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Linear Programming 145 Therefore the solution is Profits = 45 (10) + 55 (15) = 450 + 825 = 1275 7.3.2 Problem: Solve by Simplex and Graphical Method. Maximise Z = 5x1 + 4x2 Subject to 3xI + 2x2 d 50 x1 + x2 d 22 Simplex Method Z = 5x1 + 4x2 + 0.S1 + 0.S2 3x1 + 2x2 + 1.S1 + 0.S2 = 50 X1 + X2 + 0.S1 + 1.S2 = 22 where, sl and s2 are constants-slack variables Basis x1 x2 s1 s2 bi ai/bij 0 50 50/33 = 1.66 s10 3* 2 1 1 22 22 0 s20 1 1 0 22 0 cj 5 4 0 soln 0 0 50 'j 5 4 0 n c1 = 5 z1 = 0 '1 = 5 c2 = 4 z2 = 0 '2 = 4 c3 = 0 z3 = 0 '3 = 0 c4 = 0 z4 = 0 '4 = 0 Basis x1 x2 s1 s2 bi ai/bij x10 1 2/3 1/3 0 50/3 = 16.66 25 s20 0 1/3 * -1/3 1 16/3 = 5.33 cj 5 4 0 0 m 16 soln 25 0 0 16 'j 0 2/3 -5/3 0 CU IDOL SELF LEARNING MATERIAL (SLM)

146 Business Mathematics and Statistics Old row - row element in pivot column 1 - 1 - 1× 1 = 0 = 1/3 2/3 × 1 -1/3 1 0- 1/3 × 1= 16/3 1- 0x 1 = bi 22 - = 6 z1 = 5 50/3 x 1 16 z2 = 10/3 '1 = 0 x2 s1 s2 z3 = 5/3 '2 = 2/3 -2 1 z4 = 0 '3 = -5/3 3 0 '4 = 0 0 1 Basis 0 -2 x15 x1 -2 6 x24 cj 101 = soln = 'j 0 1 -1 = = 540 = 6 16 0 0 0 -1 n Old row - row element in pivot column 1 2/3 - 0 × 2/3 1/3 0 - 1 × 2/3 50/3 - -1 × 2/3 - 3 × 2/3 - 16 × 2/3 z1 = 5 D1 = 0 z2 = 4 D2 = 0 z3 = 1 D3 = -1 z4 = 2 D4 = -2 x1 = 6 x2 = 16 Z = Profit= 5 (6) + 4 (16) = 94 CU IDOL SELF LEARNING MATERIAL (SLM)

Linear Programming 147 7.3.3 Graphical Method Intercept for Coordinates Constraints x2 axis x1 axis x2 axis xl axis 3x1 + 2x2 = 50 [x, = 0] [x2 = 0] x, + x2 = 22 25 50/3 = 16.66 (25,0) (0,50/3) (22,0) (0,22) 22 22 The line drawn with the above terminal points intersect at x1 = 6 x2 = 16 Z = 5(6) + 4(16) Z = 94 is the maximum value. 26 Scale: x-axis l cm = 2 units y-axis 1 cm = 2 units 24 22 20 18 16 x2 14 12 10 8 6 4 2 0 2 4 6 8 10 12 14 16 18 20 22 24 x1 Fig. 7.2 CU IDOL SELF LEARNING MATERIAL (SLM)

148 Business Mathematics and Statistics Problem 6: A firm makes two types of furniture: Chairs and tables. The profit contribution from each product as calculated by the accounting department is ` 20 per chair and ` 30 per table. Both products are processed on the three machines M1 , M2 and M3 The time required by each product and total time available per week on each machine are as follows. Machine Chair Table Available Hours Per Week M1 3 3 36 M2 5 2 50 M3 2 6 60 How should manufacture schedule his production in order to maximise profit. Solution: We formulate L.P.P. with the objective function and the constraint inequalities as under: Let X be the number of chairs to be manufactured. Y be the number of tables to be manufactured The L.P.P. can be formulated as maximise P = 20x + 30y Subject to 3x + 3y d 36 5x + 2y d 50 2x + 6y d 60 x,yt0 Let SI, S2, S3 be the slack variables them we have 3x + 3y + 1.51S1 + 0.5S2 + 0.5S3 = 36 5x + 2y + 0.5S1 + 1.5S2 + 0.5S3 = 50 2x + 6y + 0.5S1 + 0.5S2 + 1.5S3 = 60 CU IDOL SELF LEARNING MATERIAL (SLM)

Linear Programming 149 Simplex table is: Cj Qty 20 30 0 00 xy S1 S2 S3 0 S1 36 33 1 00 0 S2 50 52 0 10 0 S3 60 26 0 01 Zj 0 00 0 00 Cj - Zj 20 30 0 00 Key Column Out going variable is S3 and in coming is X2 the row corresponding to X2 will be as under on dividing the key row by 6. (the key element) 30 X2 10 1/3 1 0 0 1/6 Also the row corresponding to Sl , will have the elements obtained as under: New number = Old no minus (Corresponding element in the key column x Corresponding number in key row) Row Sl Row S2 36 - ( 3 × 10 ) = 6 50 - ( 2 × 10 ) = 30 3 - ( 3 × 1/3 ) = 2 5 - ( 2 × 1/3) = 13/3 3 - (3 × 1) = 0 2 - ( 2 × 1) = 0 1- (3 × 0) = 1 0 - (2 × 0) = 0 0 - (3 × 0) = 0 1 - (2 × 0) = 1 0 - ( 3 × 1/6 ) = -1/2 0 - ( 2 × 1/6 ) = -1/3 Cj Prod Mix Qty Xl X2 S1 S2 S3 20 0 S1 6 13/3 0 I 0 -112 0 S2 30 1/3 I 0 1 -1/3 30 X2 10 0 0 1/6 Zj 300 10 30 0 05 Cj - Zj 10 0 0 0 -5 n Key Column CU IDOL SELF LEARNING MATERIAL (SLM)

150 Business Mathematics and Statistics Cj Prod Mix Qty X1 X2 S1 S2 S3 20 Xl 3 1 0 Y2 0 -1/4 30 X2 9 0 1 -1/6 0 1I4 0 S2 17 0 0 -13/6 3/4 0 Zj 330 20 30 5 0 5/2 Cj - Zj 0 0 -5 -5/2 ? X1 = 3 units X2 = 9 units and profit P = 20x + 30y = 20(3) + 30(9) = 60 + 270 = 330 ? Profits = ` 330 Special Cases in Linear Programming z Four special cases and difficulties arise at times when using the graphical approach to solving LP problems ˆ Infeasibility ˆ Unboundedness ˆ Redundancy ˆ Alternate Optimal Solutions No feasible solution z No feasible solution ˆ Exists when there is no solution to the problem that satisfies all the constraint equations ˆ No feasible solution region exists ˆ This is a common occurrence in the real world ˆ Generally one or more constraints are relaxed until a solution is found CU IDOL SELF LEARNING MATERIAL (SLM)

Linear Programming 151 Example: Unboundness Case z Unboundedness ˆ Sometimes a linear program will not have a finite solution ˆ In a maximization problem, one or more solution variables, and the profit, can be made infinitely large without violating any constraints ˆ In a graphical solution, the feasible region will be open ended ˆ This usually means the problem has been formulated improperly CU IDOL SELF LEARNING MATERIAL (SLM)

152 Business Mathematics and Statistics Example of Unboundness case Redundancy Case z Redundancy ˆ A redundant constraint is one that does not affect the feasible solution region ˆ One or more constraints may be more binding ˆ This is a very common occurrence in the real world ˆ It causes no particular problems, but eliminating redundant constraints simplifies the model CU IDOL SELF LEARNING MATERIAL (SLM)

Linear Programming 153 Example of Redundancy Alternate Optimal Solutions z Alternate Optimal Solutions ˆ Occasionally two or more optimal solutions may exist ˆ Graphically this occurs when the objective function’s isoprofit or isocost line runs perfectly parallel to one of the constraints ˆ This actually allows management great flexibility in deciding which combination to select as the profit is the same at each alternate solution CU IDOL SELF LEARNING MATERIAL (SLM)

