Statistics (Questions-Answers) Eyer) celia Ce) B.L. AGARWAL Premier12‘Copyright © 2003 New Age International (P) Ltd., Publishers Second Edition : 2003 Reprint : 2005 NEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS. 4835/24, Ansari Road, Daryaganj, ‘New Delhi - 110 002 Visit us at : www.newagepublishers.com Offices at : Bangalore, Chennai, Cochin, Guwahati, Hyderabad, Jalandhar, Kolkata, Lucknow, Mumbai and Ranchi This book or any part thereof may not be reproduced in any form without tho written permission of the publisher. This book cannot be sold outside the country to which it is consigned by the publisher without the prior permission of the publisher. Rs. 275.00 ISBN : 81-224-1458-3 . 345678910 Published by New Ago International (P) Ltd., 4835/24, Ansari Road, Daryaganj, New Delhi-110 002 and printed in India at Taj Press, New Delhi-110 064 typesetted at Pagetek, Delhi-110 092.Contents Preface to the Second Editian Preface ta the First Edition Chapter 1 Chapter 2 Statistics Short essay type questions on: Definitions of = s istics Statistical prospective 2 Divi oT 2 Funeti f statist Methods of collection of data _3 Types of statistical data 3 Statistical survey 3 Questionnaire and schedule 3 Kinds of statistical investigation 4 Enrors_4 Laws _¢ Approximation of values 5 Primary and secondary data_ 5 Editing of data_ 5 Bilin the blanks 6 Multiple choice questions 7 Answers JO Suggested reading Classification, Tabulation and Frequency Distribution vii 12 Short essay type questions on: Classification 12 Grouping of data_/3 Grouping error 13 Tabulation _/4 Frequency J/5Chapter 3 ‘Chapter 4 Cumulative frequency 15 ‘Types of series 15 Stem and leaf display of data_15 Fill in the blanks 15 Answers 20 Suggested reading 20 Diagramatic and Graphical Representation Short essay type questions on: Advantages of diagrams _2/ Bar diagram 2/ Sub-divided bar diagram 27 Multiple bar diagram 22 Deviation bar diagram _22 Duo-directional bar diagram 22 Paired bar diagram 23 ‘Sliding bar diagram 23 Broken bar diagram 23 Line diagram 24 Pic chart 24 Histogram 25 Frequency polygon _25 Frequency curve 25 Semilogarithmic graph 26 Ogive curve 26 Lorenz curve 27 Pictograms 27 Column chant with circular base 28 Step bar chart 28 Overlapping bar chart__28 Fill in the blanks 29 Maltiple choice questions 30 Answers 35 Suggested reading 35 Measures of Central Tendency Concept and definitions 36 Properties of central tendency 36 Classification of averages 36 Functions 26 Limitations 37 Arithmetic mean _37 Weighted mean 38 Moving average 38 Progressive average 38 CONTENTSCONTENTS Chapter 5 Chapter 6 ‘Composite average 38 Pooled mean 39 Harmonic mean 40 Relation between A.M., G.M. and H.M. 4! Quadratic mean__ 47 Median 42 Mode 42 Quartiles 43 Deciles 43 Percentiles 43 EilLin the blanks 44 Multiple choice questions 47 Answers $4 Suggested reading 59 Measures of Dispersion, Skewness and Kurtosis ‘Concept and statements 60 Requisites of a good measure 6 Uses 61 Range 6) Coefficient of range 61 Interquartile range 6 Percentile range 6/ Quartile deviation 6 Empirical relation between $.D.. Q.D. and M.D._ 65 Skewness 66 Kurtosis 67 Sheppard's corrections _68 Ellin the banks 69 Multiple choice questions 72 Answers 87 Suggested reading 87 Elementary Probability Short essay type questions on: Concept 83 83‘Chapter 7 CONTENTS Events 84 complementary 83 simple 84 equal 84 compound 84 mutually exclusive 84 primary 4 derived 84 Unision and intersection of cvenis 84 Impossible event 85 Exhaustive 85 Pairwise independent events 85 Definitions of probability: classical 85 statistical _86 Additive law of probability 86 Conditional probability 86 Bayes" probability 87 D’ Alembert’s paradox _87 Fillin the blanks 88 Multiple choice questions 90 Answers /OJ Suggested reading /05 iable, Mathematical Expectation and Probability Distributions a Random variable 06 discrete 102 continuous 107 Distribution function 107, Cumulative distribution function 107 Probability mass function _/07 Step function _J07 Probability density function 107 Probability density curve _/07 Mathematical expectation _/08 