terpretato It should also have a algebrac ormula so at deret people compute e average rom same gures, ey get e same aswer 4 It should be lable or algebrac mapulatos A measure o cetral tedecy should be lable or e algebrac mapulatos I ere are two sets o data ad e dvdual ormato s avalable or bo set, e oe ca be able to d e ormato regardg e combed set also e someg s mssg 5 It should be least aected by samplg luctuatos We should preer a tool whch has a samplg stablty I oer words, we select 0 deret groups o observatos rom same populato ad compute e average o each group, e we should epect to get appromately e same values There may be lttle derece because o e samplg luctuato oly 6 It should be based o all e observatos I ay measure o cetral tedecy s used to aalyse e data, t s desrable at each ad every observato s used or ts calculato 7 It should be possble to calculate eve or ope-ed class tervals A measure o cetral tedecy should able to be calculated or e data w ope ed classes 8 It should ot be aected by etremely small or etremely large observatos It s assumed at each ad every observato lueces e value o e average I oe or two very small or very large observatos aect e average e eer crease or decrease ts value largely, e e average caot be cosder as a good average 3 Deret Measures o Cetral Tedecy The ollowg are e varous measures o cetral tedecy: Armetc Mea Weghted Mea 3 Meda 4 Mode 5 Geometrc Mea 6 Harmoc Mea 4 Partto Values Quartles Decles 3 Percetles Measures o Cetral Tedecy 9

4 Aalyss o Quattatve Data 0 3 ARITHMETIC MEA Armetc mea (also called mea) s deed as e sum o all e observatos dvded by e umber o observatos Armetc mea (AM) may be calculated or e ollowg two types o data: For Ugrouped Data For ugrouped data, armetc mea may be computed by applyg ay o e ollowg meods: () Drect Meod Maematcally,,,, are e observatos e er mea s ) ( X 3 X I s e requecy o (=,,, ), e ormula or armetc mea would be X X () Short-cut Meod The armetc mea ca also be calculated by tag devatos rom ay arbtrary pot A, whch e ormula shall be d A X where, d = A I s e requecy o (=,,, ), e ormula or armetc mea would be, d A X A d, where Here, s e umber o dstct observatos e dstrbuto ote: Usually e short-cut meod s used whe data are large Eample : Calculate mea o e weghts o ve studets 54, 56, 70, 45, 50 ( g) Soluto: I we deote e weght o studets by e mea s obtaed by

5 X Thus, X Thereore, average weght o studets s 55 g Eample : Compute armetc mea o e weght o studets or e gve data Eample by usg shortcut meod Soluto: For shortcut meod, we use ollowg ormula d X A, where d = - A I 50 s tae as e assumed value A e gve data Eample e, or e calculato o d we prepare ollowg table: Measures o Cetral Tedecy We have A = 50 e, d = -A = = = = =0 d 5 X A = 50 + = = 55 5 Eample 3: Calculate armetc mea or e ollowg data: d 5 Soluto: We have e ollowg requecy dstrbuto:

6 Aalyss o Quattatve Data Armetc Mea, X X E) Fd e armetc mea o e ollowg observatos: 5, 8,, 5, 0, 30 E) For e ollowg dscrete requecy dstrbuto d armetc mea: For Grouped Data Drect Meod I s e requecy o ( =,,, ) where s e md value o e class terval, e ormula or armetc mea would be X, X where, = Short-cut Meod, d A X A d, where Here, would be represetg e requecy o e class, s e md-value o e class ad s e umber o classes Eample 4: For e ollowg data, calculate armetc mea usg drect meod: Class Iterval Frequecy Wages ( Rs) o o worers