154 Business Mathematics and Statistics Example of Alternate Solutions (2) Multiple Solution Multiple solutions of a linear programming problem are solutions each of which maximize or minimize the objective function under Simplex Method. Under Simplex Method, the existence of multiple optimal solutions is indicated by a situation under which a non-basic variable in the final simplex table showing optimal solution to a problem, has a net zero contribution. In other words, if at least one of the non-basic variable in the (Ci ± =j) row of the final simplex table has a zero value, it indicates that there is more than one optimal solution. (3) Unbounded Solution Sample If we consider Maximize (x + y) Subject to x-yet1 x+yt2 x, y t 0 CU IDOL SELF LEARNING MATERIAL (SLM)

Linear Programming 155 The feasible region is as follows: Objective Function 2 Feasible Region 12 3 In this case, you can see we can move as much as we want the objective function in the growing sense of x and y coordinates without leaving the feasible region Therefore, objective function can grow too into feasible region, so we are in an unbounded solution case for this problem. 4. Redundant Constraints A redundant constraint is a constraint that can be removed from a system of linear constraints without changing the feasible region. Consider the following system of nonnegative linear inequality constraints and n variables (m t n): AX d b, X t O ZKHUH $ H Rmxn, b H Rm, X H Rn, and O H Rn /HW $iX d bi be the ith constant of the system (2.1) and let S = {X H Rn/ AiX d bi, X t O} be the feasible region associated with system (2.1). /HW 6k = {X H Rn / AiX d bi, X t O, i = k} be the feasible region associated with the system of equations AiX d bi, i = 1, 2…, m, i = k. The kth constraint AkX d bk (1 < k < m) is redundant for the system (2.1) if and only if S = Sk. CU IDOL SELF LEARNING MATERIAL (SLM)

156 Business Mathematics and Statistics Definition 2.1. Redundant constraints can be classified as weakly and strongly redundant constraints. Weakly Redundant Constraints The constraint AiX < bi LV ZHDNO\\ UHGXQGDQW LI LW LV UHGXQGDQW DQG $iX = bi IRU VRPH ; H S. Strongly Redundant Constraints The constraint AiX < bi LV VWURQJO\\ UHGXQGDQW LI LW LV UHGXQGDQW DQG $iX < bi IRU DOO ; H S. Binding Constraint Binding constraint is the one which passes through the optimal solution point. It is also called a relevant constraint. Nonbinding Constraint Nonbinding constraint is the one which does not pass through the optimal solution point. But it can determine the boundary of the feasible region. Example 2.2: Consider the following linear inequality constraints: (1) 2x1 + 1x2 d 8, (2) 4x1 + 0x2 d 15, (3) 1x1 + 3x2 d 9, (4) 1x1 + 2x2 d 14, (5) 1x2 d 4 (6) 1x1 + 1x2 d 5, ZKHUH In Figure 1, the region OABCD is the feasible region and the vertex C is the optimal point. The constraints (1), (2), (3) and (6) are binding, (4) and (5) are strictly redundant. The 2nd constraint is non-binding. Among the binding constraints, (6) is weakly redundant. CU IDOL SELF LEARNING MATERIAL (SLM)

Linear Programming 157 Figure 1 7.3.5 The Concept of Duality: Given a linear programming problem it is possible to find its replica which is also a linear programming problem. The given L.P is called the ‘Primal’ and its replica is known as Dual. As the two problems are replicas of each other, it follows that when the primal problem is of the maximization type, the dual would be of the minimmisation type and vice versa. Interpretation of the Primal Problem: Given each z = unit values, of the output Cj and also given the upper limit in regard to the availability of each input bj, then we have to determine how much of each output Xj be produced so as to maximize the value of the total output. Interpretation of the Dual problem: If the values of each input (bi) are given along with a lower limit of the unit value of each output Cj then we are required to determine the unit values that should be assigned to each input yJ so as to minimize that value of the total input. 7.3.6 Problem: Maximise P = 6x1 + 7x2 Subject to 2x1 + 3x2 d 12 CU IDOL SELF LEARNING MATERIAL (SLM)

158 Business Mathematics and Statistics 2x1 + x2 d 8 x1 + x2 t 0 As explained above the dual of this L.P is: D = 12y1 + 8y2 2y1 + 2y2 t 6 3y1 + y2 t 7 y1, y2 t 0 7.3.7 The Primal Dual Relationship can be shown as follows: Primal Dual Maximise Minimise P = 6x1 + 7x2 C = 12y1 + 8Y2 Subject to 2x1 + 3x2 d 12 2y1 - 2y2 t 6 2x1 + 1x2 d 8 3y1 + 1 y2 t 7 X1, X2 t 0 Y1, Y2 t 0 The above can be represented in a matrix form as under: Primal Dual §X1· §Y1· Minimise = [6, 7]¨ ¸ Minimise D = [12, 8] ¨ ¸ ©X2¹ ©Y2¹ Subject to [2, 3] [x1] < [12] [2 2] [y1] > [6] [2, 1] [x2] < [8] [3, 1] [y2] > [7] 7.4 Summary linear Programming is a method used by the managers/authorities to make decision in business/ enterprise. the subject of Operation Research is concerned with the analysis, condensation and presentation of data for the process of decision-making. LP is one of the mathematical techniques involves the optimisation of a function which is called the objective function. CU IDOL SELF LEARNING MATERIAL (SLM)

Linear Programming 159 LP problems can be solved by using one of the following methods: (i) The Graphic Method (ii) The simplex Method The graphic method is to solve the problem by identifying it and draw the graph with decision variables and the objective function along the constraints. The optimal solution can be obtained. The simplex method: It is a systematic procedure (analgorithm) for solving LP Problems. 7.5 Key Words/Abbreviations Simplex Procedure Terminology as shown in Steps 1 to 17, given in 7.3. 7.6 Learning Activity Work out examples given in 7.7 after studying the related illustrative examples. 7.7 Unit End Questions (MCQ and Descriptive) (A) and (B) Descriptive Type: Short Answer Type Questions 1. What is meant by ‘Linear Programming’ ? Explain its uses and limitations. 2. Explain the major assumption in ‘Linear Programming’. 3. Describe the various steps involved in the formulation a L.P. problem. 4. Explain the following terms : (a) Objective function (b) Constraints (c) Non-negativity condition (d) Feasible solution 5 Explain the various steps in solving a L.P. problem involving two decision variables, by the graphical method. CU IDOL SELF LEARNING MATERIAL (SLM)

160 Business Mathematics and Statistics 6. What are the steps involved in the simplex method for solving linear programming problems? 7. A home decorator manufactures two types of lamps, Alpha and Delta. Both these lamps require the services of a cutter and a finisher. Alpha requires 3 hours of cutter’s time and 2 hours of finisher’s time. Delta requires 2 hours of cutter’s time and 1 hour of finisher’s time. The cutter has 180 hours and finisher 110 hours time each month. If one Alpha gives a profit of `10 and a Delta a profit of ` 7, formulate this as a L.P. problem. 8. A firm makes two types of furniture : chairs and tables. The contribution for each product as calculaed by accounting department are ` 20 per chair and ` 30 per table. Both products are processed on three machines M1, M2 and M3. The time required for each product and total time available per week on each machine are as follows: Machine Chair Table Available hours M1 3 3 36 M2 5 2 50 M3 2 6 60 9. A multiplant company has three manufacturing plants, A, B and C are two markets X and Y. Production cost per piece in A, B and C is ` 1,500, ` 1,600 and ` 1,700 respectively. Selling price of X and Y are ` 4,700 respectively. Demand for X and Y is 3500 and 3600 pieces respectively. Unit transportation costs are as follows : From To X Y A 1000 1500 B 2000 3000 C 1500 2500 Formulate a Linear Programming model. 10. Solve the following I.P.P. by graphical method. Minimize C = x1 + x2 Subject to x1 + x2 t 12 5x1 + 8x2 t 74 x1 + 6x2 t 24 x1 , x2 t 0 CU IDOL SELF LEARNING MATERIAL (SLM)

