Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
©. S. Bhattacharjee Ch 5_1 ©. S. Bhattacharjee Ch 5_2 What is meant by variability? Variability refers to the extent to which the observations vary from one another from some average. A measure of variation is designed to state the extent to which the individual measures differ on an average from the mean. Continued….. ©. S. Bhattacharjee Ch 5_3 What are the purposes of measuring variation ? Measures of variation are needed for four basic purposes: To determine the reliability of an average; To serve as a basis for the control of the variability; To compare two or more series with regard to their variability; To facilitate the use of other statistical measures ©. S. Bhattacharjee Ch 5_4 What are the properties of a good measure of variation ? A good measure of variation should possess the following properties: It should be simple to understand. It should be easy to compute. It should be rigidly defined. It should be based on each and every observation of the distribution. It should be amenable to further algebraic treatment. It should have sampling stability. It should not be unduly affected by extreme observations. ©. S. Bhattacharjee Ch 5_5 What are the methods of studying variation ? The following are the important methods of studying variation: The Range The Interquartile Range or Quartile Deviation. The Average Deviation The Standard Deviation The Lorenz Curve. Of these, the first four are mathematical and the last is a graphical one. ©. S. Bhattacharjee Ch 5_6 What is meant by range ? The range is defined as the distance between the highest and lowest scores in a distribution. It may also be defined as the difference between the value of the smallest observation and the value of the largest observation included in the distribution. ©. S. Bhattacharjee Ch 5_7 What are the usages of range ? Despite serious limitations range is useful in the following cases: Quality control: Range helps check quality of a product. The object of quality control is to keep a check on the quality of the product without 100% inspection. Fluctuation in the share prices: Range is useful in studying the variations in the prices of stocks and shares and other commodities etc. Weather forecasts: The meteorological department does make use of the range in determining the difference between the minimum temperature and maximum temperature. ©. S. Bhattacharjee Ch 5_8 What are the merits of range ? Merits: Among all the methods of studying variation, range is the simplest to understand and the easiest to compute. It takes minimum time to calculate the value of range. Hence, if one is interested in getting a quick rather than a very accurate picture of variability, one may compute range. ©. S. Bhattacharjee Ch 5_9 What are the limitations of range ? Limitations: Range is not based on each and every observation of the distribution. It is subject to fluctuations of considerable magnitude from sample. Range cannot be computed in case of open-end distributions. Range cannot tell anything about the character of the distribution within two extreme observations. ©. S. Bhattacharjee Ch 5_10 Example: Observe the following three series Series A: 6, Series B: 6 46 6 46 6 46 6 46 46 46 46 46 46 46 46 Series C: 6 10 15 25 30 32 40 46 In all the three series range is the same (i.e., 46-6=40), but it does not mean that the distributions are alike. The range takes no account on the form of the distribution within the range. Range is, therefore, most unreliable as a guide to the variation of the values within a distribution. ©. S. Bhattacharjee Ch 5_11 What is meant by inter-quartile range or deviation? Inter-quartile range represents the difference between the third quartile and the first quartile. In measuring inter-quartile range the variation of extreme observations is discarded. Continued….. ©. S. Bhattacharjee Ch 5_12 What is inter-quartile range or deviation measured ? One quartile of the observations at the lower end and another quartile of the observations at the upper end of the distribution are excluded in computing the interquartile range. In other words, inter-quartile range represents the difference between the third quartile and the first quartile. Symbolically, Inerquartile range = Q3 – Q1 Very often the