Cauchy-Schwartz inequality /09 109 Gurland’s inequality on expectation 109 Moment generating function 109 Characteristic function 109 Probability generating function 110 uniform__2/0‘CONTENTS ‘Chapter 8 Bernoulli //0 Binomial //0 Poisson H12 Negative 113 Geometric {14 Polya's JIS Hypergeometric 15 —— Continuous distributions: uniform or rectangular 1/7 exponential 1/8 Laplace or double exponential 1/9 normal //9 lognormal 123 Cauchy 123 beta distributions of Tand [kind /23 gamma 125 logistic 126 Pareto 426 Weibull 126 Circular 126 Pearsonian 1/26 Sampling distributions: Chi-square 127 Student’s-t 128 Pearson’s Chi-square 128 Chi 128 Fisher's-2 J3J Snedecor’s-F 13) Fill in the blanks _/32 Multiple choice questions 138 Answers 160 Suggested reading 162 Bivariate Random Variable and Distributions Short essay type questions on: Bivari jahle: 163 discrete 163 continuous J65 int distribution funetis 6 Joint probability mass function /64 Joint probability density function /64 Marginal probability functions 164 Conditional variable 165 Conditional probability density function 165 Independence of random variables /65 163xiv Chapter 9 CONTENTS jonal expectation 165 ‘Conditional variance /66. Conditional covariance 167, Joint moment generating function 167 Joint raw moments _/67 Joint central moments_/67 Fill in the banks 172 Multiple choice questions 174 Answers 182 Suggested reading /83 Sampling Methods 184 Short essay type questions on: Sampling vs. Complete enumeration _/84 Population /84 finite 185 infinite 185 real J&S hypothetical /85 Sampling frame /85 Random sampling /85 Simple random sampling 486 Principles of sampling methods 186 Sampling and non-sampling errors 186 Purposive sampling 186 Parameter {87 a Simple random sampling with and without replacement_ /87 Sample size /87 Sample mean 188 Sample variance _/88 Standard error _J59 Finite population correction _J89 Sampling fraction _/89 Stratified random sampling 1/90 Exigen oumber of strata /9/ sample size J9/ sampling procedure [9 Allocation: equal /9F proportional /9/‘Chapter 10 optimum 19) Neyman 197 i fi {vari 192 ‘Two way stratification _/93 Controlled selection _/97 Systematic sampling 193 linear [94 circular 194 Eormulac for mean and variance _/94 Double ing 195 Cluster or area sampling _/95, Multistage sampling 1/96 ‘Two stage sampling /96 Inverse sampling 196 Sampling with probability proportional to size 198 Non-response _/98 Multiple chaice questions 202 Answers 2/2 Suggested reading 213 Theory of Estimation 214 Short essay type questions on: - Parameter 2/4 Estimate 2/4 Estimator 2/4 Properties of estimators 2/5 consistency 2/5 unbiasedness 215 mean squared error 2/5 mean squared consistency 2/6 Crammer-Rao inequality 2/6 Best asymptotically normal estimator _ 2/7 Efficiency 2/7 a 7 - nT Sufficiency 2/7 Fisher-Neyman factorization theorem 2/8 Completeness 2/8 Rao-Blackwell thearem 2/3 Invariance property of estimators 2/9 Admissibility of an estimator 2/9 Mathods of estimation: slihood ion 2 least square estimation 2/9 tncthod of moments 2/9 Best is hi Urheberrechtlich geschitztes Material‘Chapter 11 Minimum variance quadratic estimator 224 Gauss-Markov theorem _224 Method of Minimum Chi-squares 225 Bayes’ estimators 227 Mini - 27 Confidence region 237 One sided confidence interval 23 Fill 232 Multiple choice questions 236 Answers 247 Suggested reading 248 Testing Parametric Hypotheses Short essay type question on: Hypothesis 249 null alternative 250 simple 250 composite 250 ‘Types of errors: type Terror 250 type Herror 250 Power of atest 250 Level of significance 250 Size of atest 25/ Best critical region 257 One and two tailed test 257 Nonrandomized and randomized test 251 Degrees of freedom 251 Critical function 257 Optimum test 252 Most powerful test 252 Neyman-Pearson lemma 252 Minimax test 254 Unbiased test 254 Uniformly most powerful unbiased test 255 Admissible west 255 Likelihood ratio test 256CONTENTS Chapter 12 xvii Student's rest 257 Cochran's approximate t-test 258 Paired rest 260 Z-test for properties iation_258 Chi-square test in multinomial distribution 264 Contingency table 