Linear Programming 161 11. Two products L and K are processed on 3 machines m1, m2 and m3. The processing time per unit, machine availability and profit per unit are as follows: Machine Processing Time in Hrs Availability in Hrs L K m1 2 3 1500 m2 3 2 1500 m3 1 1 1000 Profit per unit in ` 10 12 Formulae a L.P.P. and solve it by samplex method 12. Minimize C = 2x + y, by graphical method. Subject to 3x + y = 3 4x + 3y t 6 x + 2y d 3 x,y d 0 13. State the dual of: Minimize Z = 50x1 - 80x2 + 140x3 Subject to x1 - x2 - 3x3 t 4 5x1 - 2x2 - 232 t 3 x1 , x2 , x3 t 0 14. A company manufacures two products. The basic time data, machine capacity and profit contribution is given in the table below : Machine Machine hrs. Required Machine hrs. Product I per unit Available per week Product II Lathes 11 80 Milling 12 120 PP Profit per unit (`) 6 8 Formulate the problem as a linear programming problem and solve it graphically to determine the quantity of each product to be manufactured to maximize profit. CU IDOL SELF LEARNING MATERIAL (SLM)

162 Business Mathematics and Statistics 15. A company makes two kinds of leather belts. Belt A is of high quality and belt B is of lower quality. The respective profits are ` 4 and ` 3 per belt. Each belt of type A requires twice as much time as a belt of type B and if all the belts were of type B, the company could make 1000 belts per day. The supply of leather is sufficient for only 800 belts (both A and B combined). Belt A requires a fancy buckle and only 400 such buckles are available per day. There are only 700 buckles a day available for type B. Determine the number of bels to be purchased for each type so as to maximize profit. Solve graphically. 7.8 References References of this unit have been given at the end of the book. ˆˆˆ CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT 8 INTRODUCTION TO STATISTICS Structure 8.0 Learning Objectives 8.1 Introduction 8.2 Definition and Scope of Statistics 8.3 Statistical Methods and Applied Statistics 8.4 Statistics and Mathematics 8.5 Statistics and Economics 8.6 Statistics and Econometrics 8.7 Use of Statistics in Trade, Commerce, Industry and Decision-making 8.8 Uses and Limitations of Statistics 8.9 Statistics, Statisticians and Citizens 8.10 Summary 8.11 Key Words/Abbreviations 8.12 Learning Activity 8.13 Unit End Questions (MCQ and Descriptive) 8.14 References 8.0 Learning Objectives After studying this unit, you will be able to: z Explain the meaning, significance and the historical background. z Grasp the essentials with reference to some definition and scope of the subject and as well as the distinction between statistical methods and applied statistics. CU IDOL SELF LEARNING MATERIAL (SLM)

164 Business Mathematics and Statistics z Differentiate the relationship between statistics and other subjects: Mathematics, Economics etc. z Analyse the use of statistics in trade, commerce, industry and decision-making. z Describe the uses and limitations of statistics. z Judge the role of statistics, statisticians and citizens. z On the basis of the overall study of this unit, students can prepare a brief summary. 8.1 Introduction to Statistics In a broad sense, statistics was born when man first began to count and express his ideas and sensible facts in quantitative terms — that is, in numbers. Civilization’s progress could have been held at bay but for the discovery of numbers, for indisputably its growth through the centuries has been nurtured by numbers. The use of statistics dates back to ancient times when the Pharaohs and the Hebrews took censuses of population and wealth from time to time. Ancient kings maintained records on population, wealth, area of land, crop yields, livestock, births and deaths etc. in their kingdoms. As a result, statistics came to be known as the ‘Science of Kings’. Statistics were collected even in ancient India, during the reign of the Mauryan and the Gupta kings. According to Kautilya’s treatise on statecraft, the Arthashastra; the Mauryan kings undertook the task of collecting statistics on population, agriculture etc. from time to time. The Mughal emperors, too, maintained statistics of population, land, agriculture etc.; the Ain-i Akbari, for instance, provides statistical details during the reign of Emperor Akbar (1556-1605). The Latin word ‘Status’ and the Italian word “Statista” both mean a political state and it appears that the word statistics has been derived from both of them. The definition of statistics became broader and broader as it developed from century to century. The emergence or statistics in the modern form is largely due to the influence, keen interest and work of great mathematicians of the eighteenth and nineteenth centuries. At present, statistics is used in all fields of human activity and has come to be regarded as indispensable to the study of many sciences, especially the social sciences. CU IDOL SELF LEARNING MATERIAL (SLM)

Introduction to Statistics 165 8.2 Definition and Scope of Statistics When used in the singular, the word ‘statistics’ means the subject of statistics. In the plural, it means quantitative information or numerical data. Various authors defined statistics in different ways, emphasizing various viewpoints. A.L Bowley, for instance, has given four definitions: “Statistics is the science of the measurement of social organisms guarded as whole in all its manifestations.” “Statistics may be called the science of counting.” “Statistics are numerical statements of facts in any department of enquiry, placed in relation to each other.” “Statistics may rightly be called the science of averages.” Each of these definitions stresses only one viewpoint or aspect of the variety and richness of statistics, for it is not only concerned with applications to sociology, averages, counting or figures but with many other things. Hence, these definitions have their own limitations. Bowley also defines statistics as the ‘Science of estimates and probabilities.” While there’s no doubt that in a large number of cases it deals with estimates and probabilities, but again these are not the only methods or aspects of statistics. Webster (Dictionary Definition) defines statistics as “Classified facts respecting the condition of people in the state — especially those facts which can be stated in a table or tables of numbers or in any tabular or classified arrangement.” This definition is inadequate as it stresses only on one aspect of statistics: classification of facts — in a table or tables of numbers-regarding the condition of people in the state. It does not mention its applications in other fields like biology, physics, sociology, economics, meteorology or astronomy. According to Udney Yule: “By statistics we mean quantitative data affected to a marked extent by multiplicity of causes.” This definition, too, emphasizes on only one aspect of statistics, omitting its other important characteristics. W.I. King defines it thus: “The science of statistics is the method of judging collective natural or social phenomena from the results obtained by the analysis of an enumeration or collection of estimates.” Though this definition is a remarkable one, it could have emphasized the all-pervading or all-embracing aspects of statistics. CU IDOL SELF LEARNING MATERIAL (SLM)

166 Business Mathematics and Statistics According to Horace Secrist: “By statistics we mean aggregates of facts affected to a marked extent by multiplicity of causes numerically expressed, enumerated or estimated according to reasonable standards of accuracy, collected in a systematic manner for a predetermined purpose and placed in relation to each other.” This definition is comprehensive and exhaustive and covers most of the aspects of statistics. It’s certainly very close to the definition of modem statistics but yet, it could have been broader. In this modern scientific, technological, industrial, space-conquering, planet-probing and atomic age, statistics and statistical techniques have entered a new epoch and are being increasingly and advantageously used in every sphere of human knowledge. Just as a few mathematical equations symbolize the unknown depths of hidden facts, so also statistical facts provide a good deal of information. And with the growing realization of the indispensability of statistical knowledge — it being all-pervading and all-embracing — its definition will broaden and glitter at the hands of capable statisticians. However, it must be accepted that it is not easy for any capable statistician to come up with a definition of statistics that will remain relevant for ever. But talented authors and statisticians are coming up with more elegant, refined, comprehensive and exhaustive definitions from time to time. The modem view of statistics is that it is not just the art of collecting, sorting, classifying, grouping, summing up and using numerical facts. It is also ‘a science that provides the tools and techniques that can be used advantageously to measure and analyse facts in any sphere of human knowledge’. Though reputed authorities on the subject have given remarkable definitions, one more may be added to them: “Statistics, in the singular, is the subject that explains all the devices of collection, presentation, analysis and inference of numerical facts and figures in any field of knowledge, and in the plural, stand for any sensible statements of numerical facts which can be analysed, interpreted, compared and related if possible.” 8.3 Statistical Methods and Applied Statistics Statistics is divided into two main divisions: statistical methods and applied statistics. Statistical methods or the theory of statistics is also known as mathematical statistics as it developed mainly through mathematics. The theory of statistics has the mathematical theory of probability as its CU IDOL SELF LEARNING MATERIAL (SLM)