7 Soluto: We have e ollowg dstrbuto: Class Iterval Md Value Frequecy Mea = ow let us solve oe eercse ==8 = 760/8 = 743 = 760 Measures o Cetral Tedecy E3) Fd armetc mea o e dstrbuto o mars gve below: Mars o o studets Propertes o Armetc Mea Armetc mea ullls most o e propertes o a good average ecept e last two It s partcularly useul whe we are dealg w a sample as t s least aected by samplg luctuatos It s e most popular average ad should always be our rst choce uless ere s a strog reaso or ot usg t Three algebrac propertes o mea are gve below: Property : Sum o devatos o observatos rom er mea s zero Devato s also called dsperso at wll be dscussed detal Ut o s bloc Proo: We have to prove ( mea) = 0 The sum o devatos o observatos,,, rom er mea s = 0 Property : Sum o squares o devatos tae rom mea s least comparso to e same tae rom ay oer average Proo: We have A A where, A s a assumed mea / Meda / Mode 3

8 Aalyss o Quattatve Data ( ) A ( ) ( A) ( A) A ( ) ( A) 0 (By Property ) ( A) 0 ( A) That meas e sum o squares o devatos tae rom mea s least comparso to e same tae rom ay oer average Property 3: Armetc mea s aected by bo e chage o org ad scale a Proo: I u, h where, a ad h are costat The a h u a h a h X a h 3 Merts ad Demerts o Armetc Mea Merts o Armetc Mea U It utlzes all e observatos; It s rgdly deed; u 3 It s easy to uderstad ad compute; ad 4 It ca be used or urer maematcal treatmets Demerts o Armetc Mea It s badly aected by etremely small or etremely large values; It caot be calculated or ope ed class tervals; ad 3 It s geerally ot preerred or hghly sewed dstrbutos 4 WEIGHTED MEA u 4 Weght here reers to e mportace o a value a dstrbuto A smple logc s at a umber s as mportat e dstrbuto as e umber o tmes t appears So, e requecy o a umber ca also be ts weght But ere may be oer stuatos where we have to determe e weght based o some oer reasos For eample, e umber o gs whch rus were made

9 may be cosdered as weght because rus (50 or 00 or 00) show er mportace Calculatg e weghted mea o scores o several gs o a player, we may tae e streg o e oppoet (as judged by e proporto o matches lost by a team agast e oppoet) as e correspodg weght Hgher e proporto stroger would be e oppoet ad hece more would be e weght I has a weght w, e weghted mea s deed as: w X W or all =,, 3,, w Measures o Cetral Tedecy 5 MEDIA Meda s at value o e varable whch dvdes e whole dstrbuto to two equal parts Here, t may be oted at e data should be arraged ascedg or descedg order o magtude Whe e umber o observatos s odd e e meda s e mddle value o e data For eve umber o observatos, ere wll be two mddle values So we tae e armetc mea o ese two mddle values umber o e observatos below ad above e meda, are same Meda s ot aected by etremely large or etremely small values (as t correspods to e mddle value) ad t s also ot aected by ope ed class tervals I such stuatos, t s preerable comparso to mea It s also useul whe e dstrbuto s sewed (asymmetrc) Sewess wll be dscussed Ut 4 o s bloc Meda or Ugrouped Data Maematcally,,,, are e observatos e or obtag e meda rst o all we have to arrage ese values eer ascedg order or descedg order Whe e observatos are arraged ascedg or descedg order, e mddle value gves e meda s odd For eve umber o observatos ere wll be two mddle values So we tae e armetc mea o ese two values M d observato ; (whe s odd) M d observato observato ; ( whe s eve ) Eample 5: Fd meda o ollowg observatos: 6, 4, 3, 7, 8 Soluto: Frst we arrage e gve data ascedg order as 3, 4, 6, 7, 8 Sce, e umber o observatos e 5, s odd, so meda would be e mddle value at s 6 Eample 6: Calculate meda or e ollowg data: 7, 8, 9, 3, 4, 0 Soluto: Frst we arrage gve data ascedg order as 5

