The Institute of Chartered Accountants of Sri Lanka

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1 The Isttute of Chartered Accoutats of Sr Laka Executve Dploma Accoutg, Busess ad Strategy Quattatve Methods for Busess Studes Hadout 0: Presetato ad Aalyss of data Presetato of Data Arragg Data The arragemet or classfcato of the data some sort of systematc way of trasformg the data to formato. Classfcato of Qualtatve data Whe the umber of categores are small t s farly easy. However, whe the umber of categores are farly large, the basc classfcato volves tabulato Tabulato of Data The process of placg classfed data to tabular form s kow as tabulato. A table s a symmetrc arragemet of statstcal data rows ad colums. Rows are horzotal arragemets whereas colums are vertcal arragemets. It may be smple, double or complex depedg upo the type of classfcato. Types of Tabulato: (1) Smple Tabulato or Oe-way Tabulato: Whe the data are tabulated to oe characterstc, t s sad to be smple tabulato or oe-way tabulato. For Example: Tabulato of data o populato of world classfed by oe characterstc lke Relgo s example of smple tabulato. ) Double Tabulato or Two-way Tabulato: Whe the data are tabulated accordg to two characterstcs at a tme. It s sad to be double tabulato or two-way tabulato. 1

2 For Example: Tabulato of data o populato of world classfed by two characterstcs lke Relgo ad Sex s example of double tabulato. (3) Complex Tabulato: Whe the data are tabulated accordg to may characterstcs, t s sad to be complex tabulato. For Example: Tabulato of data o populato of world classfed by two characterstcs lke Relgo, Sex ad Lteracy etc s example of complex tabulato Example:- The followg table depcts lst of shops that sell watchers by usg teret. Sce the lst s so log to be preseted as t s, t s requred to preset the data a summery table. TYPE OF SHOP FREQUENCY RELATIVE FREQUENCY (%) JEWELLERY SHOPS FASHION SHOPS SUPERMARKETS OTHER TOTAL Classfcato of Quattatve data There are dfferet ways of classfyg quattatve data. It s bascally deped o the amout of data. Smple Data If the few values are occurred, t s ot requred to group the data. We called that ugrouped data. Frequecy Dstrbutos A dstrbuto s a collecto, array or group of umercal values. A frequecy dstrbuto s a lst of data classes or categorze alog wth the umber of values that fall to each.

3 Eg 1. The umber of customer arrvals to a shop each 15 mutes are show below: Eg. the umbers of package holders sold oe week by each of depedet travel agets were: Produce the grouped frequecy dstrbuto to preset the data, eg. Marks obtaed for mathematcs by 50 studets a school are lsted as follows Prepare a frequecy dstrbuto usg equal class tervals 30-34, ad so o. Relatve Frequecy Relatve frequeces are calculated by dvdg the actual frequecy for each class by the total umber of observatos beg classfed. Multply the relatve frequecy by 100 to arrve at percetage relatve frequecy. Cumulatve frequecy dstrbuto Cumulatve frequecy dstrbuto shows the total umber of occurreces that le above or below certa key values. There are two types of dstrbutos 1. Less tha Cumulatve frequecy dstrbuto.. More tha Cumulatve frequecy dstrbuto. A). Dagrammatc represetato Bar dagram Pe chart Pctogram 3