interquartile range is reduced to the form of the semi-interquartile range or quartile deviation by dividing it by 2. ©. S. Bhattacharjee Ch 5_13 The formula for computing inter-quartile deviation is stated as under: Q.D. Q Q1 3 2 Q.D. = Quartile deviation Quartile deviation gives the average amount by which the two Quartiles differ from the median. In asymmetrical distribution, the two quartiles (Q1 and Q3 ) are equidistant from the median, i.e., Median ± Q.D. covers exactly 50 per cent of the observations. ©. S. Bhattacharjee Ch 5_14 When quartile deviation is very small it describes high uniformity or small variation of the central 50% observations, and a high quartile deviation means that the variation among the central observations is large. Quartile deviation is an absolute measure of variation. The relative measure corresponding to this measure, called the coefficient of quartile deviation, is calculated as follows: Q Q1 Co-efficient of Quartile deviation 3 Q3 Q1 Coefficient of quartile deviation can be used to compare the degree of variation in different distributions. ©. S. Bhattacharjee Ch 5_15 How is quartile deviation computed? The process of computing quartile deviation is very simple. It is computed based on the values of the upper and lower quartiles. The following illustration would clarify the procedure. Example: You are given the frequency distribution of 292 workers of a factory according to their average weekly income. Calculate quartile deviation and its coefficient from the following data: Continued………… ©. S. Bhattacharjee Ch 5_16 Example: Weekly Income No. of workers Below 1350 8 1350-1370 16 1370-1390 39 1390-1410 58 1410-1430 60 1430-1450 40 1450-1470 22 1470-1490 15 1490-1510 15 1510-1530 9 1530 & above 10 Continued………… ©. S. Bhattacharjee Ch 5_17 Example: Calculation of Quartile deviation Weekly Income No. of workers c.f. Below 1350 8 8 1350-1370 16 24 1370-1390 39 63 1390-1410 58 121 1410-1430 60 181 1430-1450 40 221 1450-1470 22 243 1470-1490 15 258 1490-1510 15 273 1510-1530 9 282 1530 & above 10 292 N = 292 ©. S. Bhattacharjee Continued………… Ch 5_18 Example: N 292 146th observation Median = Size of th observation 2 2 Median lies in the class 1410 - 1430 N p.c. f Medain L 2 i f 146 121 1410 20 60 1410 8 333 1418.333 N 292 Q1 Size of th observation 73rd observation 4 4 Q1 lies in the class 1390 1410. ©. S. Bhattacharjee Continued………… Ch 5_19 Example: N p.c. f . Q1 L 4 i f 73 63 1390 20 1390 3 448 1393 448 58 3N Q3 Size of th observation 4 3 292 219 th observation Q3 lies in the class 1430 1449 4 3N p.c. f . 219 181 Q3 L 4 i 1430 20 1430 19 1449 f 40 Q Q1 1449 1393 448 55 552 55552 Coeffiecnt of Q.D. 3 0 020. Q3 Q1 1449 1393 448 2842 448 2842 448 ©. S. Bhattacharjee Ch 5_20 What are the merits of quartile deviation? Merits: In certain respects it is superior to range as a measure of variation It has a special utility in measuring variation in case of open-end distributions or one in which the data may be ranked but measured quantitatively. It is also useful in erratic or highly skewed distributions, where the other measures of variation would be warped by extreme value. The quartile deviation is not affected by the presence of extreme values. ©. S. Bhattacharjee Ch 5_21 What are the limitations of quartile deviation? Limitations: Quartile deviation ignores 50% items, i.e., the first 25% and the last 25%. As the value of quartile deviation does not depend upon every observation it cannot be regarded as a good method of measuring variation. It is not capable of mathematical manipulation. Its value is very much affected by sampling fluctuations. It is in fact not a measure of variation as it really does not show the scatter around an average but rather a distance on a scale, i.e., quartile deviation is not itself measured from an average, but it is a positional average. ©. S. Bhattacharjee Ch 5_22 What is average deviation? Average deviation refers to the average of the absolute deviations of the scores around the mean. It is obtained by calculating the absolute deviations of each observation from median ( or mean), and then averaging these deviations by taking their arithmetic mean. How is it calculated? Continued……. ©. S. Bhattacharjee Ch 5_23 Ungrouped data The formula for average deviation may be written as: A.D.