10 Aalyss o Quattatve Data 3, 4, 7, 8, 9, 0 Here, umber o observatos () = 6 (eve) So we get e meda by M M d d 6 3 rd observato 6 observato observato 4 observato E4) Fd e meda o e ollowg values: () 0, 6, 5,, 3,, 8 () 0, 6,, 3, 8, 5,, 5 observato observato For Ugrouped Data (whe requeces are gve) I are e deret value o varable w requeces cumulatve requeces rom e meda s deed by e we calculate M d = Value o varable correspodg to requecy = cumulatve ote: I / s ot e eact cumulatve requecy e value o e varable correspodg to et cumulatve requeces s e meda Eample 7: Fd Meda rom e gve requecy dstrbuto Soluto: Frst we d cumulatve requecy c

11 M d = Value o e varable correspodg to e 9 cumulatve requecy = Value o e varable correspodg to 95 sce 95 s ot amog c So, e et cumulatve requecy s ad e value o varable agast cumulatve requecy s 40 So meda s 40 Measures o Cetral Tedecy E5) Fd e meda o e ollowg requecy dstrbuto: Md Values oo studets Meda or Grouped Data For class terval, rst we d cumulatve requeces rom e gve requeces ad use e ollowg ormula or calculatg e meda: C Meda L h where, L = lower class lmt o e meda class, = total requecy, C = cumulatve requecy o e pre-meda class, = requecy o e meda class, ad h = wd o e meda class Meda class s e class whch e (/) observato alls I / s ot amog ay cumulatve requecy e et class to e / wll be cosdered as meda class Eample 8: Calculate meda or e data gve Eample 4 Soluto: We rst orm a cumulatve requecy table (cumulatve requecy o a class gves e umber o observatos less a e upper lmt o e class; strctly speag, s s called cumulatve requecy o less a type; we also have cumulatve requecy o more a type whch gves e umber o observatos greater a or equal to e lower lmt o a class): Class Iterval Frequecy Cumulatve Frequecy (< type)

12 Aalyss o Quattatve Data Sce 4 s ot amog e cumulatve requecy so e class w et cumulatve requecy e 5, whch s 0-30, s e meda class We have L = lower class lmt o e meda class = 0 = total requecy = 8 C = cumulatve requecy o e pre meda class =8 = requecy o e meda class = 7 h = wd o meda class = 0 ow substtutg all ese values e ormula o Meda Thereore, meda s 857 C Meda L h 4 8 M d = E6) Fd Meda or e ollowg requecy dstrbuto: Mars oo studets E7) Fd e mssg requecy whe meda s gve as Rs 50 Epedture (Rs) o o amles Merts ad Demerts o Meda Merts o Meda It s rgdly deed; It s easy to uderstad ad compute; 3 It s ot aected by etremely small or etremely large values; ad 4 It ca be calculated eve or ope ed classes (le less a 0 or 50 ad above ) Demerts o Meda I case o eve umber o observatos we get oly a estmate o e meda by tag e mea o e two mddle values We do t get ts eact value; It does ot utlze all e observatos The meda o,, 3 s I e observato 3 s replaced by ay umber hgher a or equal to ad e umber s replaced by ay umber lower a or equal to, e meda value wll be uaected Ths meas ad 3 are ot beg utlzed;

13 3 It s ot ameable to algebrac treatmet; ad 4 It s aected by samplg luctuatos Measures o Cetral Tedecy 6 MODE Hghest requet observato e dstrbuto s ow as mode I oer words, mode s at observato a dstrbuto whch has e mamum requecy For eample, whe we say at e average sze o shoes sold a shop s 7 t s e modal sze whch s sold most requetly For Ugrouped Data Maematcally,,,, are e observatos ad some o e observato are repeated e data, say s repeated hghest tmes e we ca say e would be e mode value Eample 9: Fd mode value or e gve data,, 3, 4, 7, 7, 7, 7, 9, 0,, Soluto: Frst we prepare requecy table as Ths table shows at 7 have e mamum requecy Thus, mode s 7 E8) Fd e model sze or e ollowg tems: 4, 7, 6, 5, 4, 7, 8, 3, 7,, 7, 6,,, 5 For Grouped Data: Data where several classes are gve, ollowg ormula o e mode s used 0 M 0 L h 0 where, L = lower class lmt o e modal class, 0 = requecy o e modal class, = requecy o e pre-modal class, = requecy o e post-modal class, ad h = wd o e modal class Modal class s at class whch has e mamum requecy Eample 0: For e data gve Eample 4, calculate mode Soluto: Here e requecy dstrbuto s 9