4 B). Graphc represetato Hstogram A hstogram s "a represetato of a frequecy dstrbuto by meas of rectagles whose wdths represet class tervals ad whose areas are proportoal to the correspodg frequeces." How Shall We Look at Hstograms? Of course, part of the power of hstograms s that they allow us to aalyze extremely large datasets by reducg them to a sgle graph that ca show prmary, secodary ad tertary peaks data as well as gve a vsual represetato of the statstcal sgfcace of those peaks. Frequecy polygo Frequecy polygos are a graphcal devce for uderstadg the shapes of dstrbutos. They serve the same purpose as hstograms, but are especally helpful for comparg sets of data. Frequecy polygos are also a good choce for dsplayg cumulatve frequecy dstrbutos. To create a frequecy polygo, start just as for hstograms, by choosg a class terval. The draw a X-axs represetg the values of the scores your data. Mark the mddle of each class terval wth a tck mark, ad label t wth the mddle value represeted by the class. Draw the Y-axs to dcate the frequecy of each class. Place a pot the mddle of each class terval at the heght correspodg to ts frequecy. Fally, coect the pots. You should clude oe class terval below the lowest value your data ad oe above the hghest value. The graph wll the touch the X-axs o both sdes. Cumulatve Frequecy Curve (Ogve) Ths s the graphcal represetato of cumulatve frequecy dstrbuto. There are two types of ogves, amely Less tha ogve ad More tha ogve. I a ogve data may be expressed usg a sgle le. A ogve (a cumulatve le graph) s best used whe you wat to dsplay the total at ay gve tme. The relatve slopes from pot to pot wll dcate greater or lesser creases; for example, a steeper slope meas a greater crease tha a more gradual slope. A ogve, however, s ot the deal graphc for showg comparsos betwee 4

5 categores because t smply combes the values each category, thus dcatg a accumulato (a growg or lesseg total). Percetage ogve: Eg. Stem ad leaf dagram The stem ad leaf plot s a devce to group data whle dsplayg most of the orgal data. Each score s cosdered to have two parts ( e. Stem ad leaf) The leadg dgt of a score s called stem. Eg. Costruct the stem ad leaf dagram for the followg data. 13, 46, 87, 16, 91, 5, 44, 7,, 10, 76, 3, 65, 3, 35, 43, 59, 75, 56, 64, 8, 36, 47, 53, 70, 8, 49. Eg. Daly sales of Alpha Compay s gve bellow: (Rs, 000) Show the above data a stem ad leaf dsplay. Eg. Daly sales of Beta Compay s gve bellow: (Rs, 000) Show the above data a stem ad leaf dsplay. Eg. Show the daly sales of both Alpha Compay ad Beta Compay a same stem ad leaf dsplay. 5

6 Aalyss of Data Measure of Cetral Locato To vestgate a set of quattatve data s useful to defe umercal measures that descrbe mportat features of the data. Oe of the mportat ways of descrbg the group of measuremets, whether t be a sample or populato, s by the use of a average. A sgle umber that s used to represets a set of umbers s called a average. Whe the data s arraged creasg or decreasg order magtude, the average value les at the ceter or close to the ceter of ths set. Hece the measuremet of the average s kow as measuremet of cetral tedecy. The most commoly used masseurs of cetral locato are Mea Meda Mode Mea By Summg all of the observatos ad dvdg by the umber of observatos ca obta a arthmetc mea. For a set of observatos x1, x, x3, x X 1 Arthmetc mea = If the observatos x1, x, x3, x occur wth frequeces f1, f, f3, f the Mea Example: f1x1 fx... f fx x f f f... f f The umbers of employees at fve dfferet stores are 3,5,6,4,6. Fd the mea umber of employees for the fve stores. Mea X

7 . Fd the mea of the umbers 8,6,6,5,1,9,5,8,8,8. Tabulate the data as follows: Number (x) Frequecy (y) fx f 10 fx 75 Mea fx f 10 Weghted Mea I calculatos of smple mea, all tems a seres are gve equal mportace. But practcal lfe, t may ot to so. I case some tems the dstrbuto carry more mportace tha others, the smple mea s ot the true represetatve average. To have a represetatve average such a case weghts are assged to each tem/value equal to t mportace. Weghts assg to varous values are ether estmate or arbtrarly fxed ad are ot the actual frequeces as gve a frequecy dstrbuto. Weghted mea s defed as the average obtaed by multplyg the varous values a seres by certa values kow as weghts ad the by dvdg the total of products so obtaed by the total weght. Example: 1. A tervew was coducted to detfy sutable caddate for the posts of Accoutats assstat. The marks obtaed for the test are as follows, Mathematcs 60, Ecoomcs 50, Accouts 34, Law 40. The weghts assged for are,3,4,1 respectvely. Calculate the Average marks of the studet. (60) (350) (434) (1 40) Weght mea X