( Med .) X Med N If the distribution is symmetrical the average (mean or median) ± average deviation is the range that will include 57.5 per cent of the observation in the series. If it is moderately skewed, then we may expect approximately 57.5 per cent of the observations to fall within this range. Hence if average deviation is small, the distribution is highly compact or uniform, since more than half of the cases are concentrated within a small range around the mean. ©. S. Bhattacharjee Ch 5_24 Ungrouped data The relative measure corresponding to the average deviation, called the coefficient of average deviation, is obtained, by dividing average deviation by the particular average used in computing average deviation. Thus, if average deviation has been computed from median, the coefficient of average deviation shall be obtained by dividing average deviation by the median. A.D. Coefficient of A.D.Med . Median If mean has been used while calculating the value of average deviation, in such a case coefficient of average deviation is obtained by dividing average deviation by the mean. ©. S. Bhattacharjee Ch 5_25 Example: Calculate the average deviation and coefficient of average deviation of the two income groups of five and seven workers working in two different branches of a firm: Branch 1 Income (Tk) 4,000 4,200 Branch II Income (Tk) 3,000 4,000 4,400 4,600 4,200 4,400 4,600 4,800 4,800 5,800 ©. S. Bhattacharjee Continued….. Ch 5_26 Calculation of Average deviation Branch 1 │X- Med Income (Tk) Branch II Med.=4,400 Income (Tk) │X- Med Med.= 4,400 4,000 400 3,000 1,400 4,200 200 4,000 400 4,400 0 4,200 200 4,600 200 4,400 0 4,800 400 4,600 200 4,800 400 5,800 1,400 N= 5 │X- Med =1,200 N=7 │X- Med = 4000 Continued….. ©. S. Bhattacharjee Ch 5_27 Brach I: X Med . 1200 A.D. 240 N 5 A.D. 240 Coeff . of A.D. 0 054 Median 4,400 X Med . 4,000 A.D. 571 43 Brach II: N 7 57143 coeff . of A.D. 0 13 4,400 ©. S. Bhattacharjee Ch 5_28 Grouped data In case of grouped data, the formula for calculating average deviation is : A.D.( Med .) f X Med N Continued……….. ©. S. Bhattacharjee Ch 5_29 Example: Calculation of Average Deviation from mean from the following data: Sales (in thousand Tk) No. of days 10 – 20 3 20 – 30 6 30 – 40 11 40 – 50 3 50 – 60 2 Continued…….. ©. S. Bhattacharjee Ch 5_30 Calculation of Average deviation Sales (in thousand Tk) m.p X f X 35 fd 10 XX f XX (=d) 10 – 20 15 3 –2 –6 18 54 20 – 30 25 6 –1 –6 8 48 30 – 40 35 11 0 0 2 22 40 – 50 45 3 +1 +3 12 36 50 – 60 55 2 +2 +4 22 44 N = 25 fd = –5 f X X = 204 Continued…… ©. S. Bhattacharjee Ch 5_31 f A.D. X A XX N fd N 5 i 35 10 35 2 33 25 204 A.D. 8 16 25 Thus the average sales are Tk. 33 thousand per day and the average deviation of sales is Tk. 8.16 thousand. ©. S. Bhattacharjee Ch 5_32 What are the areas suitable for use of average deviation? It is especially effective in reports presented to the general public or to groups not familiar with statistical methods. This measure is useful for small samples with no elaborate analysis required. Research has found in its work on forecasting business cycles, that the average deviation is the most practical measure of variation to use for this purpose. ©. S. Bhattacharjee Ch 5_33 What are the merits of average deviation? Merits: The outstanding advantage of the average deviation is its relative simplicity. It is simple to understand and easy to compute. Any one familiar with the concept of the average can readily appreciate the meaning of the average deviation. It is based on each and every observation of the data. Consequently change in the value of any observation would change the value of average deviation. ©. S. Bhattacharjee Ch 5_34 What are the merits of average deviation? Merits: Average deviation is less affected by the values of extremes observation. Since deviations are taken from a central value, comparison about formation of different distributions can easily be made. ©. S. Bhattacharjee Ch 5_35 What are the limitations of average deviation? Limitations: The greatest drawback of this method is that algebraic signs are ignored while taking the deviations of the items. If the signs of the deviations are not ignored, the net sum of the deviations will be zero if the reference point is the mean, or approximately zero if the reference