14 Aalyss o Quattatve Data Class Iterval Frequecy Correspodg to hghest requecy 9 model class s ad we have L = 40, = 9, o 7, 4, h 0 Applyg e ormula, Mode = = 486 E9) Calculate mode rom e data gve E6) 6 Relatoshp betwee Mea, Meda ad Mode For a symmetrcal dstrbuto e mea, meda ad mode cocde But e dstrbuto s moderately asymmetrcal, ere s a emprcal relatoshp betwee em The relatoshp s Mea Mode = 3 (Mea Meda) Mode = 3 Meda Mea ote: Usg s ormula, we ca calculate mea/meda/mode oer two o em are ow E0) I a asymmetrcal dstrbuto e mode ad mea are 354 ad 386 respectvely Calculate e meda 6 Merts ad Demerts o Mode Merts o Mode Mode s e easest average to uderstad ad also easy to calculate; It s ot aected by etreme values; 3 It ca be calculated or ope ed classes; 4 As ar as e modal class s cormed e pre-modal class ad e post modal class are o equal wd; ad 5 Mode ca be calculated eve e oer classes are o uequal wd 0 Demerts o Mode It s ot rgdly deed A dstrbuto ca have more a oe mode; It s ot utlzg all e observatos; 3 It s ot ameable to algebrac treatmet; ad 4 It s greatly aected by samplg luctuatos

15 7 GEOMETRIC MEA Measures o Cetral Tedecy The geometrc mea (GM) o observatos s deed as e - root o e product o e observatos It s useul or averagg ratos or proportos It s e deal average or calculatg de umbers (de umbers are ecoomc barometers whch relect e chage prces or commodty cosumpto e curret perod w respect to some base perod tae as stadard) It als to gve e correct average a observato s zero or egatve For Ugrouped Data I,,, are e observatos o a varable X e er geometrc mea s GM Tag log o bo sdes GM log GM = log log GM = log log log GM = Atlog log Eample : Fd geometrc mea o, 4, 8 Soluto: GM GM 3 Thus, geometrc mea s 4 E) Fd e GM o 4, 8, 6 E) Calculate GM o 5, 5, 5, 35 For Grouped data I,, are values (or md values case o class tervals) o a varable X w requeces,,, e where = GM GM Tag log o bo sdes

16 Aalyss o Quattatve Data log GM = log log GM = log log log GM = Atlog log Eample : For e data Eample 4, calculate geometrc mea Soluto: Class Md Value Frequecy log log () () = 8 log Usg e ormula GM = Atlog log 3875 = Atlog 8 = Atlog (3669) = 38 E3) Calculate GM o e ollowg dstrbuto Class Frequecy Merts ad Demerts o Geometrc Mea Merts o Geometrc Mea It s rgdly deed; It utlzes all e observatos; 3 It s ameable to algebrac treatmet (e reader should very at GM ad GM are Geometrc Meas o two seres-seres o sze ad Seres o sze m respectvely, e Geometrc Mea o e combed seres s gve by Log GM = ( GM + m GM ) / ( + m); 4 It gves more weght to small tems; ad 5 It s ot aected greatly by samplg luctuatos