8 Meda It s defed as the value that has a equal umber of observatos o ether sde of t whe the observatos are arraged ascedg or descedg order. I ths case there are odd umber of observatos. Whe there are eve umber of observatos, the arthmetc mea of the two mddle values ca be take as meda. Example: Ages of elderly people the home are 90,87,84,78 ad 63. The meda s 84. Cosder the frequecy dstrbuto (grouped data) Meda for the grouped data Meda : Where : / fa l c fb l: lower lmt of the meda class : Number of observatos f : summato of frequeces up to meda class a Mode f : frequecy of the meda class b The mode s the most commo observato the data. If there are two most commo values the the dstrbuto s sad to be bmodal ad t has two separate peaks. The mode does ot exst always. Example: Fd the mode of 9,10,5,9,7,9,6,8,10,11,9 Mode s 9 Mode for the grouped data 1 Mode = l 1 c Where l : Lower lmt of the mode class 1: Frequecy dfferece betwee mode class ad class below to the mode class : Frequecy dfferece betwee mode class ad class above to the mode class C : Class wdth 8

9 Measure of dsperso Rage The rage s the dfferece betwee the maxmum ad mmum values a data set. Rage = Maxmum value Mmum value Example: The largest mothly retur of a orgazato from Jauary 1980 to March 005 s 4.56 percet ad the smallest s 9.73 percet. The rage of returs s therefore 7.9 percet (4.56 percet - ( 9.73 percet)). Quartles ad the sem- terquartle rage Box Plot Mea devace Devato from the mea ca be calculated as follows X X Mea Devato = 1 I geeral the Mea devato ca be calculated as follows by cosderg frequeces Mea Devato = 1 f X f X Stadard devace The stadard devace measures the amout by whch the values a data collecto dffer from mea. Populato Stadard devato N 1 ( x ) Sample Stadard devato S N 1 ( x x) 1 Or presece of frequeces 9

10 Stadard devato N 1 f ( x x) f Varace The varace s the average of the squared dffereces betwee the data values ad the mea. Populato Varace N 1 ( x ) Sample Varace S N 1 ( x x) 1 Varace for frequecy dstrbuto Varace N 1 f ( x x) Coeffcet of Varato f Skewess Skewess s a measure of the degree of asymmetry of a dstrbuto. If the left tal (tal at small ed of the dstrbuto) s more proouced tha the rght tal (tal at the large ed of the dstrbuto), the fucto s sad to have egatve skewess. If the reverse s true, t has postve skewess. If the two are equal, t has zero skewess The Pearso mode skewess s defed by or 10

11 Exercses 1. Number of defectve tems 1000 uts samples are as follows: a. Determe the mode ad meda of ths dstrbuto b. Calculate the mea of the dstrbuto ad compare t wth mode ad meda. c. Draw a bar chart to represet the dstrbuto.. A stadard test were admstered to 30 studets to determe ther IQ scores. These scores are recorded as follows. Class terval (CI) Frequecy (f) 115 to less tha Compute 1. A Hstogram. A Frequecy Polygo 3. Less tha Cumulatve frequecy dagram (Ogve) 4. More tha Cumulatve frequecy dagram (Ogve) 11

12 3. The followg frequecy dstrbuto represets the umber of abset days of a employees durg a year. Number of days Number of employees Costruct a cumulatve frequecy dagram for the above data.. How may employees were abset for less tha 3 days durg the year. 3. Hoe may employees were abset for more tha 8 days durg the year. 4. Draw a frequecy polygo for ths data. 4. A large retaler s studyg the lead tme (tme betwee the recept of the order ad shpmet of the merchadse) for a sample of 40 orders receved the prevous moth. The lead tme days s reported as follows: Lead tme (days) Frequecy Total 40 a) Draw a frequecy polygo for ths data. b) Draw a hstogram for ths data. c) How may orders were delvered less tha 1 days? d) Covert the frequecy dstrbuto to less tha cumulatve frequecy dstrbuto. e) About 65% of the orders were delvered less tha how may days? f) How may orders had a lead tme of 1 days or more? 1

