The Institute of Chartered Accountants of Sri Lanka

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1 The Insttute of Chartered Accountants of Sr Lanka Postgraduate Dploma n Accountng, Busness and Strategy Quanttatve Methods for Busness Studes Handout 0: Presentaton and Analyss of data Tables and Charts for Categorcal Data. One way tabulaton It s a table wth two columns. One column lsts the categores, and the other for the frequences or percentages wth whch the tems n the categores occur. Example: Two way tabulaton When the data are tabulated accordng to two characterstcs at a tme, t s sad to be double tabulaton or two-way tabulaton. Example: Complex Tabulaton When the data are tabulated accordng to many characterstcs, t s sad to be complex tabulaton. Example Product Sales. (Rs 000s) Quantty Dscount Proft (Rs 000s) A B C D

2 Pe Charts A pe chart s a crcle wth wedges cut of varyng szes marked out lke slces of a pe or pzza. The relatve szes of the wedges correspond to the relatve frequences of the categores. How Do You Spend the Holday's 5% 7% 5% 38% 45% At home wth famly Travel to vst famly Vacaton Catchng up on work Other Smple bar Chart In bar charts, each category of data s depcted by a bar, the heght of whch represents the frequency or percentage of observatons fallng nto a category. Multple bar chart In a multple bar chart two or more sets of nter-related data are represented facltatng comparson between more than one phenomena. Stacked Bar chart In a stacked bar chart, data seres are stacked one on top of the other n vertcal columns.

3 Pctogram A pctogram s a statstcal graphc n whch the sze of the pcture s ntended to represent the frequences or sze of the values beng represented. Organzng Numercal data. As the number of observatons gets large, t becomes more and more dffcult to focus on the major features n a set of data. We need ways to organze the observatons so that we can better understand what nformaton the data s conveyng. Large sets of data are presented under groups to facltate better presentaton. Presentng Numercal data n tables and graphs Frequency Dstrbuton A frequency dstrbuton s a table n whch the data s arranged nto convenently establshed, numercally ordered class groupngs or categores. Frequency Dstrbuton for Ungrouped Data When the number of values n a varable s small, an ungrouped frequency dstrbuton would be approprate. Exercse 0 famles were surveyed to fnd how many chldren they had. The data obtaned were as follows. 0,,3,1,1,3,4,,0,3,4,,,1,0,4,1,,,3 Construct a frequency dstrbuton. Frequency Dstrbuton for Grouped Data Sometmes t s mpractcal to prepare a frequency dstrbuton table usng ungrouped data. Ths s partcularly true when there s a large number of observatons. In cases such cases t s better to collect the observatons nto groups or classes wth clearly defned upper and lower lmts. The followng steps have to be followed to construct a frequency dstrbuton for a grouped set of data. Fnd the range of the dstrbuton. (e dfference between hghest and lowest ) Select class ntervals of a convenent sze. For most dstrbutons about 6 to 1 classes wll be suffcent. Usually nterval wdths of 5, 10, 0 and so on are sutable. Mark the number of values fallng wthn each class nterval usng tally marks and construct the frequency dstrbuton table.

4 Important defntons relatng to frequency dstrbuton. Lower class lmt These are the smallest numbers that can actually belong to dfferent classes. Upper class lmt These are the largest numbers that can actually belong to dfferent classes Class wdth Ths s the dfference between two consecutve lower class lmts or upper class lmts. Cumulatve Frequency dstrbutons The cumulatve frequency of a class s the sum of frequences for that class and all prevous classes. Relatve Frequency Dstrbuton The relatve frequency of a class can be found by dvdng the class frequency by the total of all frequences. Ths can also be gven as a percentage. Relatve frequency = class frequency / total frequency Frequency Hstogram A hstogram s a column or bar graph of a frequency table. However, unlke bar graphs, there are no gaps between adjacent bars. Frequency Polygon A frequency polygon s a lne graph drawn by jonng the md-ponts of the tops of each rectangle n a hstogram wth straght lnes. To construct a frequency polygon frequences should be plotted aganst the md ponts of each nterval and joned wth a straght lne graph. Ogve or Cumulatve Frequency Graph An ogve s the graphcal presentaton of a cumulatve frequency dstrbuton. Ogves are used when t s consdered more useful to determne the number (or proporton) of data tems that fall above or below a partcular value rather than wthn a gven nterval. There are types of ogves - less than and greater than ogves. The followng steps have to be followed to construct a less than ogve. 1 Compute the cumulatve frequences of the dstrbuton n ascendng order. Prepare a graph wth the cumulatve frequency on the vertcal axs and the class ntervals on the horzontal axs. 3 After plottng the frst pont, the respectve cumulatve frequences must be plotted aganst the upper lmts of each class. 4 Jon all the ponts wth straght lnes. Exercse 1. The manager of Jerry s salon recently asked hs last 50 customers to punch a tme card when they frst arrved at the shop and to punch out rght after they pad for ther har cut. He then used the data on the cards to measure how long t took Jerry and hs har dressers to cut har n order to schedule ther appontment ntervals. The followng data were tabulated.