17 Demerts o Geometrc Mea Dcult to uderstad ad calculate; ad It becomes magary or a odd umber o egatve observatos ad becomes zero or udeed a sgle observato s zero Measures o Cetral Tedecy 8 HARMOIC MEA The harmoc mea (HM) s deed as e recprocal (verse) o e armetc mea o e recprocals o e observatos o a set For Ugrouped Data I,,, are e observatos o a varable X, e er harmoc mea s HM HM Eample 3: Calculate e Harmoc mea o, 3, 5, 7 Soluto: Formula or harmoc mea s HM HM = 4 / ( ) = 4/676 = 39 E4) Calculate e Harmoc Mea o, 4, 6 For Grouped Data I,, are values (or md values case o class tervals) o a varable X w er correspodg requeces,,,, e HM 3

18 Aalyss o Quattatve Data HM where, = Whe equal dstaces are travelled at deret speeds, e average speed s calculated by e harmoc mea It caot be calculated a observato s zero Eample 4: For e data gve Eample 4, calculate harmoc mea Soluto: Class Md Value () Frequecy () / = = 8 / = 555 Usg e ormula, HM = 8/555 = 7956 E5) Calculate harmoc mea or e gve data: Class Frequecy Merts ad Demerts o Harmoc Mea Merts o Harmoc mea It s rgdly deed; It utlzes all e observatos; 3 It s ameable to algebrac treatmet; ad 4 It gves greater mportace to small tems Demerts o Harmoc Mea Dcult to uderstad ad compute 4 8 Relatos betwee AM, GM ad HM Relato : AM GM HM Proo: Let ad be two real umbers whch are o-zero ad o egatve The

19 AM GM HM Measures o Cetral Tedecy Cosder 0 so AM GM 0 () Aga 0 0 So by equatos () & () Relato : GM = so GM HM () AM GM HM AM HM Proo: Let ad be two real umbers whch are o-zero ad o egatve The AM HM GM AM HM So GM = AM HM 9 PARTITIO VALUES Partto values are ose values o varable whch dvde e dstrbuto to a certa umber o equal parts Here t may be oted at e data should be 5

20 Aalyss o Quattatve Data arraged ascedg or descedg order o magtude Commoly used partto values are quartles, decles ad percetles For eample, quartles dvde e data to our equal parts Smlarly, decles ad percetles dvde e dstrbuto to te ad hudred equal parts, respectvely 9 Quartles Quartles dvde whole dstrbuto to our equal parts There are ree quartles- st Quartle deoted as Q, d Quartle deoted as Q ad 3 rd Quartle as Q 3, whch dvde e whole data our parts st Quartle cotas e ¼ part o data, d Quartle cotas ½ o e data ad 3 rd Quartle cotas e ¾ part o data Here, t may be oted at e data should be arraged ascedg or descedg order o magtude For Ugrouped Data For obtag e quartles rst o all we have to arrage e data eer ascedg order or descedg order o er magtude The, we d e (/4), (/) ad (3/4) placed tem e arraged data or dg out e st Quartle (Q ), d Quartle (Q ) ad 3 rd Quartle (Q 3 ) respectvely The value o e (/4) placed tem would be e st Quartle (Q ), value o (/) placed tem would be e d Quartle (Q ) ad value o (3/4) placed tem would be e 3 rd Quartle (Q 3 ) o at data Maematcally,,, are values o a varable X e st Quartle (Q ), d Quartle (Q ) ad 3 rd Quartle (Q 3 ) are deed as: Frst Quartle (Q ) = placed tem e arraged data 4 Secod Quartle (Q ) = placed tem e arraged data Thrd Quartle (Q 3 ) = 3 placed tem e arraged data 4 For Grouped Data I,,, are values (or md values case o class tervals) o a varable X w er correspodg requeces,,,, e rst o all we orm a cumulatve requecy dstrbuto Ater at we determe e quartle class as smlar as we do case o meda The quartle s deoted by Q ad deed as Q C 4 L h or where, L = lower class lmt o quartle class, h = wd o e quartle class, = total requecy,,, 3 C = cumulatve requecy o pre quartle class, ad = requeces o quartle class 6