Introduction to Statistics 167 basis. Statistical methods deal with the formulation of devices or general rules of procedure that can be employed in handling different types or forms of data. For instance, the rules for the collection of data, classification, tabulation, analysis, comparison by averages and the methods of finding the coefficient of correlation, regression equations and coefficients and index number construction are all statistical methods. The theory of statistics is further divided into statistical inference and the design of experiments. Statistical inference involves drawing inferences or valid conclusions from a sample of any population. These conclusions are concerned with the specification of the population from the information supplied by the sample. Design of experiments is concerned with designs for the collection and analysis of data. Experiments are conducted on the basis of these to test and verify the validity of hypotheses. Therefore, selecting the design is of the utmost importance: a wrong one inevitably leads to fallacious and absurd conclusions in the course of experimentation. Applied statistics deals with practical applications to business and economic data by statistical methods to tackle concrete problems like population, agriculture, industry, trade and transport, wages and prices of commodities. Economic statistics, which is a branch of applied statistics, deals with statistics or prices, production, income, expenditure, investment, sales profits etc. Applied statistics has several branches. Some or these are: (i) Business and financial statistics: These are statistics of public finance, business money, banking, currency and exchange. (ii) Population statistics: These form a part of demography, which is a collective and detailed statistical study of human life with reference to population growth and vital and social aspects. (iii) Agricultural statistics: These deal with area, yield of crops, livestock etc. (iv) Industrial statistics. (v) Labour and employment statistics. (vi) Administrative statistics. (vii) Social statistics. (viii) Trade and transport statistics. (ix) Price statistics. (x) National income statistics. CU IDOL SELF LEARNING MATERIAL (SLM)

168 Business Mathematics and Statistics Statistics of consumption, production, exports, imports, prices and wages help in framing economic policies. Economic growth is determined to a considerable extent by certain statistical measure like index numbers etc. 8.4 Statistics and Mathematics The science of statistics owes its development and progress to the mathematical theory of probability. Statistical methods or mathematical statistics have their origins in mathematics and owe a great deal to mathematicians. One of the first to sow the seeds of the idea of probability was Jerome Cardano (1501-1536), a gambler and mathematician, who came up with rules to minimize risks in gambling and precautions against cheating in his book ‘Liber De Ludo Aleae’. Another step in this direction was taken in the year 1654, when another gambler and amateur mathematician, Chevelier-de-Mere, communicated a problem in a game of dice to the French mathematician Pascal Pascal in turn passed it on to Fermat, another French mathematician. Later, these ideas were taken forward by mathematician. Jacob Bernoulli (1654-1705), who first explained the law of large numbers in his book, ‘Ars Conjectandi’ and another mathematician Daniel Bernoulli (1700-1782), who expressed the idea of ‘mortal expectation’. They laid the first foundations of the modem theory of probability. Years later, mathematicians like Laplace (1749-1827), Gauss (1777-1855) and S. Poisson (1781-1840) made significant contributions to probability and statistical theory. In this manner, eminent men — mathematicians, statisticians, social scientists and research workers in diverse fields-contributed to the development, enrichment and progress of the theory of statistics over the generations. The following are some of the notable men who contributed to the growth and development of statistics: Euler, Lagrange. De Moivre. Knapp, Lexis, Morgan, Chrystal, Bayes, Charlier, August Mcitzen, Davenport Francis, Edgeworth, Francis Galton, Karl Pearson, Udney Yule, W.W. Gossett, A.L. Bowley, Adams, W.I. King, W. Pearson and R.A. Fisher. 8.5 Statistics and Economics There is a great deal of inter-relationship and interdependence between statistics and economics. Economists have to use mathematical methods and statistical procedures to make more intensive, accurate and detailed studies of economic problems. According to economist Alfred Marshall: “Statistics are the straw out of which I, like every other economist, have to make bricks.” CU IDOL SELF LEARNING MATERIAL (SLM)

Introduction to Statistics 169 One of the earliest to use statistics in economics was an Englishman called John Graunt (1620-1674) who wrote his book Natural and Political Observation ... upon the Bills of Mortality after thoroughly studying the records of births and deaths in the cities of England in the year 1662. A few years later, in 1676, Sir William Petty (1623-1685) published his Political Arithmetic in which he insisted upon the use of statistical methods in studying social phenomena. In 1871, W.S. Jevons (1835-1882) published his Theory of Political Economy in which he said political economy could be made an exact science by using more detailed commercial statistics. Further, the Historical School (1843-1883) also advocated the use of statistical techniques in economics. Alfred Marshall, Pareto, Lord Keynes and F.Y. Edgeworth relied on a lot of statistical detail in their works on economics. 8.6 Statistics and Econometrics Statistics is also closely related to econometrics, a subject of recent origin. Econometrics is more or less a sensible synthesis of economic theories, their mathematical symbols and expressions and certain statistical procedures that verify these theories in other words, it is the formulation of economic theories in mathematical terms and the examination of these theories in the light of statistical methodology to find out whether they can be accepted or rejected in practical and real- life situations. Econometrics is to be distinguished from statistical economics, mathematical statistics and mathematical economics, all of which are quite different from each other. While statistical economics deals with economic data in quantitative terms, mathematical statistics is concerned with the general rules of procedure and devices that can be used to handle different types of quantitative data. Mathematical economics explains economic theories in mathematical terms. 8.7 Use of Statistics in Trade, Commerce, Industry and Decision-making Wherever there is anything to be counted — in any area of trade, commerce or industry-you can find statistics there. Statistics provide the businessman a beacon of light that he can use to profit and progress. A businessman with a sound statistical knowledge of his entire business may do well. Without it, he will perish. CU IDOL SELF LEARNING MATERIAL (SLM)

170 Business Mathematics and Statistics Today many companies maintain their own statistics departments to evaluate their business .policies on profits, sales, consumer preferences, manufacturing costs, product quality, productivity and consumer goodwill. Any economic planning without accurate and reliable statistics is unthinkable. Statistical charts, graphs and diagrams on quantitative data act as visual aids, making business figures and facts more significant and easily intelligible. In order to succeed in trade and commerce, it is vital to have correct estimates, probabilities and favourable forecasts based on statistics that have been collected systematically, presented accurately and interpreted wisely. Whether a person is a banker, an investor, a manufacturer or a stock exchange broker, he would be benefited immensely if he possesses statistical knowledge relating to monetary savings, rates of interest, investment markets, stock markets etc. Statistics is also useful for insurance companies. Using the Law of Statistical Regularity of statistics of births and deaths due to various causes-their actuaries can calculate Insurance premiums. Statistics also contribute a great deal to human welfare. Modern statistical techniques are the key to industrial progress because when these techniques — quality control, market research etc. are properly used they lead to increased profits, reduce wastage of materials and labour and enable businessmen to take informed decisions while selling and purchasing. The tools of statistical analysis may not lead us to peaks of precision or to depths of illusion, but they certainly do show us the practical levels of human expectation in quantitative terms. Statistical methods provide analytical results that help ion making decisions. Both small and large companies can profit by maintaining statistical departments and employing expert statisticians. Such departments collect, record and analyse data relating to the industry, prepare statistical charts, diagrams and maps, design and execute market research surveys, furnish periodical progress reports and give sensible statistical advice to business executives. Undoubtedly, statistics and statisticians are indispensable to trade, commerce and industry. Statistics provide feasible alterations for the manager enabling him to select base alternative. 8.8 Uses and Limitations of Statistics Uses: (i) A given mass of complex quantitative data can be simplified by using statistical methods. The results can be numerically measured, comparisons can be made and relationships observed. CU IDOL SELF LEARNING MATERIAL (SLM)

Introduction to Statistics 171 (ii) Statistics enriches knowledge and widens human experience. (iii) Planning, which is critical in the twenty-first century, is to a large extent dependent on statistical results. Statistics not only helps to lay the foundations, it also provides the brick and mortar of all efficient and wise planning. (iv) Businessmen can increase profits by using statistics extensively. Thus commerce flourishes in statistics and statistics flourish in commerce. (v) The diverse applications of statistical methods are so stupendous that people of each generation and men in every field can gain immensely and lose nothing. The modem age is remarkably statistical in character. Limitations: (i) Everybody of knowledge has its own limitations and so also statistics. Statistics is concerned with quantitative data and not with descriptive or qualitative facts. (ii) Statistics as a science deals with groups of numerical items and not with individual items. The group, not the individual item, is its target. (iii) Statistics, more or less, are numerical results expressed as approximations and estimates. (tv) Statistical laws are not as accurate or exact as pure scientific laws but are quite dependable on an average, and to a large extent. After all statistics is considered a social science. (v) It is not always possible to study all aspects of a problem only through statistics for statistical facts and figures throw light only on some areas. (vi) The sum and substance of all statistics is that they are all a number of numbers. Consequently they are likely to be wrongly interpreted and misused by non-experts. Numbers do not lie, but it may not be wrong to say that some persons may lie with numbers. After all, numbers are as innocent as flowers. 8.9 Statistics, Statisticians and Citizens Statistics is not only of immense use to economists, businessmen, planning commissions, scientific investigators and so on, it is useful to ordinary citizens as well. People today are bombarded by statistical facts and figures in newspapers, magazines, advertisements, TV, radio and so on but it is interesting to see how far they understand them. In a democratic country, a citizen can function CU IDOL SELF LEARNING MATERIAL (SLM)