point is median. The method may not give us very accurate results. The reason is that average deviation gives us best results when deviations are taken from median. But median is not a satisfactory measure when the degree of variability in a series is very high. ©. S. Bhattacharjee Continued……. Ch 5_36 What are the limitations of average deviation? Limitations: Compute average deviation from mean is also not desirable because the sum of the deviations from mean ( ignoring signs) is greater than the sum of the deviations from median (ignoring signs). If average deviation is computed from mode that also does not solve the problem because the value of mode cannot always be determined. It is not capable of further algebraic treatment. It is rarely used in sociological and business studies. ©. S. Bhattacharjee Continued……. Ch 5_37 What is meant by Standard Deviation? Standard deviation is the square root of the squared deviations of the scores around the mean divided by N. S represents standard deviation of a sample; ∂, the standard deviation of a population. Standard deviation is also known as root mean square deviation for the reason that it is the square root of the means of square deviations from the arithmetic mean. The formula for measuring standard deviation is as follows : X X 2 ©. S. Bhattacharjee N Ch 5_38 What is meant by Variance? This refers to the squared deviations of the scores around the mean divided by N. A measure of dispersion is used primarily in inferential statistics and also in correlation and regression techniques; S2 represents the variance of a sample ; ∂2 , the variance of a population. If we square standard deviation, we get what is called Variance. Hence Variance or Variance 2 ©. S. Bhattacharjee Ch 5_39 How is standard deviation calculated? Ungrouped data Standard deviation may be computed by applying any of the following two methods: By taking deviations from the actual mean By taking deviations from an assumed mean Continued…….. ©. S. Bhattacharjee Ch 5_40 How is standard deviation calculated? Ungrouped data By taking deviations from the actual mean: When deviations are taken from the actual mean, the following formula is applied: X X 2 N If we calculate standard deviation without taking deviations, the above formula after simplification (opening the brackets) can be used and is given by: Continued…….. ©. S. Bhattacharjee Ch 5_41 Formula: X N 2 X N 2 or 2 X N X 2 By taking deviations from an assumed mean: When the actual mean is in fractions, say 87.297, it would be too cumbersome to take deviations from it and then find squares of these deviations. In such a case either the mean may be approximated or else the deviations be taken from an assumed mean and the necessary adjustment be made in the value of standard deviation. ©. S. Bhattacharjee Ch 5_42 How is standard deviation calculated ? The former method of approximation is less accurate and therefore, invariably in such a case deviations are taken from assumed mean. When deviations are taken from assumed mean the following formula is applied: Where ©. S. Bhattacharjee d d N N 2 2 d X A Ch 5_43 Example: Find the standard deviation from the weekly wages of ten workers working in a factory: Workers Weekly wages (Tk) A 1320 B 1310 C 1315 D 1322 E 1326 F 1340 G 1325 H 1321 I 1320 j 1331 ©. S. Bhattacharjee Ch 5_44 Calculations of Standard Deviation X X X X Workers Weekly wages (Tk) A 1320 -3 9 B 1310 - 13 169 C 1315 -8 64 D 1322 -1 1 E 1326 +3 9 F 1340 +17 289 G 1325 +2 4 H 1321 -2 4 I 1320 -3 9 J 1331 +8 64 N= 10 x=13230 ©. S. Bhattacharjee X X = 0 Continued……. 2 XX 2 = 622 Ch 5_45 X X 2 We know N ...............(i ) dXA Since X Ad X Ad Subtracting X from X , we get X X d d Continued……. ©. S. Bhattacharjee Ch 5_46 Substituting the value of X X d d 2 N X X N 2 X X N in (i), mentioned d 2 N d N 2 13230 Tk .1323 10 622 62.2 10 7 89 If, in the above question, deviations are taken from 1320 instead of the actual mean 1323, the assumed mean method will be applied and the calculations would be as follows: ©. S. Bhattacharjee Continued……. Ch 5_47 Calculation of standard deviation (assumed mean method) Workers Weekly wages (Tk) X A d d2 A = 1320 A 1320 0 0 B 1310 -10 100 C 1315 -5 25 D 1322 +2 4 E 1326 +6 36 F 1340 +20 400 G 1325 +5 25 H 1321 +1 1 I 1320 0 0 j 