21 deotes quartle class It s e class whch observato alls 4 cumulatve requecy It s easy to see at e secod quartle ( = ) s e meda 9 Decles Decles dvde whole dstrbuto to te equal parts There are e decles D, D,,D 9 are ow as st Decle, d Decle,,9 Decle respectvely ad Decle cotas e (/0) part o data Here, t may be oted at e data should be arraged ascedg or descedg order o magtude For Ugrouped Data Measures o Cetral Tedecy For obtag e decles rst o all we have to arrage e data eer ascedg order or descedg order o er magtude The, we d e (/0), (/0),,, (9/0) placed tem e arraged data or dg out e st decle (D ), d decle (D ),, 9 decle (D 9 ) respectvely The value o e (/0) placed tem would be e st decle, value o (/0) placed tem would be e d decle Smlarly, e value o (9/0) placed tem would be e 9 decle o at data Maematcally,,, are values o a varable X e e decle s deed as: Decle (D ) = For Grouped Data placed tem e arraged data ( =,, 3,,9) 0 I,,, are values (or md values case o class tervals) o a varable X w er correspodg requeces,,,, e rst o all we orm a cumulatve requecy dstrbuto Ater at we determe e decles class as smlar as we do case o quartles The decle s deoted by D ad gve by D L C 0 h or where, L = lower class lmt o decle class, h = wd o e decle class, = total requecy,,,,9 C = cumulatve requecy o pre decle class; ad = requecy o decle class deotes decle class It s e class whch observato alls 0 cumulatve requecy It s easy to see at e quartle ( =5) s e meda 7

22 Aalyss o Quattatve Data 93 Percetles Percetles dvde whole dstrbuto to 00 equal parts There are ety e percetles P, P,,P 99 are ow as st percetle, d percetle,,99 percetle ad percetle cotas e (/00) part o data Here, t may be oted at e data should be arraged ascedg or descedg order o magtude For Ugrouped Data For obtag e percetles rst o all we have to arrage e data eer ascedg order or descedg order o er magtude The, we d e (/00), (/00),, (99/00) placed tem e arraged data or dg out e st percetle (P ), d percetle (P ),,99 percetle (P 99 ) respectvely The value o e (/00) placed tem would be e st percetle, value o (/00) placed tem would be e d percetle Smlarly, e value o (99/00) placed tem would be e 99 percetle o at data Maematcally,,, are values o a varable X e e percetle s deed as: Percetle (P ) = placed tem e arraged data( =,,,99) 00 For Grouped Data I,,, are values (or md values case o class tervals) o a varable X w er correspodg requeces,,,, e rst o all we orm a cumulatve requecy dstrbuto Ater at we determe e percetle class as smlar as we do case o meda The percetle s deoted by P ad gve by P C 00 L h or where, L = lower lmt o percetle class, h = wd o e percetle class, = total requecy,,,,99 C = cumulatve requecy o pre prcetle class; ad = requecy o percetle class deotes percetle class It s e class whch 00 alls cumulatve requecy It s easy to see at e te percetle ( =50) s e meda observato Eample 5: For e data gve Eample 4, calculate e rst ad rd quartles Soluto: Frst we d cumulatve requecy gve e ollowg cumulatve requecy table: 8

23 Measures o Cetral Tedecy Class Iterval Frequecy Cumulatve Frequecy (< type) = 8 Here, /4 = 8/4 = 7 The 7 observato alls e class 0-0 So, s s e rst quartle class 3/4 = observato alls class 30-40, so t s e rd quartle class For rst quartle L = 0, = 5, C = 3, = Q = 0 0 = 8 5 For rd quartle L = 30, = 9, C = 5 5 Q 3 = 30 0 = E6) Calculate e rst ad rd quartles or e data gve E6) 0 SUMMARY I s ut, we have dscussed: How to descrbe a average; The utlty o a average; 3 The propertes o a good average; 4 The deret types o averages alog w er merts ad demerts; ad 5 The deret ds o partto values SOLUTIOS / ASWERS E) For calculatg e armetc mea, we add all e observatos ad dvde by 6 as ollows: X 5 6 9