265 Incomplete contingency table _ 265 Structural and random zeros 265, Explanatory variable 266 ‘Test of independence of attributes _266 Yates’ correction _ 266 Fisher's exact test 268 Coefficient of contingency 268 Bartletts’ test 269 Fatest 269 Test of equality af wo population variances 269 test of equality of several population means 269 Analysis of variance 270 Bayes’ test 270 Sequential probability ratio test 270 Fill in the blanks | 27/ Multiple choice questions _277 Answers 289 Suggested reading 290 Nonparametric Statistical Methods 292 Short essay type questions on: General discussion 292 Asymptotic relative efficiency 293 Power efficiency 293 Tied observations 293 Kolmogorov-Smimov test for one sample _295 ordinary sign test_ 295 Wilcoxon's signed rank test_296 Run test 297 Kolmogorov-Smirnov two sample test 298 Wilcoxon's test for matched pairs 298 Median test 298 Wald-Wolfowitz run test 307 Mann-Whitney U-test JO/ Mcnemar’s test 303 Mood's test 304xviii ‘Chapter 13 CONTENTS Moses’ test. 305 Cochran's Q-test 306 Kruskal-Wallis one way analysis 307 Friedman's two way analysis of variance 309 Multiple comparisons in Friedmans’ test 309 Jonckheere-Terpstra test 3/0 Page’s test 321 Spearman's rank correlation 3/2 Kendall's rank correlation 3/2 Coefficient of concordance 3/4 Nonparametric approach in regression analysis 316 Brown and Mood’s test 3/7 Confidence band 3/9 Fill in the blanks 3/9 Multiple choice questions 323 Answers 335 Suggested reading 336 Regression and Correlation Methods 338 Short essay type questions on; General discussion 338 Regression model 338 Scatter diagram 339 Estimation of parameters of a regreassion linc 340 Properties of regression coefficients 340 Two regression lines: point of iniersection 341 angle between two regression lines 34/ Coding 342 ‘Test of significance of regression coefficient 342 ‘Test of significance of the intercept 342 Regression function 343 confidence limits of Band a 343 Test of linearity of regression 344 ‘Weighted regression 345 ‘Curvilinear regression 346 ing of orthogonal polynomial 347 “ Correlation 347CONTENTS Chapter 14 Properties of correlation coefficient_ 350 Probable error 351 Confidence limits for P_352 ‘Test of significance of: come Miclent 252 aspecified value of P_ 352 Equality of two corr. cocffs. 352 Equality of more than two corr cocffs 352 Coefficient af concurrent deviation 353 Correlation coefficient using variance of the difference 355, Correlation cocfficient by the method of Icast squares 356 Correlation ratio 357 Intraclass correlation 357 Biserial correlation 358 Tetrachoric correlation 358 Multiple linear regression 359 ‘Test of significance of partial regression coefficient(s) 361 Regression in trivariate population 363 Partial correlation 364 Pari comelation 265 Spurious correlation 366 Fill in the blanks 366 Multiple choice questions 372 Answers 387 Suggested reading 388 Measures of Association of Attributes Short essay type questions on: General discussion and notations 390 Order of a class 390 Class frequency 392 Inconsistency of data 392 Kinds of association 392 Methods of measuring association: Proportion methad 393 Method of probability 393 Yule's coefficient of association 394 Coefficient of colligation 394 Partial association 394 Ilusory association 394 Coefficient of contingency 395 Tschuprow’s cocfficient 395 Fill in the banks 395 390Chapter 15- Chapter 16 Muhiple choice questions 393 Answers 404 Suggested reading 404 Interpolation and Extrapolation 405 Short essay type questions on: a ii 7 0S Assumptions and uses 405 Interpolation methods: graphical method 406 binomial expansion method 406 parabolic curve method 406 Finite differences 408 Divided differences 408 Diagonal difference table 408 ‘Central difference table 408 Divided difference table 409 Newton's formula of advancing differences 4/0 Newton's backward formula 41] Newton-Gauss forward formula 4// Newton-Gauss backward formula 4#// Newton's method of backward differences 412 Newton's method of divided differences #/2 Lagrange's interpolation formula 4/2 Inverse interpolation: Lagrange’s interpolation