172 Business Mathematics and Statistics efficiently only when he is able to understand these facts and figures. Statistical education therefore is very important. As citizens adopt a more scientific viewpoint of life, they will be able to correctly obtain, reduce, correlate and interpret various kinds of statistical information and facts strictly on a mathematical basis. But if they are not able to do so, they may distrust statistics and condemn them as “tissues of falsehood”. No remark can be more damaging to the fair subject of statistics than that of Mark Twain who is said to have remarked that “there are three degrees of comparison in lying- lies, damned lies and statistics”. It is for this reason that statisticians should possess broad knowledge, a lot of experience, common sense and judgement. They should exercise the self-restraint of an artist, be accurate, painstaking and have a passion for accuracy in all their calculations, generalizations and conclusions. They should thoroughly understand human nature and do research to discover new facts in their field. Last but not the least, they should be thoroughly acquainted with the latest trends and developments in their field, and have a desire to work well, efficiently and patiently without being slovenly even for a single movement while working with numerical facts. Figures do not lie, but figures· obtained haphazardly, compiled unscientifically and analysed incompetently would lead to a general distrust in statistics. If statistical methods are not properly used, the fault lies not with the science of statistics but with the person who uses them. According to Wallis and Roberts, “He who accepts statistics indiscriminately will be often duped unnecessarily. But he who distrusts statistics indiscriminately will often be ignorant unnecessarily.” 8.10 Summary ‘Statistics in the singular, is the subject that explains all the devices of collection, presentation, analysis and inference of numerical facts and figures in any field of knowledge, and in the plural, stand for any sensible statements of numerical facts, which can be analysed, interpreted, compared and related if possible.” The two main division of statistics are: Statistical Methods and Applied Statistics. There is much relationship and inter-dependence between statistics and other subjects. These subjects are: Mathematics, Economics, Econometrics, etc. Statistics is of much use in trade, commerce, industry as also in decision making (refer 8.7). It has many uses and some limitations as explained in (8.8). CU IDOL SELF LEARNING MATERIAL (SLM)

Introduction to Statistics 173 8.11 Key Words/Abbreviations Classification, tabulation, Inference, Interpretation, decision-making, Applied, Statisticians, multiplicity, correlated, Econometrics. 8.12 Learning Activity 1. Give any two definitions of statistics and explain them in detail. 2. Explain the importance of the subject of statistics with reference to the statement. “There is hardly any field which does not fall within the scope of statistics.” 8.13 Unit End Questions (MCQ and Descriptive) A. Descriptive Type: Short Answer Type Questions 1. Discuss the statement: statistics is both a science and an art. 2. Explain the relationship between statistics and some other important sciences. 3. “Statistics are the straw out of which like every other economist, have to make bricks.” — Alfred Marshall. Explain in the light of the above observation the relation between economics and statistics. How far is it correct to say that the science of economics is becoming statistical in its methods? 4. Describe briefly the scope and utility of statistics in the field of trade and commerce. 5. Discuss the meaning and scope of statistics, bringing out its importance, particularly in the field of trade and commerce. 6. Explain the uses and limitations of statistics. CU IDOL SELF LEARNING MATERIAL (SLM)

174 Business Mathematics and Statistics 7. “It is never safe to take published statistics at their face value without knowing their meanings and limitations and it is always necessary to criticize the arguments that can be based on them.” — A.L. Bowley. Explain this statement. 8. Discuss the importance of ‘statistics in the fields of economics, business and commerce. 9. “Statistics are aggregates of facts, affected to a marked extent by a multiplicity of causes, numerically expressed, enumerated or estimated according to reasonable standards of accuracy, collected in a systematic manner for a predetermined purpose and placed in relation to each other.’ Discuss the above statement. 10. Write an essay on “Statistics in the service of trade and commerce.” 11. Define statistics and show how it can help in spreading scientific knowledge, the establishment of a sound business and the formulation of a plan for national economic development. 12. Comment on the following definitions of statistics: (a) “Statistics is the science of averages.” (b) “Statistics is the science of estimates and probabilities.” 13. Write an essay on the importance of statistics in a planned economy. 14. “There is hardly any field which does not fall within the scope of statistics,” says an author. Corroborate this statement. 15. Discuss the importance of the study of statistics and show how it can help the extension of scientific knowledge, the establishment of a sound business and the introduction of social and political reforms. 16. Comment on the following: “Statistics is always concerned with mass phenomena and never with a single observation.” 17. “Statistics helps business by its employment as a tool of industrial research and as an important factor in scientific research.” Elucidate. CU IDOL SELF LEARNING MATERIAL (SLM)

Introduction to Statistics 175 18. “The science of statistics is a most useful servant but only of great value to those who understand its proper use.” — W.I. King. Comment on the above statement. 19. Describe briefly the use of statistics in various spheres of human activity. 20. “Statistics only furnishes a tool, necessary though imperfect, which is dangerous in the hands of those who do not know its use and deficiencies.” Discuss the above statement. B. Multiple Choice/Objective Type Questions 1. Statistics may rightly be called the science of __________. (a) Averages (b) Limitations (c) Application (d) None of these 2. The statistics that deals with area, yield of crops, livestock etc. is ___________________ statistics. (a) Population (b) Agricultural (c) Industrial (d) Price 3. Statistics enriches knowledge and widens __________ experiences. (a) Social (b) Accounting (c) Human (d) None of these 4. The Mughal emperor maintained statistical detail in the name of __________. (a) Ain-i Akbari (b) Emperor Akbar (c) Akbari (d) None of these Answers (1) (a); (2) (b); (3) (c); (4) (a) 8.14 References References of this unit have been given at the end of the book. ˆˆˆ CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT 9 CLASSIFICATION AND TABULATION OF DATA Structure 9.0 Learning Objectives 9.1 Introduction 9.2 Classification of Data — Its Meaning and Objectives 9.3 Number of Classes 9.4 Methods of Classification 9.5 Problem 9.6 Tabulation 9.7 Types of Tabulation 9.8 Rules and Precautions on Tabulation 9.9 Illustrative Examples 9.10 Summary 9.11 Key Words/Abbreviations 9.12 Learning Activity 9.13 Unit End Questions (MCQ and Descriptive) 9.14 References 9.0 Learning Objectives After studying this unit, you will be able to: z Meaning of classification, its categories, types and methods of classification, and numerical examples on exclusive method and inclusive method. CU IDOL SELF LEARNING MATERIAL (SLM)

Classification and Tabulation of Data 177 z Meaning of tabulation, components, general purpose and special purpose tables. z Types of tables, rules and precautions regarding tabulation. z Various illustrative examples on tabulation. 9.1 Introduction Statistical data consists of a number of characteristics. It is generally heterogeneous. To interpret such a data, it is necessary to classify the same into homogeneous classes. Therefore, classification of quantitative data is a fundamental step that leads to simplification and clarification. The different methods of classification are explained in 9.4 along with the problem that follows, are of much importance. Tabulation of data is the step that is necessary for an appropriate presentation of data in rows and column under various headings. The various types of tabulation as explained in 9.7, along with the rules and presentations deserve careful study. Illustrative examples explain the process of tabulation. 9.2 Classification of Data — Its Meaning and Objectives Classification is the process of sorting ‘similar’ things from among a group of objects with different characteristics In other words, heterogeneous data is divided into separate homogeneous classes according to characteristics that exist amongst different individuals or quantities constituting the data.’ Thus, fundamentally classification is dependent upon similarities and resemblance among the items in the data. The main object of classification” is to present vividly, in a simplified and quickly intelligible form, a mass of complex data. Without condensing details in a classified form it is difficult to compare quickly, interpret thoroughly and analyse properly different sets of quantitative and qualitative phenomena. The basic requirements of good classification are stability, non-ambiguity, flexibility and comparability. 9.2.1 Descriptive and Quantitative Classification Depending on the characteristics of the data, they can be broadly categorized into two separate and distinct groups — descriptive and numerical. Descriptive characteristics are those that can be CU IDOL SELF LEARNING MATERIAL (SLM)