5 Form the frequency dstrbuton and percentage dstrbuton.. Plot the hstogram. Plot the frequency polygon. v. Form the cumulatve percentage dstrbuton. v. Plot the cumulatve percentage polygon. v. On the bass of the results (a)-(e), comment on the tme gap to be kept between two consecutve appontments for harcuts.. Moore Travel, a natonwde travel agency, offers specal rates on certan Carbbean cruses to senor ctzens. The presdent of Moore Travel wants addtonal nformaton on the ages of those people takng cruses. A random sample of 40 customers takng a cruse last year revealed these ages: Organze the data nto a frequency dstrbuton, usng 7 classes and 15 as the lower lmt of the frst class.. Where do the data tend to cluster?. Determne the relatve frequency dstrbuton. v. Descrbe the dstrbuton. The Stem and Leaf Dsplay A stem-and-leaf dagram, also called a stem-and-leaf plot, s a dagram that quckly summarzes data whle mantanng the ndvdual data ponts. In such a dagram, the "stem" s a column of the unque elements of data after removng the last dgt. The fnal dgts ("leaves") of each column are then placed n a row next to the approprate column and sorted n numercal order. Exercse 1. A bank wants to fnd the number of tmes a partcular automated teller machne (ATM) s used each day. The followng s the number of tmes t was used durng each of the last 30 days. Develop a stem-and-leaf dsplay. Summarze the data on the number of tmes the machne was used:

6 How many tmes was the ATM used on a typcal day? What were the largest and the smallest number of tmes the ATM was used? Around what values dd the number of tmes the ATM was used, tend to cluster?. The back to back stem and leaf plot below shows the LDL cholesterol levels (n mllgram per declter mg/dl) of two groups of people, smokers and non-smokers. The dgts n the stem represents the hundreds and tens and the dgt n the leaf represents the ones. For or example 11 8 = 118 and so on.. People wth a cholesterol level of 19 or less are sad to have a near deal level of cholesterol. How many people, n each group, have a near deal level of cholesterol?. People wth a cholesterol level between 130 and 159 nclusve are sad to be n the border hgh. How many people, n each group, are n the border hgh?. People wth a cholesterol level between 160 and 189 nclusve are sad to have a hgh level of cholesterol. How many people, n each group, have a hgh level of cholesterol? v. People wth a cholesterol level of 190 or above are sad to have a very hgh level of cholesterol. How many people, n each group, have a very hgh level of cholesterol? v. Comparng the two groups, whch group has more people wth a hgher level of cholesterol?

7 Measures of Central Tendency and Dsperson Measures of Central Tendency A measure of central tendency s a value at the center or mddle of a data set. It condenses the mass of data nto one sngle value and enables us to get an dea of the entre set of data. It also enables comparson of two or more sets of data. Mean The mean s computed by dvdng the sum of the values of each and every observaton by the total number of observatons. Mean for populaton: Mean for sample = Where, = symbol representng summaton, x = the set of values n the sample set, n = number of values n the sample set. When mean s gven for grouped data, the mdpont of every class nterval s consdered to be representng the values wthn the class. The mean s gven as, _ fx x f Where, f = the frequency of each class x = md pont of each class X n 1 n X N 1 N X Medan Medan s the mddle value n a range of values arranged n sequence by sze. When the data s arranged n the ascendng or descendng order, Medan = the [(n+1)/]th value. The medan for a grouped set of data s the (n/)th term. To calculate the medan for grouped data, the cumulatve frequency dstrbuton has to be constructed frst and then the followng formula can be appled. n Medan = L + C f Where, L = real lower lmt of the medan class, n = number of tems of data,