1331 +11 121 d=30 d2 =712 N= 10 ©. S. Bhattacharjee Continued……. Ch 5_48 d N 2 d N 2 2 712 30 71 2 9 62 2 7 89 10 10 Thus the answer remains the same by both the methods. It should be noted that when actual mean is not a whole number, the assumed mean method should be preferred because it simplifies calculations. ©. S. Bhattacharjee Ch 5_49 Grouped data In grouped frequency distribution, standard deviation can be calculated by applying any of the following two methods: By taking deviations from actual mean. By taking deviations from assumed mean. ©. S. Bhattacharjee Continued…….. Ch 5_50 Grouped data Deviations taken from actual mean: When deviations are taken from actual mean, the following formula is used: f X X 2 N If we calculate standard deviation without taking deviations, then this formula after simplification (opening the brackets ) can be used and is given by ©. S. Bhattacharjee fX N 2 fX N 2 or Continued…….. fX 2 N X 2 Ch 5_51 Grouped data Deviations taken from assumed mean: When deviations are taken from assumed mean, the following formula is applied : fd N 2 fd i, where N 2 xA d i ©. S. Bhattacharjee Continued…….. Ch 5_52 Example: A purchasing agent obtained samples of 60 watt bulbs from two companies. He had the samples tested in his own laboratory for length of life with the following results: Length of life (in hours) 1,700 and 1,900 and 2,100 and 2,300 and 2,500 and under under under under under ©. S. Bhattacharjee 1,900 2,100 2,300 2,500 2,700 Samples from Company A Company B 10 3 16 20 8 40 12 3 6 2 Continued…….. Ch 5_53 Example: 1. Which Company’s bulbs do you think are better in terms of average life? 2. If prices of both types are the same, which company’s bulbs would you buy and why? ©. S. Bhattacharjee Continued…….. Ch 5_54 Example: Length of life (in hours) Sample from Co. A Midpoint Samples from Co. B f d fd fd2 f d fd fd2 1,700– 1,900 1800 10 –2 –20 40 3 –2 –6 12 1,900–2,100 2000 16 –1 –16 16 40 –1 – 40 40 2,100–2,300 2200 20 0 0 0 12 0 0 0 2,300–2,500 2400 8 1 +8 8 3 1 +3 3 2,500–2,700 2600 6 2 +12 24 2 2 +4 8 N=60 d=0 fd = –16 fd2 =88 N=60 d=0 fd = –39 fd2 = 63 d XA , where Assumed mean i Here, A = 2200 i = 200 ©. S. Bhattacharjee Continued…….. Ch 5_55 Example: Here, N = 60 A = 2,200 For Company A : fd = - 16 fd2 = 88 fd 16 200 2,146 67 i 2,200 X A 60 N fd N 2 fd i N 2 2 88 16 200 60 60 1 467 0 071 200 1 182 200 236 4 236 4 100 11 per cent. C.V . 100 2146 67 X ©. S. Bhattacharjee Continued…….. Ch 5_56 fd 39 fd 63 For Company B : 2 A 2200 39 X 2200 200 2200 130 2070 60 2 63 39 200 60 60 1 05 42 200 794 200 158 8 158 8 C.V . 100 7 67 per cent. 2070 ©. S. Bhattacharjee Continued…….. Ch 5_57 Illustration :18 You are given the data pertaining to kilowatt hours electricity consumed by 100 persons in Deli. Consumption (K. Wait hours) No. of users 0 but less than 10 6 10 but less than 20 25 20 but less than 30 36 30 but less than 40 20 40 but less than 50 13 Calculate the mean and the standard deviation. ©. S. Bhattacharjee Continued…….. Ch 5_58 Solution: Calculation of Mean and standard Deviation (Taking deviation from assumed mean) Consumption K. wait hours m.p (X) No. of Users (f) X 25 fd2 fd c.f. 10 d 0–10 5 6 –2 –12 24 6 10–20 15 25 –1 –25 25 31 20–30 25 36 0 0 0 67 30–40 35 20 +1 +20 20 87 40–50 45 13 +2 +26 52 100 N=100 ©. S. Bhattacharjee fd =9 Continued…….. fd2=121 Ch 5_59 (i) fd X A i 25 N 2 fd fd 121 9 i 10 N 100 100 N 2 (ii) 9 10 25.9k . wait hours 100 2 1.21 .008 10 1.096 10 10.96 ©. S. Bhattacharjee Ch 5_60 Calculation of Mean and the Standard Deviation (Taking deviation from assumed mean) Consumption Midpoi K. wait hours nt X No. of Users f fX X X X X 2 f X X 2 0–10 5 6 30 –2 436.81 2620.6 10–20 15 25 375 –10.9 118.81 2970.25 20–30 25 36 900 –0.9 0.81 29.16 30–40 35 20 700 9.1 82.81 1656.20 40–50 45 13 585 19.1 364.81 4742.53 N=100 fX= 2590 ©. S. Bhattacharjee Continued…….. f X X 2 12019 Ch 5_61 fX X 2590 100 25.9 N f X X N 2 12019 100 120.19 10.96 ©. S. Bhattacharjee Ch 5_62 1. Since average length of life is greater in case of company A, hence bulbs of company A are better. 