formula 4/2 Central interpolation 4/3 Sterling's formula 4/3 Bessel's formula 4/4 Inverse interpolation: Lagrange’s method 415 Iterative method 4/5 Successive approximation method 4/5 Fill in the blanks 4/6 Multiple choice questions 478 Answers 424 Suggested reading 424 Time Series Analysis - 425 Short essay type questions on: Defini F Editing of data_426 Secular trend 426 Segsonal variation 427 Cyclic variation 427 Irregular variations 427CONTENTS Chapter 17 ; Methods of measuring linear trend: graphic method 428 semi-average method 428 Moving average method 428 Least square method 429 Shift of origin 430 Curvilinear end 434 Methods of measuring seasonal variations: simple average method 433 ratio to trend method 433 ratio to moving average method 433 specific and typical seasonals 434 link relative method 434 Methods of measuring cyclic variations: residual method 435 first difference method 435 percentage ratio method 435 direct method 435 reference cycle analysis 435 harmonic analysis 435 Measurement of irregular variations 436 Fill in the blanks 437 Multiple choice questions 440 Answers 446 Suggested reading 447 Index Numbers Short essay type questions on: Definitions 448 Uses 449 Limitations and lecunae 449 Price index numbers 450 Value and diffusion index 450 Problems involved in the construction of index numbers 450 Laspeyre’s index number 45/ Paasche's index number 452 Drobish-Bowley index number 452 Walsh (Fisher's ideal) index number 452 Marshall and Edgeworth index number 452° Kelly's fixed weight formula 453 Eonmula error 453 Homogeneity error 454 Time reversal test 454 Factor reversal test 454 Circular test_455,Chapter 18 Chapter 19 Chain base method 455 Base shifting 456 Splicing 457 Consumer price index 459 Explicit and implicit weights_ 467 ‘Whole-sale price index 46 Index of industrial production 462 Gross national product_ 462 Eillin the blanks 464 Multiple choice questions 467 Answers 474 Suggested reading 475 Business Forecasting Short essay type questions on: ‘Definitions 426 Historical analysis 476, Analysis of current economic conditions 477 Methods of forecasting: Naive method 477 specific historical analogy method 477 lead-lag relationship 477 Diffusion index 478 action-reaction theory 478 factor listing method 479 cross-cut analysis theory 479 opinion polling 480 exponential smoothing 480 econometric method 48 Forecasting agencies 48! Fill in the blanks 482 Multiple choice questions 483 Answers 486 Suggested reading 487 Statistical Techniques in Quality Control Short essay type questions on: Chance factor 488 Assignable causes 488 Shewhart control charts: X-charr 490 o-chan 490 R=chan 497 Control charts for fraction defectives (p-chart)_ 495 ‘ontral s for fects (c= 496, 476 488CONTENTS Chapter 20 Advantages of statistical quality control 499 —— Specification limits _ 499 Acceptance sampling plan 500 Inspection by attributes 50/ Inspection by variables 50/ Definitions and explanations for: major defect 50f minor defect S507 P * tisk 50 ci "s tisk 502 Acceptance quality level_502 Lot tolerance percentage defective 502 Average outgoing quality 502 Blind sampling 502 Average sample number and curve 502 Operating characteristic function and curve 503 Single sampling inspection plan 503 Dodge and Roming OC curves 504 Double sampling inspection plan 505 Sequential sampling plan 505 Sequential probability ratio test 505 Eill_in the blanks 508 Multiple choice questions SJ? Answers $17 Suggested reading 5/8 Vital and Population Statistics Short essay type questions on: Uses_$20 Collection of vi istics 520 registration method 52 census enumeration method 522 survey method 522 sampling registration system 522 Formulae for estimation of population 523 Mital rates $24 Crude death rate "52.5 Specific death rate 525 Standardised death rate 525 Crude birth rate 527 Age specific fertility rate 528 General marital fertility rate 528 ood 519‘Chapter 21 ‘CONTENTS Age specific marital fertility rate 529 Total marital fertility rate $29 Total fertility rate 529 Measures of population growth: crude rate of natural increase 529 vital index 529 