Mba i qt unit-2.1_measures of variations
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This document discusses various measures of variability or variation used in statistics. It defines variability as the extent to which observations vary from each other or from the average. The key measures discussed are range, interquartile range, average deviation, and standard deviation. Range is the simplest but ignores the distribution within its limits. Interquartile range excludes outliers but also ignores half the data. Average deviation measures average distance from the mean/median and indicates how compact 50% of data is. Examples are provided to demonstrate calculating and comparing these measures.
Mba i qt unit-2.1_measures of variations
- 1. Course: MBA Subject: Quantitative Techniques Unit: 2.1
- 2. Ch 5_2
- 3. Ch 5_3 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…..
- 4. Ch 5_4 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
- 5. Ch 5_5 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.
- 6. Ch 5_6 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.
- 7. Ch 5_7 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.
- 8. Ch 5_8 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.
- 9. Ch 5_9 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.
- 10. Ch 5_10 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.
- 11. Ch 5_11 Example: Observe the following three series Series A: 6, 46 46 46 46 46 46 46 Series B: 6 6 6 6 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.
- 12. Ch 5_12 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…..
- 13. Ch 5_13 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 inter- quartile 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.
- 14. Ch 5_14 2 .. 13 QQ DQ 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. The formula for computing inter-quartile deviation is stated as under:
- 15. Ch 5_15 Co-efficient of Quartile deviation 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: 13 13 QQ QQ Coefficient of quartile deviation can be used to compare the degree of variation in different distributions.
- 16. Ch 5_16 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…………
- 17. Ch 5_17 Example: Continued………… 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
- 18. Ch 5_18 Example: Continued………… 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 Calculation of Quartile deviation
- 19. Ch 5_19 Example: Continued………… nobservatiothnobservatioth N 146 2 292 2 Median = Size of Median lies in the class 1410 - 1430 .14101390 73 4 292 4 333.141833381410 20 60 121146 1410 .. 2 1 1 classtheinliesQ nobservatiordnobservatioth N ofSizeQ i f fcp N LMedain
- 21. Ch 5_21 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.
- 22. Ch 5_22 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. What are the limitations of quartile deviation?
- 23. Ch 5_23 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…….
- 24. Ch 5_24 The formula for average deviation may be written as: N MedX DA Med .)(.. 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. Ungrouped data
- 25. Ch 5_25 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. Ungrouped data Median DA DAoftCoefficien Med .. .. . 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.
- 26. Ch 5_26 Branch 1 Income (Tk) Branch II Income (Tk) 4,000 3,000 4,200 4,000 4,400 4,200 4,600 4,400 4,800 4,600 4,800 5,800 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: Example: Continued…..
- 27. Ch 5_27 Branch 1 │X- Med Income (Tk) Med.=4,400 Branch II │X- Med Income (Tk) 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 Calculation of Average deviation Continued…..
- 28. Ch 5_28 Brach I: Brach II: 130 400,4 57143 ... 43571 7 000,4. .. 0540 400,4 240.. ... 240 5 1200. .. DAofcoeff N MedX DA Median DA DAofCoeff N MedX DA
- 29. Ch 5_29 Grouped data In case of grouped data, the formula for calculating average deviation is : Continued……….. N MedXf DA Med .)(..
- 30. Ch 5_30 Example: Sales (in thousand Tk) No. of days 10 – 20 3 20 – 30 6 30 – 40 11 40 – 50 3 50 – 60 2 Continued…….. Calculation of Average Deviation from mean from the following data:
- 31. Ch 5_31 Sales (in thousand Tk) m.p X f (=d) fd 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 = 204 10 35X XX XX XX f f Calculation of Average deviation Continued……
- 32. Ch 5_32 168 25 204 .. 3323510 25 5 35 .. DA i N fd AX N XXf DA Thus the average sales are Tk. 33 thousand per day and the average deviation of sales is Tk. 8.16 thousand.
- 33. Ch 5_33 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.
- 34. Ch 5_34 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.
- 35. Ch 5_35 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.
- 36. Ch 5_36 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. Continued…….
- 37. Ch 5_37 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. Continued…….
- 38. Ch 5_38 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 : N XX 2
- 39. Ch 5_39 VarianceorVarianceHence 2 If we square standard deviation, we get what is called Variance. 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.
- 40. Ch 5_40 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……..
- 41. Ch 5_41 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: N XX 2 If we calculate standard deviation without taking deviations, the above formula after simplification (opening the brackets) can be used and is given by: Continued……..
- 42. Ch 5_42 Formula: 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. 2 22 2 X N X or N X N X
- 43. Ch 5_43 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: 22 N d N d Where AXd
- 44. Ch 5_44 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
- 45. Ch 5_45 2 XX XX Calculations of Standard Deviation Continued……. XX 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 = 0 = 622 2 XX
- 46. Ch 5_46 ).....(.......... 2 i N XX We know Since dAX dAX AXd ddXX getweXfromXgSubtractin , Continued…….
- 47. Ch 5_47 XX Substituting the value of in (i), mentioned 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: 222 N d N d N dd 897 2.62 10 622 2 N XX 1323. 10 13230 Tk N X X Continued…….