178 Business Mathematics and Statistics described in words and are expressible in qualitative terms. Numerical’ characteristics are quantitative-in nature. For instance, literacy, sex, caste and religion are descriptive characteristics. Height, weight, age, income and expenditure are numerically expressible characteristics. Descriptive or qualitative classification is termed classification according to attributes. Numerical or quantitative classification of data in certain class intervals is termed as classification in terms of classes with certain intervals, or classification according to class intervals. 9.2.2 Simple and Manifold Classification Classification based on attributes may be either simple or manifold] In the case of “simple classification, only one attribute is studied. That is, the data is classified into two separate classes under a single attribute. For instance, data collected on literacy in the country can be classified into two distinct classes: literate and illiterate. Since this process is quite simple, it is known as simple classification. On the other hand, analyzing and classifying collected data under several attributes in different classes is called manifold classification. For example, if each of the two classes literate and illiterate is divided into males and females, then there would be four classes. If classified further on a regional basis, there would be a number of other classes. Such a process of classification of data into a number of classes and classes within classes is known as manifold classification. 9.2.3 Classification According to Class Intervals Phenomena like income, heights and weights are ail quantitatively measurable and data on them can be classified into separate class intervals of uniform length. For instance, the marks obtained by a group of 50 candidates in a subject at an examination can be classified into the following classes: 0-10, 10-20, 20-30, 30-40, 40-50, 50-60, 60-70 etc. Each class has a lower and an upper limit and the number of candidates getting marks between these two limits of the same class interval is called the frequency of the respective class. To give an example, if 12 candidates get between 40 and 50 marks. 12 is the frequency of the class 40-50. CU IDOL SELF LEARNING MATERIAL (SLM)

Classification and Tabulation of Data 179 9.3 Number of Classes The number of classes into which particular data should be classified depends upon the mass of data. The larger the mass, the more should be the number of classes. Usually, data is classified into not less than six classes and not more than 20 classes, depending upon the mass and size of the data and the length of the class intervals. The fundamental object of classifying data is to get the maximum possible amount of information most precisely. According to Sturges, Rule the number of class intervals (n) = 1 + 3.322 log N, where N = total number of observations. 9.3.1 Length of Class Intervals The uniform length of class intervals depends upon the difference between the extreme items in the data-the largest item and the smallest item-and the number of classes required. For example, if in the data on marks secured by 250 candidates in a subject at an examination, 0 and 93 are the lowest and highest marks respectively and 10 classes are required, each class would then have a class interval length of 10. Ordinarily class intervals are fixed in such a way as to enable easy calculation and precision. Range Using Sturges’ Rule, length of class interval (1) = 1 3.322 log N Where Range = Highest value - lowest value. 9.3.2 Class Limits The choice of class limits is determined by the mid-value of a class interval, which should as far as possible be identical with the arithmetic average of the items occurring in that class interval. The classification of a given data should be definite and unambiguous. 9.4 Methods of Classification Exclusive Method: According to the exclusive method all items, the values of which are less than five, equal to 0 or more than zero are classified in the class 0-5. All those quantities that are equal to 5 or less than 10 are classified in the class 5-10, and so on. CU IDOL SELF LEARNING MATERIAL (SLM)

180 Business Mathematics and Statistics Exclusive Method 0-5 5-10 10-15 15-20 20-25 25-30 Inclusive Method: Here, quantities that are equal to four, less than four, equal to or more than zero are classified in the class 0-4. All items that are equal to or more than five and less than or equal to nine are classified in the class 4-9, and so on. Inclusive Method 0-4 5-9 10-14 15-19 20-24 25-29 The exclusive method of classification can be made clearer if the classes are written as below: 0 and under 5 5 and under 10 10 and under 15 15 and under 20 20 and under 25 25 and under 30 This method of classification of numerical data in class intervals is explained in the following problem: 9.5 Problem The following are the marks secured by 50 candidates at an examination (out of 50 marks). Classify this data into intervals of 0-5, 5-10, 10-15 and so on, and calculate the number of items (marks) in each class. CU IDOL SELF LEARNING MATERIAL (SLM)

Classification and Tabulation of Data 181 15, 27, 21, 18, 21, 10, 7, 0, 8, 2 12, 16, 18, 28, 24, 23, 32, 20, 24, 16 10. 16, 1, 4, 11. 20, 15, 14, 25, 34 15, 29, 30, 22, 17, 13, 3, 17, 19, 14 11, 5, 19, 15, 8, 15, 19, IS, 5, 6 Solution: The least and the greatest items in the data are a and 34 respectively, therefore the classes range from 0-5 to 30-35. These classes are written in a score sheet which is usually called the tally sheet. Tally Sheet Class Score-marking (tallies) Total 0-5 IIII 5 15-10 IIII I 6 10-15 IIII III 8 15-20 IIII IIII IIII I 16 20-25 IIII III IR 25-30 IIII 14 30-35 III 3 Total 50 In the tally sheet, four vertical strokes and the diagonal stroke over them, for example IIII - indicate a group of five numbers. When the size of data small this tally sheet method is simple, but when dealing with complex and very extensive type of data, mechanical devices can be used for quick and precise classification. 9.6 Tabulation Tabulation is the process of arranging given quantitative data based on similarities and common characteristics in certain rows and columns so as to present the data vividly for quick intelligibility, easy comparability and visual appeal. 9.6.1 Components of a Statistical Table A statistical table comprises a title, a headnote, a stub head and stub details, captions “and columns under the captions, field of the table under different column heads, footnotes and source notes. CU IDOL SELF LEARNING MATERIAL (SLM)

182 Business Mathematics and Statistics Here’s a sample: Title: Students studying in different classes in X, Y, Z Commerce College. Head-Note: Data relates to the academic year 1998-99. Class Number of Students ... (Caution) Males Females ... (Column n heads) Pre-Degree Com. 354 212 F.Y. B.Com. 256 201 S.Y. B.Com. 204 193 T.Y. B.Com. 156 144 M.Com. 48* 12* Total 1,018 762 *These figures for the M.Com class Include both the junior M.Com. Students and as well as senior. Source: X, Y, Z Commerce College. 9.6.2 Purpose of Statistical Tables Statistical tables are of two types: general purpose table and special purpose. 9.6.3 General Purpose Table This is primarily meant to present the entire original data on a subject. Such presentation of numerical data in a tabular form is especially useful as a source of information and reference for constructing different special purpose tables. 9.6.4 Special Purpose Table As its name implies, this is a statistical table that specially presents and emphasizes certain phases or significant aspects of the information given in a general purpose table. Presenting data in a special table not only makes it easy to understand specific data, it also facilitates easy comparison and clear-cut interpretation. 3.6 Types of Tabulation (a) One-way Table (single tabulation) A one-way table gives answers to questions about one characteristic of the data. For instance, the following table is a one-way table. CU IDOL SELF LEARNING MATERIAL (SLM)

Classification and Tabulation of Data 183 One-way Table Population of Different States of the Indian Union State Population Total (b) Two-way Table (Double Tabulation) A two-way table gives information about two inter-related characteristics of a particular type of data. Two-way Table Population of different States of the Indian Union State Population Males Females Total Total (c) Three-way Table (Triple Tabulation) A three-way table answers questions relating to three inter-related characteristics of a given data. Population sex-wise and Literacy in Different States of the Indian Union State Population Males Females Total Literate Illiterate Total Literate Illiterate Total Literate Total Illiterate Total CU IDOL SELF LEARNING MATERIAL (SLM)