gross reproduction rate 530 net reproduction rate 53/ Replacement index 533 Life table 533 Construction of life table 534 Unemployment rate 535 ‘Stable and stationary population 535 Abridged life table 536 Central mortality rate 537 Force of mortality 537 Fillin the blanks 539 Multiple choice questions 542 Answers _349 Suggested reading 549 Basic Experimental Designs 550 Short essay type questions on; Definitions: experimental unit 550 iteatment 550 uirements of a good design 550 ization 55 lication 557 Iocal control S5¢ Experimental error _552 Determination of number of replications 552 Size and shape of experimental units 553 Relative efficiency of designs 554 Analysis of variance 554 Statistical models for experimental designs 555 ‘Tests for ordered treatment means: least significant difference 557 Student-Newman Kuels test_ 557 Duncan's multiple range test_ 558 Tukeys’ test 558 Contrasts (comparisons) 558 Pairwise and nonpairwise contrasts 559 Priori and posteriori contrasts 559 Systematic designs 562 Randomized designs 562‘CONTENTS xxv Completely randomized design 562 Nested or hierarchical classification 567 Randomized block design (RBD) 567 Missing plot technique in RBD_57/ Preliminary test of significance (PTS) 573 Latin square design (LSD) 574 Mutually orthogonal Latin squares (MOLS) 576 Greaco-Latin square 576 Missing plot technique in LSD 577 Cross over design 580 Factorial experiments 58/ Main effect $82 Interaction 582 Methods of estimation and analysis through 583 contrasts S87 modulo technique 584 cross tables 585 Yates’ method 586 Expected mean squares and their role__590 Complete confounding 59/ Partial confounding 59? Confounding through: contrasts 592 modulo technique $92 Confounding in asymmetrical factorial experiments 593 Fractional replication 595 Resolution of fractional replication 596 Split plot design 596 Split-split plot design 602 Split block or strip plot design 604 Confounding in split plot design 605 Eillin the blanks 605 Multiple choice questions 642 Answers 627. Suggested reading 629.Urheberrechtlich geschitztes MaterialChapter 1 Statistics in General SECTION-A Short Essay Type Questions Definc statistics as given be Sir R.A. Fisher. Ans, Sir R.A. Fisher defined statistics as, “The science of statistics is essentially a branch of applied mathematics and may be regarded as mathematics applied to observational data.” Q.2 Give the definitions of statistics given by ALL. Bowley, Lovitt, W.A. Wallis and H.V. Roberts and CH, Meyers. Ans, The definitions given by statisti in the question are quoted below: A.L, Bowley: (@) Statistics is the device for abbreviating and classifying the statements and making clear the relations. jans named ics is the science of measurement of social phenomenon regarded as a whole in all its manifestations. (iy Statistics is the numerical statement of facts in any department of enquiry, placed in relation to cach other. ii) Lovitt: Statistics is the science which deals with the collecting, classifying, presenting, comparing and interpreting numerical data collected to throw light on any sphere of enquiry. W.A. Wallis and H.V. Roberts: Statistics is not a body of substantive knowledge, but a body of methods obtaining knowledge. Cecil H. Meyers: Statistics may be defined as a science of numerical information which employs the processes of measurement and collection, classi- fication, analysis, decision-making and communi- cation of results in a manner understandable and verifiable by other. - Q.3. Give the statements given by A.L. Bowley and William A, Spurr and Charles P. Bonini. Ans, A.L. Bowley: Great numbersare not counted correctly to a unit, they are estimated. William A, Spurr and Charles P. Bonini: Not all numbers are statistical, Logarithms, for instance, are mere abstract numbers. Statistical data are concrete numbers, which represent objects. Q. 4. Quote statements about statistics made by A.B. Waugh, A.L. Boddington, Whipple, Tippet, W.1. King, Marshall, Yule and Kendall, R.A. Fisher, A.M. Mood, Disraeli and Darrel Huf. A.E, Waugh: The purpose of statistical methods is to simplify great