- 48. Ch 5_48 Workers Weekly wages (Tk) A = 1320 d2 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 N= 10 d=30 d2 =712 Calculation of standard deviation (assumed mean method) Continued……. dAX
- 49. Ch 5_49 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. 8972629271 10 30 10 712 222 N d N d
- 50. Ch 5_50 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. Continued……..
- 51. Ch 5_51 Grouped data Deviations taken from actual mean: When deviations are taken from actual mean, the following formula is used: Continued…….. If we calculate standard deviation without taking deviations, then this formula after simplification (opening the brackets ) can be used and is given by 2 222 X N fX or N fX N fX N XXf 2
- 52. Ch 5_52 Grouped data Deviations taken from assumed mean: When deviations are taken from assumed mean, the following formula is applied : Continued…….. , 22 i N fd N fd i Ax d where
- 53. Ch 5_53 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: Example: Length of life (in hours) Samples from Company A Company B 1,700 and under 1,900 10 3 1,900 and under 2,100 16 40 2,100 and under 2,300 20 12 2,300 and under 2,500 8 3 2,500 and under 2,700 6 2 Continued……..
- 54. Ch 5_54 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? Example: Continued……..
- 55. Ch 5_55 Example: Continued…….. Sample from Co. ALength of life (in hours) Midpoint Samples from Co. A 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=6 0 d=0 fd = –39 fd2 = 63 Samples from Co. B meanAssumedwhere i AX d , Here, A = 2200 i = 200
- 56. Ch 5_56 Example: Continued…….. .11100 672146 4236 100.. 4236200182120007104671 200 60 16 60 88 67146,2200 60 16 200,2 222 centper X VC i N fd N fd i N fd AX For Company A : Here, N = 60 A = 2,200 fd = - 16 fd2 = 88
- 57. Ch 5_57 For Company B : Continued…….. .677100 2070 8158 .. 8158200794 20042051 200 60 39 60 63 20701302200200 60 39 2200 2 centperVC X 2200 63 39 2 A fd fd
- 58. Ch 5_58 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 Illustration :18 You are given the data pertaining to kilowatt hours electricity consumed by 100 persons in Deli. Calculate the mean and the standard deviation. Continued……..
- 59. Ch 5_59 Solution: Calculation of Mean and standard Deviation (Taking deviation from assumed mean) 10 25XConsumption K. wait hours m.p (X) No. of Users (f) d fd fd2 c.f. 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 fd =9 fd2=121 Continued……..
- 60. Ch 5_60 hourswaitki N fd AX .9.2510 100 9 25 96.1010096.110008.21.1 10 100 9 100 121 222 i N fd N fd (i) (ii)
- 61. Ch 5_61 Calculation of Mean and the Standard Deviation (Taking deviation from assumed mean) XX 2 XX 2 XXf 12019 2 XXf Consumption K. wait hours Midpoi nt X No. of Users f fX 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 Continued……..
- 63. Ch 5_63 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.
- 64. Ch 5_64 For two firms A and B belonging to same industry, the following details are available: Number of Employees Average monthly wage: Standard deviation: Firm A 100 Tk. 4,800 Tk. 600 Firm B 200 Tk. 5,100 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. Example:
- 65. Ch 5_65 i. For finding out which firm pays larger amount, we have to find out X. Firm A : N = 100, Firm B : N = 200, X X = 4800, X=100×4800 =4,80,000 = 5100, X=200×5100 =10,20,000 Hence firm B pays larger amount as monthly wages. XNXor N X X
- 66. Ch 5_66 ii. For finding out which firm pays greater variability in the distribution of wages, we have to calculate coefficient of variation. Since coefficient of variation is greater in case of firm A, hence it shows greater variability in the distribution of wages. 5012100 4800 600 100.. X VC Firm A : 5910100 5100 540 100.. X VC Firm B :
- 67. Ch 5_67 iii. Combined average weekly wage: 21 2211 12 NN XNXN X = 4800,1N = 100, 1X 2N = 200, 2X = 5100, 000,5. 300 1020000480000 200100 51002004800100 12 TK X
- 68. Ch 5_68 Combined Standard Deviation 540,200,600,100 2211 NN d1= 1X 12X d2 = = 5100 - 5000 =10012X d1 = 1X = 4800 - 5000 =200 2X Continued……. 21 2 2 2 2 1 1 2 2 2 2 1 1 12 NN NNNN dd
- 69. Ch 5_69 25.578 300 000,320100 300 20000004000000583200036000000 200100 100200200100540200600100 2222 12 Hence the combined standard deviation is Tk. 578.25.
- 70. Ch 5_70 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
- 71. Ch 5_71 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.
- 72. Ch 5_72 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…..
- 73. Ch 5_73 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.
- 74. Ch 5_74 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.
- 75. Ch 5_75 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 Continued……
- 76. Ch 5_76 Continued…….. 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 Calculation for drawing the Lorenz curve Solution:
- 77. Ch 5_77 0.0 20.0 40.0 60.0 80.0 100.0 120.0 1 20 46 60 79 92 98 100 Lorenz curve Per Cent of Companies PerCentofProfits
- 78. References Quantitative Techniques, by CR Kothari, Vikas publication Fundamentals of Statistics by SC Guta Publisher Sultan Chand Quantitative Techniques in management by N.D. Vohra Publisher: Tata Mcgraw hill