184 Business Mathematics and Statistics (d) Higher Order Table (Manifold Tabulation) This table gives information under several main heads and sub- heads on questions relating to a number of inter-related characteristics. 9.8 Rules and Precautions on Tabulation 1. Every statistical table should be given an appropriate title to indicate the nature of the data. The title should be simple, intelligible and unambiguous and should not be too lengthy or too short. 2. If necessary the title may be further explained with a suitable headnote. 3. Different types of data require different types of tabulation. It has to be decided at the outset whether one or more tables would be necessary to fit in the data precisely and suitably. A single simple table is appealing to the eye provided it is prepared properly. Several tables or a large table make comparisons difficult. 4. The stub heads and the main heads should be consistent with the nature of the data and be very clear. 5. The main headings under the caption should be as few as possible in keeping with the requirements of space and type of data. If the main headings are few, comparison between different sets of data becomes easy. 6. The entire arrangement of data should be appropriate, compact and self-explanatory so that it’s not necessary to re-arrange the data in any manner. 7. Comparisons between different figures — such as totals and averages are easier if they are arranged vertically and not horizontally. 8. In order to show important parts of the data (under main heads) distinctly, it is necessary to draw thick double or multiple ruled lines. 9. Depending upon the nature of the data, items in the stub column may be arranged according to: (i) Alphabetical order. (ii) Geographical importance. (iii) Customary classification. CU IDOL SELF LEARNING MATERIAL (SLM)

Classification and Tabulation of Data 185 (iv) Chronological order. (v) Size or magnitude. 10. Figures in the data that are estimates, approximate or revised should be indicated by an alphabet, asterisk, number or any other symbol. An explanation should be given in the footnote. 11. The different units used in the data should be indicated in the column heads. For example: ‘figures in rupees’, ‘figures in meters’, and so on. 12. The source of the data should be indicated under the foot- note. It is necessary to mention the source for further references and other details and also for indicating the reliability of the data. 9.9 Illustrative Examples Problem 1: The data given below relate to the heights and weights of 20 persons. Construct a two-way frequency table with class intervals 62-64 inches, 64-66 inches and so on and 115-125lb, 125-135lb, and so on. S. No. Height Weight S. No. Height Weight 1 70 170 11 70 163 67 139 2 65 135 12 63 122 68 134 3 65 136 13 67 140 69 132 4 64 137 14 66 120 68 148 5 69 148 15 67 129 67 152 6 63 124 16 7 65 117 17 8 70 128 18 9 71 143 19 10 62 129 20 CU IDOL SELF LEARNING MATERIAL (SLM)

186 Business Mathematics and Statistics Solution: Height Weight in lbs Total inches 115-125 125-135 135-145 145-155 155-165 165-175 62-64 II I 3 64-66 66-68 I III 4 68-70 70-72 I I II I 5 II II 4 II I I4 Total 4 5 6 3 1 1 20 Problem 2: Convert the following data into a frequency table. 10 students get less than 19 marks 25 Students get less than 20 marks 50 Students get less than 30 marks 78 Students get less than 40 marks 110 Students get less than 50 marks 130 Students get less than 60 marks 148 Students get less than 70 marks 160 Students get less than 80 marks 170 Students get less than 90 marks 175 Students get less than 100 marks Solution: The following is the two-way table showing the tally marks and the frequencies. 10-20 26 - 10 = 16 20-30 50 - 26 = 24 3040 78 - 50 = 28 40-50 110 - 78 = 32 50-60 130 - 110 = 20 60-70 148 - 130= 18 70-80 160 - 148 = 12 80-90 170 - 160 = 10 90-100 175 - 170 = 5 CU IDOL SELF LEARNING MATERIAL (SLM)

Classification and Tabulation of Data 187 Problem 3: Make a frequency distribution taking the variable as the number of letters in a word in the extract given below: “Mathematics renders its best service through the immediate furthering of rigorous thought and spirit of invention.” Solution: The numbers of letters in the different words are as follows: Mathematics (11) Furthering (10) Renders (7) Of (2) Its (3) Rigorous (8) Best (4) Thought (7) Service (7) And (3) Through (7) The (3) The (3) Spirit (6) Immediate (9) Of (2) No. of letters in a word Tallies No. of words (Variable) (Frequency) 2 II 2 3 IIII 4 4 1 5 I 0 6 1 7 I 4 8 IIII 1 9 2 10 I 1 11 II 1 I Total I 17 Problem 4: Draw up a table to show the following information: (a) Number of students according to sex in pre-degree, F.Y.B.Com., S.Y.B.Com, T.Y.B.Com. and M.Com. classes. (b) During the years 1990-91, 1991-92, 1992-93. Total number of students each year in each class. CU IDOL SELF LEARNING MATERIAL (SLM)

188 Business Mathematics and Statistics Solution: Students sex-wise in different classes during the years 1990-91 to 1992-93. Class 1990-91 1991-92 1992-93 Males Female Total Males Female Total Males Female Total F.Y.B.Com. S.Y.B.Com. T.Y. B.Com. M.Com. Total Problem 5: Present the following information in a tabular form and suggest a suitable title. The production of rice in Maharashtra in 1962-63 was 10.95 lakh tonnes, the lowest since 1955-56. In 1963-64 however it showed a spectacular recovery and reached 15.14 lakh tonnes. During 1963-64 wheat and bajra output decreased. Bajra production, which was. 5.5 lakh tonnes in 1962-63, declined to 4.51 lakh tonnes in 1963-64. The production of wheat also fell from 4.63 lakh tonnes in 1962-63 to 3.44 lakh tonnes in 1963- 64. The area under pulses showed a decreasing trend; production in 1963-64 was 22,000 tonnes less than figure of 8.89 lakh tonnes in 1962-63. Solution: Production of certain crops in Maharashtra State During the years 1962-63 and 1963-64. (Figures in lakh tonnes) Food Crop Year Rice 1962-63 1963-64 Wheat Bajra 10.95 15.14 Pulses 4.63 3.44 Total 5.50 4.51 8.89 8.67 29.97 31.76 Problem 6: The total number of students in a certain commerce college in 1999-00 was 1,800. The ratio of the number of, boys to girls was 5:1 in that year. The next year the number of boys decreased by 5% and the number of girls increased by 65% when compared to their respective strengths in 199. In the year 2001- 02 the total number of students was 2,100 and girls comprised 33% of this total. CU IDOL SELF LEARNING MATERIAL (SLM)

Classification and Tabulation of Data 189 Solution: Academic No. of No. of Total 1999-00 1500 300 1800 2000-01 1425 495 1920 2001-02 1407 693 2100 Table showing the number of boys and girls in a certain commerce college during the years 1999-00 to 2001-02. Problem 7: Prepare a neat table, paying proper attention to headings, double lines, spaces etc. showing fully the information in the following report as clearly as possible. “During the quinquennium 1935-39, there were in Great Britain 1,775 cases of industrial diseases made up of 677 cases of lead poisoning, 111 of other poisoning, 144 of anthrax, and 843 of gassing. The number of deaths reported was 20 per cent of the cases for all the four diseases taken together, that for lead poisoning was 135, for other poisoning 25, and that for anthrax was 30.” “During the next quinquennium 1940-44, the total number of cases reported was 2,807. But lead poisoning cases reported fell, by 351 and anthrax cases by 35. Other poisoning cases increased by 748 between the two periods. The number of deaths reported decreased by 45 for lead poisoning, but fell only by two for anthrax from the pre-war to the post-war quinquennium. In the later period, 52 deaths were reported for poisoning other than lead poisoning. The total number of deaths reported in 1940-44 including those from gassing was 64 greater than in 1935-39.” Solution: Table Showing the Number of Cases and Deaths due to Industrial Diseases in Great Britain Type of No. of 1935-1939 No. of 1940-1944 Industrial Cases No. of Percentage of Cases No. of Percentage of Deaths Deaths to Cases Deaths Deaths to Cases Disease Lead Poisoning 677 135 19.9 326 90 27.6 Other 111 25 22.5 859 52 6.1 Anthrax 144 ~O 20.8 109 28 19.6 1513 249 25.7 Gassing 843 165 1200 2807 119 16.49 Total 1775 355 14.9 Quinquenniums 1935-39 and 1940-44. CU IDOL SELF LEARNING MATERIAL (SLM)