bodies of numerical data. ALL. Boddington: The essence of statistics is not mere counting but comparison.2 ‘Whipple: Statistics enables one to enlarge his Horizon. L.H.C. Tippet: (i) Planning is the order of the day and without statistics planning is inconceivable. (ii) Statistics is both a science and an art. W.L. King: (i) The science of statistics is a most useful servant, but only of great valuc to those who understand its proper use. (ii) The science of statistics is the method of judging collective, natural or social pheno- mena from the results obtained by the analysis of enumeration or collection of estimates. Marshall: Statistics are the straw out of which I like every other economist have to make bricks. Yule and Kendall: Statistics is nota science, itis a scientific method. R.A. Fisher: Statistics isa branch of applied mathe- matics which specialises in data. A.M. Mood: Statistics provides tools and techniques for research workers. Disraeli; There are three kinds of lies: lies, damned lies and statistics. Darrel Hef: A well wrapped statistics is better than Hitlers biglie it misleads, yet it cannot be pinned on you. Q.5. Which definition of statistics is considered to be the best. Ans, The definition of statistics given by R.A. Fisher is considered to be the best and most exact. Q.6. Give in afew words the statistical perspective. Ans, Statistical perspective is the invaluable compendium which gather all the facts, figures, objective survey and fascinating remembrances to a assimilable record. Q7. Ans. Following are the main divisions of statistics: (i) Mathematical or theoretical statistics: It covers development of statistical distributions, experimental designs, sampling designs, etc. Mention main divisions of statistics. PROGRAMMED STATISTICS (ii) Statistical methods or functions: It covers collection, tabulation, analysis and inter- pretation of data, etc, (iii) Descriptive statistics: Classification and diagramatic representation of data, (iv) Inferential staristics: To draw conclusion about population on the basis of sample drawn from it, - (v) Applied statistics: It mainly covers popu- lation, census, national income, production, business statistics, industrial statistics, quality control, biostatistics, ete. Q. 8. What are the limitations of statistics? Ans. Broadly the limitations of statistics are: {i) Statistics deals with quantitative data only. ‘Even qualitative information is converted into numerical data by the method of ranking, ‘scoring or scaling. (ii) Statistics is true on an average only. (iii) Statistics deals with the masses, not an individual. No statistics is applicable for a single observation. Statistical results are correct in a general sense. They are always subject to certain amount of error. (iv) (¥) Statistics is only a means to draw conclusions about masses or population but not a panacea to all sort of problems. Statistics can be misused in many ways. Q.9 What are different types of investigations? Ans. There are two types of investigations, namely: (i) investigation through census method Gi) investigation through sample methods. Q.10 What does census method imply? Ans. Census method means to include each and every unit or object of the population under reference for enquiry or observation. For example, to know the ional income, we have to include every individual or unit which contributes towards the national income. Q.11 What is meant by investig: sample method? (wi) ion through‘STATISTICS IN GENERAL Ans. In sample method, an investigator has to select some units from the population about which conclusions have to be drawn and take observations ‘on the selected units. The results obtained from sample values are applicable to the population as a whole. For instance, to know the average age at marriage, an investigator selects an adequate number of married couples and arrive at an average age of marriage which is considered to be the average age of marriage for the whole population. Q.12 Ans, @ (ii) (ii) (iv) Qi Ans, data: @ G@ (iii) (iv) (vy) (vi) (vii) Q14 ‘What are four main functions of statistics? Four