13 5. The followg are the scores for the md-term exam gve to 13 studets statstcs. 4,4,68,80,75,54,6,89,7,80,80,75,65 Calculate: a. Mea b. Meda c. Mode 6. The followg data represets the mothly salary ( Rs. 000) of seve employees of the factory. 6,30,6,9,8,60,X. the mea salary of a employee was computed as Rs Compute X. 7. Because of a specal sale o me s suts, a survey dcated that 4 suts were sold betwee 10:00 am ad 11:00 am o the sale day at Macy s. These suts were of the followg szes: 4, 38, 4, 48, 4, 4, 45, 4 46, 43, 50, 38, 4, 43, 4, 36 39, 41, 4, 45, 39, 39, 49, 4 Compute a) The mea b) The mode c) The meda Whch of these three measures of cetral tedecy most accurately represets the average sut sze. 8. The followg fgures represets the weghts ( pouds) of 10 ewbor babes at flushg hosptal o a gve day: 8, 6, 7, 7, 7, 5, 9, 7, 8, 6 a) Compete the mea, the mode ad the meda. b) What s the rage of the data? c) Compute the stadard devato. d) Compute the coeffcet of varato. e) Are these data skewed? If so how? 9. A elevator s desg to carry a maxmum load of 3,00 pouds. If 18 passegers are rdg the elevator wth a average weght of 154 pouds, s there ay dager that the elevator mght be over loaded? 13

14 10. I a car assembly plat, the cars were dagostcally checked after assembly ad before shppg them to the dealers. All such cars wth ay defect were retured for extra work. The umber of such detectve cars retur oe day for a 16 day perod s gve below: 30, 34, 10, 16, 8, 9,,, 6, 3, 5, 10, 15, 10, 8, 4 a) Fd the average umber of defectve cars retured for extra work per day. b) Fd the meda umber of defectve cars per day. c) Fd the mode for defectve cars per day. d) Compute the stadard devato. 11. Mddle aged persos were ecouraged to have ther blood pressure checked free of charge o a gve day by the Nursg Departmet of our college. The systolc blood pressure of the frst 0 persos s recorded as follows: 10, 118, 15, 160, 150, 134, 15, 145, 135, , 14, 139, 156, 135, 140, 16, 136, 148, 130 a) Compute the mea, the mode ad the meda systolc blood pressure. b) What s the rage of the data? c) Compute the terquartle rage. d) Compute the coeffcet of skewess. 1. The followg dstrbuto represets the perod moths that De Hard car batteres lasted before beg replaced. Class Iterval (CI) ( f ) 45 ad upto ad upto ad upto ad upto ad upto Total 14

15 a) Compute the average umber of moths that a battery lasted. b) Compute the meda of the data. c) Compute the varace ad the stadard devato of the data. d) Compute the terquartle rage. e) Compute the coeffcet of varato. f) Compute the coeffcet of skewess. 13. Professor Alexader 50 studets hs Statstcs class. I the fal exam, the marks obtaed by hs studet rage from low of 0 to a hgh of 98. He arraged the scores to 4 groups wth a class terval of 0 pots. The data s preseted a frequecy dstrbuto as follows: Class Iterval (CI) ( f ) 0 ad less tha ad less tha ad less tha ad less tha Total a) Fd the average score the fal exam. b) If Professor Alexader fals every studets who gets a score of 40 or less, ca we fd, from the dstrbuto as gve, the percetage of studets that faled the exam? c) Compute the varace ad the stadard devato of ths data. d) Fd the meda score form ths data. 15