2. Coefficient of variation is less for company B. Hence if prices are same, we will prefer to buy company B’s bulbs because their burning hours are more uniform. ©. S. Bhattacharjee Ch 5_63 Example: For two firms A and B belonging to same industry, the following details are available: Firm B Firm A Number of Employees 100 200 Average monthly wage: Tk. 4,800 Tk. 5,100 Standard deviation: Tk. 600 Tk. 540 Find i. Which firm pays out larger amount as wages? ii. Which firm shows greater variability in the distribution of wages? iii. Find average monthly wage and the standard deviation of the wages of all employees in both the firms. ©. S. Bhattacharjee Ch 5_64 i. For finding out which firm pays larger amount, we have to find out X. X X or X NX N Firm A : N = 100, X = 4800, X=100×4800 =4,80,000 Firm B : N = 200, X = 5100, X=200×5100 =10,20,000 Hence firm B pays larger amount as monthly wages. ©. S. Bhattacharjee Ch 5_65 ii. For finding out which firm pays greater variability in the distribution of wages, we have to calculate coefficient of variation. Firm A : 600 C.V . 100 100 12 50 X 4800 Firm B : 540 C.V . 100 100 10 59 X 5100 Since coefficient of variation is greater in case of firm A, hence it shows greater variability in the distribution of wages. ©. S. Bhattacharjee Ch 5_66 iii. Combined average weekly wage: N1 X 1 N 2 X 2 X 12 N1 N 2 N1 = 100, X 1 = 4800, N 2 = 200, X 2 = 5100, X 12 100 4800 200 5100 100 200 480000 1020000 TK .5,000 300 ©. S. Bhattacharjee Ch 5_67 Combined Standard Deviation 12 N1 1 2 N2 2 2 N1 d 2 N 2 1 d2 2 N1 N 2 N1 100, 1 600, N 2 200, 2 540 d1== X 1 d2 = X2 X 12 X 12 = 4800 - 5000 =200 = 5100 - 5000 =100 Continued……. ©. S. Bhattacharjee Ch 5_68 12 100600 200 540 100 200 200 100 100 200 2 2 2 2 36000000 5832000 4000000 2000000 300 100 320,000 578.25 300 Hence the combined standard deviation is Tk. 578.25. ©. S. Bhattacharjee Ch 5_69 Which measure of variation to use? The choice of a suitable measure depends on the following two factors: The type of data available The purpose of investigation ©. S. Bhattacharjee Ch 5_70 What is Lorenz curve? It is cumulative percentage curve in which the percentage of items is combined with the percentage of other things as wealth, profit, turnover, etc. The Lorenz curve is a graphic method of studying variation. ©. S. Bhattacharjee Ch 5_71 What is the procedure of drawing the Lorenz curve? While drawing the Lorenz curve the following procedure is used: The size of items and frequencies are both cumulated and the percentages are obtained for the various commutative values. On the X-axis, start from 0 to 100 and take the per cent of variable. On the Y-axis, start from 0 to 100 and take the per cent of variable. Continued….. ©. S. Bhattacharjee Ch 5_72 What is the procedure of drawing the Lorenz curve? Draw a diagonal line joining 0 with 100. This is known as line of equal distribution. Any point on this line shows the same per cent on X as on Y. Plot the various points corresponding to X and Y and join them. The distribution so obtained, unless it is exactly equal, will always curve below the diagonal line. ©. S. Bhattacharjee Ch 5_73 How is interpretation of the Lorenz curve done? If two curves of distribution are shown on the Lorenz presentation, the curve that is farthest from the diagonal line represents the greater inequality. Clearly the line of actual distribution can never cross the line of equal distribution. ©. S. Bhattacharjee Ch 5_74 Example: In the following table is given the number of companies belonging to two areas A and B according to the amount of profits earned by them. Draw in the same diagram their Lorenz curves and interpret them. Profits earned in Tk.'000 No. of companies Area A Area B 6 6 2 25 11 38 60 13 52 84 14 28 105 15 38 150 17 26 170 10 12 400 14 4 ©. S. Bhattacharjee Continued…… Ch 5_75 Solution: Calculation for drawing the Lorenz curve Profit Profits earned in Tk. ‘000 Cumulativ e Profits Cumulativ e Percentag e No. of Companies Cumulative Number Cumulative Percentage No. of Companies Cumulative Number Cumulative Percentage 6 6 0.6 6 6 6 2 2 1 25 31 3.1 11 17 17 38 40 20 60 91 9.1 13 30 30 52 92 46 84 175 17.5 14 44 44 28 120 60 105 280 28.0 15 59 59 38 158 79 150 430 43.0 17 76 76 26 184 92 170 600 60.0 10 86 86 12 196 98 400 1000 100.0 14 100 100 4 200 100 ©. S. Bhattacharjee Continued…….. Ch 5_76 Lorenz curve 120.0 100.0 Per Cent of Profits 80.0 60.0 40.0 20.0 Per Cent of Companies ©. S. Bhattacharjee 100 98 92 79 60 46 20 1 0.0 Ch 5_77 ©. S. Bhattacharjee Ch 5_78