functions of statistics are: Collection of data; Presentation of data; Analysis of data; and Interpretation of results. Give different methods of collection of data. Following are the methods of collection of Direct personal enquiry method; Indirect oral investigation; By filling of schedules: By mailed questionnaires; Information from local agents and corres- pondents; By old records; and By direct observational methed. Name two kinds of statistical data and describe them in brief, Ans. (i) Gi) Qs ‘Two kinds of statistical data are: Primary data: Primary data are those which are collected from the units or individuals directly and these data have never been used for any purpose earlier. Secondary data: The data, which had been collected by some individual or agency and statistically treated to draw certain conclu- sions. Again the same data are used and analysed to extract some other information, are termed as secondary data. What are the requisites of a reliable data? Ans, @ (iy Giiy (iv) Q. 16 ‘The requisites of a reliable data are: Tt should be complete; It should be consistent; It should be accurate; and It should be homogencaus in respect of unit of information. ‘What precautions should be taken in the planning of a statistical survey? Ans. Following precautions are to be taken in the planning of a survey: @ i) (iii (iv) Q17 Purpose: First a clear-cut abjective of the survey should be spelled out. Scope of survey: Different aspects to be covered to achieve the fixed objectives should clearly be explained. Definition af terms: All the terms involved in a survey should be defined without ambiguity so that no unit is likely to fall in more than ‘one calcgory. Siating the hypothesis: Hypothesis to be tested from the data collected by way of survey should be laid down in accordance with the objectives of the survey. Give briefly the characteristics of a good questionnaire or a schedule. Ans. Characteristics of a good questionnaire or a schedule are: @ ai) Giiy tiv) w) (ip Q.18 Ans. Number of questions should be such that it extracts all information required for the report. Each question should have almost all alter- native answers. The question should be clear and without any ambiguity. All questions should be mutually exclusive in nature. ‘Some very personal questions be avoided. ‘Questionnaire or schedule should not be very lengthy and time-consuming, Name five fields where statistics is inevitable. Broadly five fields where statistics is inevi- table can be named as follows:(i) Scientific research; (ii), Economic analysis; Gii) Planning: (iv) Business and commerce; and (v) Forecasting and projection. Q. 19 Mention different kinds of statistical investi- gations. Ans. Different kinds of statistical investigations are: w (ii) (ii) (iv) (v) (wi) (vii) Surveys or experiments; Surveys through census or sample enquiry; Confidential or open enquiry; Direct of indirect enquiry; Original or repetitive enquiry; Regular or ad hoe enquiry; and Limited or extensive enquiry. Q. 20 What is an absolute biased error? Ans. When the figures are rounded straight way to the nearest lowest unit of rounding or to the nearest highest unit of rounding, the difference between actual and estimated (rounded) values in the two cases are called biased errors. For cxampie, if we round the value 357 to the nearest 100, the nearest lower value is 300 and nearest higher value is 400. In case I, Absolute biased error = 357 - 300 = 57. In case I, Absolute biased error = 357 — 400 = - 43. If there are two or more values in a set, the sum of absolute errors is taken, Q. 21 What is an absolute unbiased error? If the values are rounded as per the rules of rounding, ie., a given value is rounded to nearest lower value of the unit of rounding in case it is less than half of the unit of rounding and to next higher value of the unit of rounding if it is more than half of the unit of rounding, it is called unbiased error. The difference between the actual value and the estimated (rounded) value is called absolute unbiased error. If, there are two or more values in a set, then the sum of the