8 C = cumulatve frequency of the class pror to the medan class, f = frequency of the medan class, = class nterval wdth. Mode Mode s the most frequently occurrng fgure n an ungrouped set of data. For grouped data, mode can be estmated usng the followng formula. Mode = L + d1 d1 + d Where, L = real lower lmt of the modal class, d1 = dfference n frequences between the modal class and the proceedng class. d = dfference n frequences between the modal class and the followng class. = class nterval wdth. Exercses 1. Refer the har cuttng data wth respect to Jerry s salon gven on page 4. Compute the mean, medan and mode.. The followng frequency dstrbuton shows the annual proft levels n last fnancal year wth respect to 100 small enterprses n a cty. Prevous studes show that average monthly proft of small enterprses for the prevous fnancal year was Rs 40,000. Proft (Rs 000) Number of enterprses total 100. Fnd the mean and medan and mode and comment on the proft values compared to the prevous year... Fnd the proft below whch, the lowest 5% of the enterprses fall.. Draw the cumulatve percentage dstrbuton and fnd the answer for part graphcally. The weghted mean Weghted mean s used n stuatons where scores vary n ther degree of mportance. In such stuatons, dfferent weghts are attached to dfferent scores. A weght s a value correspondng to how much the score s counted. Gven a lst of scores x1, x, xn and correspondng lst of weghts w1, w,.wn the weghted mean s obtaned by the formulae

9 Weghted mean = Σ(w. x) Σw Exercse The fnal score of a course s computed as a weghted mean wth the weghts 10% for md term test, 30% for assgnment and 60% for the wrtten exam. A student scored 80 marks for the md term test, 70 for the assgnment and 60 for the wrtten exam. Fnd the fnal mark obtaned for the course. Measures of Non Central Tendency Quartles. Quartles are employed partcularly when summersng or descrbng the propertes of large sets of numercal data. The quartles are descrptve measures that splts the ordered data nto four quarters. In an ungrouped set of scores, Q1 = [(n+1)/4]th value and Q3 = [3(n+1)/4]th value. IQR for grouped data Q1 (lower quartle) = L + n/4 -C FQ1 Q (medan) = L + n/ -C fq Q3 (upper quartle) = L + 3n/4 -C FQ3 where L = real lower lmt of the quartle class, n = total number of observatons n the entre data set, C = cumulatve frequency n the class mmedately before the quartle class, fq = frequency of the relevant quartle class, = the length of the real class nterval of the relevant quartle class. Exercse 1. A manufacturer of flashlght batteres took a sample of 13 batteres from a day s producton and used them contnuously untl they were draned. The number of hours they were used untl falure were 34, 46, 317, 545, 64, 451, 1049, 631, 51, 66, 49, 56, 98. Compute the mean, medan, mode and md range(average of hghest and lowest).. Lookng at the dstrbuton of tmes to falure, whch measures do you thnk s best?. In what ways would ths nformaton be useful to the manufacturer? Dscuss. v. Usng the nformaton above, what would you advce f the manufacturer wanted to be able to say n advertsements that these batteres should last 400 hours?

10 Measures of Varaton Measures of varaton descrbe the spread of ndvdual values around the central value. Range The range s the dfference between the hghest value and the lowest value n a set of data. Interquartle Range The nter-quartle range (IQR) = Q3 - Q1 where Q1 = lower quartle, Q3 = upper quartle. IQR, unlke the range, s not affected by the extreme values. It shows the spread of the mddle 50% of data. The fve number summary and Box plot The fve numbers that help descrbe the center, spread and shape of data are: Xsmallest, Frst Quartle (Q1), Medan (Q), Thrd Quartle (Q3) and Xlargest Box plot s a graphcal dsplay of the data based on the fve-number summary: 5% of data 5% 5% 5% of data of data of data Xsmallest Q1 Medan Q3 Xlargest Mean Absolute Devaton The mean devaton s the average of the absolute devatons taken from the mean. Consderng sample data, Mean devaton = x- x n Varance Average of squared devatons of values from the mean Varance of populaton: N 1 (X N ) Varance of the sample : S n 1 (X X) n -1 Varance for grouped data: S n 1 f(x n - 1 X)