190 Business Mathematics and Statistics Problem 8: Present the following information in a suitable table: The total rural population of India is 2,948 lakhs of which 2.404 lakhs belong to the agricultural classes. Of the total urban population of 618 lakhs, 531 lakhs belong to non-agricultural classes. Of the rural agricultural classes, 687 lakhs are self-supporting persons, 1,414 lakhs are non-earning dependents and 303 lakhs are earning dependents. The rural non- agricultural population comprises 544 lakhs of which 170 lakhs are self supporting persons, 326 lakhs are non-earning dependent and 48 lakhs are earning dependents. In urban agricultural classes 23 lakhs are self-supporting, 56 lakhs are non-earning dependents and 8 lakhs are earning dependents. The urban non-agricultural population comprises of 531 lakhs of persons of whom 153 lakhs are self- supporting persons, 347 lakhs are non-earning dependents and 31 lakhs are earning dependents. Solution: (Figures in lakhs) Table Showing India’s Population According to Living Area, Occupation and Earning Capacity Earning Agricultural Rural Total Agricultural Urban Grand Capacity classes classes Total Total Non- Non- Agricultural Agricultural classes classes Self 687 170 857 23 153 176 1033 supporting 1414 326 1740 56 347 403 2143 303 48 351 8 31 39 390 Non-earning 2404 544 2948 87 531 618 3566 Dependant Earning Dependents Grand Total Problem 9: Present the following information in a suitable tabular form “In 1971, the gross value of production of the world’s 3,600 million people was probably around 1,550 billion, making an average income of around 430 per head. About 400 billion of the 1,550 billion was produced by the 200 million people of the United States, giving them an average per capita income of 2,000 a year. About another 500 billion was produced by the 500 million people of the other rich countries of the free world (Western Europe, Japan and what used to be called the old British Dominions) giving them an average income of £ 1,000 a year. There is also a world middle class of over 600 million people with an average per head income of 750; they are led CU IDOL SELF LEARNING MATERIAL (SLM)

Classification and Tabulation of Data 191 in numbers by the Soviet Union and other Communist European countries, but they include some swiftly developing countries of Pacific Asia (e.g., Singapore, South Korea, Taiwan), some very developable countries of Southern Europe and Near East and North Africa littoral. This still leaves about 2,300 million of the world’s people who are miserably sharing the other 200 billion a year of gross world product, an average per head that is below 100 a year. Gross World Product Countries Population Total GNP GNP per head (million) (£ billion) (£) United States 200 400 2,000 500 500 1,000 “Other rich” (western Europe, Japan, ‘old’ dominions) 600 450 750 “Middle class” (Communist Europe, 2,300 200 85 Mediterranean, some Pacific-Asia, some 3600 1550 * Latin America) Poor Countries Total Average income per head 430 9.10 Summary Classification is the process of sorting out similar things from among a group of objects with different characteristics. In other words, heterogeneous data is divided into separate homogenous classes according to characteristics that exist among different individuals or quantities constituting the data. The main object of classification is to present vividly, in a simplified and quickly intelligible form, a mass complex data. Depending on the characteristics of the data, they can be broadly categorized into two separate and distinct groups — Descriptive and Numerical. Classification based on attributes maybe either simple or manifold. Phenomena like income, heights and weights are all quantitatively measurable and, data on them can be classified into separate class intervals of uniform length. CU IDOL SELF LEARNING MATERIAL (SLM)

192 Business Mathematics and Statistics Tabulation is the process of arranging given quantitative data based on similarities and common characteristics in certain rows and columns so as to present the data vividly for quick intelligibility, easy comparability and visual appeal. A statistical table comprises a title, a headnote, a stub head and stub details, captions and columns under the captions, field of the table under different column heads, footnotes and source notes. Statistical tables are of two types: General purpose table and Special purpose table. Different types of data require different types of tabulation. It has to be decided at the outset whether one or more tables would be necessary to fit in the data precisely and suitably. A single simple table is appealing to the eye provided it is prepared properly. Several tables or a large table make comparisons difficult. The entire arrangement of data should be appropriate, compact and self-explanatory so that it’s not necessary to re-arrange the data in any manner. Depending upon the nature of the data, items in the stub column may be arranged according to: (i) Alphabetical order (ii) Geographical importance (iii) Customary classification (iv) Chronological order (v) Size of magnitude The source of the data should be indicated under the footnote. It is necessary to mention the source for further references and other details and also for indicating the reliability of the data. 9.11 Key Words/Abbreviations Exclusive and Inclusive methods, single, double, triple, manifold, tabulation, alphabetical, customary and chronological. CU IDOL SELF LEARNING MATERIAL (SLM)

Classification and Tabulation of Data 193 9.12 Learning Activity 1. Tabulate the following information. “In 1990 the total number of readers was 46,000 and they borrowed some 16,000 volumes. In 2000 the number of books borrowed increased by 4,000 and borrowers by 50%, The classification was on the basis of three sections: literature, fiction and illustrated news. There were 10,000 and 30,000 readers in the literature and fiction sections respectively in the year 1990. In the same year 2,000 and 10,000 books were lent in the illustrated news and fiction sections respectively. Marked changes were seen in 2000. There were 7,000 and 42,000 readers in the literature and fiction sections respectively, while 4,000 and 13,000 books were lent in the illustrated news and fiction sections respectively.” 2. The city of Timbuctoo was divided into three areas: the administrative districts, other urban districts and the rural districts. A survey of housing conditions was carried out and the following information was gathered. There were 6,77,100 buildings of which 1,76,100 were in the: rural districts. Of the buildings in other urban districts 4,06,400 were inhabited and 4,500 were under construction. In the administrative districts 4 buildings were uninhabited and 500 were under construction of the total of 61,600. The total number of buildings in the city that were under construction were 62 and those uninhabited were 44,900. Tabulate the above information so as to give the maximum possible information. How many buildings were under construction in rural areas? 3. Tabulate the following: “In a trip organised by a college, there were 80 persons each of whom paid ` 15.50 on an average. There were 60 students each of whom paid ` 16, while members of the teaching staff were charged a higher rate. The number of servants was six (all male) and they were not charged anything. Ladies constituted 20% of the total, of which one was a lady staff member.” 4. Tabulate the following information: “In two towns Sand T the total population figures are 76,000 and 62,000 respectively, of which 60% in Sand 64% in T are males. Of the males in S, 60% are married, 30% CU IDOL SELF LEARNING MATERIAL (SLM)

194 Business Mathematics and Statistics unmarried and 10% are widowers. Of the males in T, the corresponding figures of married, unmarried and widowers are 50%, 25% and 25% respectively, in both towns the number of married women is exactly equal to the number of married men. Further the number of widows and unmarried females is the same in S and as well as in T.” 5. Draft a tabular form representing the information contained in the following extract relating to industrial accidents: “In 1929 there were 19,077 non-fatal compensated cases which might be classed as permanent injury, while there were 80,107 classed as temporary injury. For 1930 these figures were 22,434 and 86,106 respectively. For 1931 there were 21,761 and 70,0 respectively. The total accidents for the three years were 523,604, 471,510, 419,072 respectively. The total deaths (included in the former accident totals) were 2,093, 2,006, 1,793 respectively of which total compensation was paid to 1,217, 1,308 and 1,241 respectively. The number of total accidents for which compensation was paid was 100,401 in 1929, 109,848 in 1930 and 102,985 in 1931.” 6. Tabulate the following data: A survey was conducted amongst one lakh spectators visiting on a particular day cinema houses showing criminal, social, historical, comic and mythological films. The proportion of male to female spectators under survey was three to two. It indicated that while the respective percentages of spectators seeing criminal, social and historical films was sixteen, twenty-six and eighteen, the actual number of female viewers seeing these types was four thousand six hundred, twelve thousand two hundred, and seven thousand eight hundred respectively. The remaining two types of films namely comic and mythological were seen by forty percent and one percent of the male spectators. The number of female spectators seeing mythological films was four thousand four hundred. 7. Present the following data in tabular form: “The total strength of a college in 1960-61 was 1,100 students distributes as under: F.Y. Com Class-400: 1. Com. Class-300; Jr. B. Com. Class-200; Sr. B. Com. Class- 200. In 1961-62 there was a fall of 10% in the strength of each of the first three classes as compared with their strength in 1960-61 and the Sr. B. Com. Class had a strength of 200. In 1962-63, there was an increase of 10% in each of the four classes as compared CU IDOL SELF LEARNING MATERIAL (SLM)