24 Aalyss o Quattatve Data Usg short-cut meod suppose e assumed mea A = d = A d 0 d 0 X A = 5 = 5 6 E) We have e ollowg requecy dstrbuto: Wages o o Worers = X Usg short cut meod w assumed mea A = 30 d = -30 d = 50 d =50 30

25 X A 6 d E3) We have e ollowg requecy dstrbuto: Measures o Cetral Tedecy Mars o o Studets Md Pots () X 5 Usg short-cut meod Mars 5 d d =A d 5 ow d X A h X 5 = E4) () Here = 7, ater arragg ascedg order, we get, 3, 6, 8, 0,, 5 ad e Meda value wll be 3

26 Aalyss o Quattatve Data E5) Meda = value o tem = value o 4 tem = 8 () Here = 8, so arragg ascedg order we get e values as, 3, 5, 6, 8, 0,, 5 ad ereore observato observato Md 4 Observato 5 Observato 6 8 M d = 7 Mars o o studets Cumulatve Frequecy ow, Meda = Value o e varable correspodg to e cumulatve requecy 00 = Value o e varable correspodg to e cumulatve requecy = Value o e varable correspodg to e 50 cumulatve requecy, Sce 50 s ot amog cumulatve requecy so e et cumulatve requecy s 67 ad e value o varable agast 67 s 30 Thereore 30 s e meda E6) Frst we shall calculate e cumulatve requecy dstrbuto Mars Cumulatve Frequecy = C = = Here 40,

27 Sce, 40 s ot e cumulatve requecy so, e class correspodg to e et cumulatve requecy 50 s meda class Thus s meda class Meda L C h = = 35 E7) Frst we shall calculate e cumulatve requecy dstrbuto Measures o Cetral Tedecy Class Cumulatve Frequecy Meda class s sce meda value s gve 50, Meda L C h = = 30 4 = 30-8 = E8) Frst we shall orm e requecy dstrbuto Sze Frequecy 4 From e requecy dstrbuto we see at sze 7 occurs w mamum requecy o 4 hece mode s 7 33

28 Aalyss o Quattatve Data E9) Frst we shall orm e requecy dstrbuto Mars Here mode class s correspodg to e hghest requecy 0 Mode 0 M0 L h = = 3384 E0) We have Mode = 3 Meda Mea 354 = 3 Meda (386) 3 Meda = Meda = 6/3 = 3753 E) We have GM = E) Frst we shall orm e requecy dstrbuto 0 log log

29 ow GM = atlog log 4797 = atlog 4 = atlog (993) =58 E3) We have e ollowg requecy dstrbuto: Measures o Cetral Tedecy Class log log log = GM = atlog log = atlog 80 = atlog (38) = 05 E4) We have e ormulae o e Harmoc mea HM By puttg e gve values HM

30 Aalyss o Quattatve Data E5) We have gve e ollowg requecy dstrbuto: Class / = Thereore by puttg e values ormulae, we get HM = 5 = 3 0 / 646 = 9 E6) Frst we shall calculate e cumulatve requecy dstrbuto Mars Cumulatve Frequecy = Here 40, Here, /4 = 80/4 = 0 The 0 observato alls e class 0-30 So, s s e rst quartle class 3/4 = 3 80/4 = 60 observato alls class 40-50, so t s e rd quartle class For rst quartle L = 0, = 5, C = 5, = 80 Q = = For rd quartle L = 40, =, C = Q 3 = 40 0=