11 Standard devaton Standard devaton shows the varaton of the fgures about the mean. It s the square root of the varance. Standard devaton for populaton = N 1 (X N ) Standard devaton of sample S n 1 (X X) n -1 Exercses 1. The followng data contans the data from 10 days and shows the number of rejected cathode ray tubes out of 10 nspected per day. Number rejected: What s the average number of rejects per day?. What s the standard devaton of the rejects?. A set of fnal examnaton grades n an ntroductory statstcs course was found to be dstrbuted wth a mean of 73 and a standard devaton of 8. Are you better off wth a grade of 81 on ths exam or a grade of 68 on a dfferent exam where the mean s 6 and standard devaton s 3? Show statstcally and explan. 3. Refer the har cuttng data wth respect to Jerry s salon gven on page 4. Compute the standard devaton. Coeffcent of varaton (Cv) The coeffcent of varaton measures the scatter n the data relatve to the mean. A lower coeffcent of varaton ndcates a lower relatve dsperson. Cv = s x Where, s - standard devaton x - mean. Exercse To set an approprate prce for a product, t s necessary to be able to estmate ts cost of producton. One element of the cost s based on the length of tme spent by workers to produce the product. The most wdely used technque for makng such a measurement s the tme study. In a tme study, the task to be studed s dvded nto measurable parts and each s measured wth a stopwatch or flmed for later analyss. For each worker, ths process s repeated many tmes for each sub task. Then the average and standard devaton of the tme requred to complete each subtask are computed for each worker.

12 The data(n mnutes) gven n the table are the result of a tme study of a producton operaton nvolvng two tasks. Worker A Worker B Repetton Subtask 1 Subtask Subtask 1 Subtask For each worker, fnd the mean and the standard devaton for each subtask.. If you could choose workers smlar to A or B to perform subtasks 1 and, whch type would you assgn to each subtask?. Explan your decson on the bass of your answer to part b above. Shape of a Dstrbuton Skewness A symmetrcal dstrbuton s represented by a curve that can be dvded by a vertcal lne nto two parts whch are mrror mages of each other. A skewed dstrbuton s represented by a curve that lacks symmetry. Dfferences among mean, medan and mode can be seen from the graphs of symmetrcal and skewed dstrbutons. Left-Skewed Mean < Medan< Mode Symmetrc Mean = Medan =Mode Rght-Skewed Mode <Medan < Mean

13 Measure of skewness Pearson s coeffcent of skewness = Exercses: 3(Mean-Medan) 1. In a factory, the tme durng workng hours n whch a machne s not operatng as a result of breakage or falure s called the downtme. The followng dstrbuton shows a sample of 100 downtmes of a certan machne (rounded to the nearest mnute):. Fnd the mean, medan and the mode. Fnd the standard devaton. Comment about the skewness of data. Data on the frequency of absenteesm n plant owned by Novel Electroncs s shown n the table below. Management expects your statstcal expertse to analyse ther employee absenteesm levels as compared wth natonal norms. Studes show that average number of days per year that employees are absent across the naton for smlar plants s about 7. Days absent per Frequency year Total 30. Calculate the mean and the medan levels of absenteesm and comment about the absenteesm level of Novel compared to the natonal norms.. Calculate the standard devaton of absenteesm.

14 . v. If the management of Novel decdes to gve a reward to employees who are n the lower 5% of absenteesm levels, fnd the level of absenteesm below whch the reward s applcable. Draw the cumulatve frequency polygon for the absenteesm levels and fnd the answer for part c graphcally. 3. A cnema s showng 3 flms, A,B and C. The ages of people watchng the flms are llustrated n the followng box and whsker plots. Descrbe the ages of people watchng the three flms. In your vew whch flm s sutable for : chldren, young adults and adults.? 4. A bank called applcatons for the post of management tranee. The applcants were asked to st for three papers: Language, Numercal sklls and general knowledge. The results of the performance for the three papers are gven below. Analyse the performance of the canddates n the three exam papers. Paper Mean Medan Std. Devaton Skewness Language Numercal General knowledge Total