absolute unbiased error is taken. Q.22 How do you estimate an average unbiased absolute error (AE). PROGRAMMED STATISTICS Ans. The Formula is, Average A.B. (unbiased) = Average error x Jn where, n= number of items in the set and Average error = The mean of the minimum and maximum values which are likely to be left over or increased in the process of rounding. For example, if we arc rounding a value to the nearest 100, the chances are that the lowest value which may be left out is 0 and maximum value which may be added is 50, Hence, the average error is (0 + 50)/2 = 25, Q. 23 Enunciate the law of statistical regularity. Ans, The law states that a reasonably large number of items selected at random from a large group of items will, on the average, be representative of the large group or population. This law is governed by the theory of probability. Q. 24 State the law of inertia of large numbers, Ans, The law of inertia states that the large aggre- gates are more stable than small ones. According to Professor A.L. Bowley, great numbers and averages resulting from them, such as we always obtain measuring social phenomena have a great inertia. Q.25 What is the law of persistence of small number? Ans. The law of persistence of small numbers states that the ratio of the small number of items having some distinguished characteristics to the total number of units in the population remains constant even through the population size is immensely increased. Q.26 Give an example of the law of persistence of small numbers. Ans, Suppose a school admits all good students. Even then some students will be of poor intelligence. ‘The ratio of such students to the total number of students will remain the same even if the number of students in the schoo! is doubled, trebled, etc. Q. 27 State the law of decreasing variation? Ans. Law of decreasing variation indicates that the variation in a sample tends to reduce as the sample size increases. Q. 28. How is the law of decreasing variation helpful in sample surveys?STATISTICS IN GENERAL Ans. Tt is the law of decreasing variation which puts an investigator on sound footing to decide about ‘the adequate sample size which is a true represen- tative of the population. Q. 29 What db you understand by approximation of values or figures? Ans. Toexpress a value or figure to a round figure which is easy to write and understand is called approximation, This is mostly done from convenience point of view. It helps in comparison of values tremendously. Q. 30. Give different methods of approximation with a brief description. Ans. Different methods of approximation are: (i) By adding figure; In this methods the given value of figure is always increased to next higher value of unit of rounding. For instance, a value 21, 357.4 will be approximated to 21,358 up to unit place, 21, 360 up to tenth place, 21,400 up to hundredth place and 22,000 up to thousandth place. By discarding figures. It is a process just reverse to the adding figures. In this method, the given value is decreased to next lower value of unit of rounding. For example, the value 21,3574 is approximated to 21,357 up to unitplace, 21,350 up to tenth place, 21,300 up to hundredth place and 21,000 up to thousandth place. (iii) Approximation to the whole number: This method is also known as rounding of figures and is an usually accepted method. This method is the best one as it minimises the error of approximation. In this method, a value is raised to the next higher value of the unit of rounding if it is more than half of the unit of rounding and is left over if it is less than half of the unit of rounding. (ii) Q.31 What are different sources of statistical errors? 5 Ans. errors: Following are the four sources of statistical Errors of origin; Errors of inadequacy; Errors of manipulation; and Envors of interpretation. Explain briefly the possible errer. Ans. In rounding of valucs to the nearest of unils, tens or hundreds or thousands, etc., a value less than half of the unit of rounding is left over and greater than half of the unit of rounding is increased to the next higher anit. Thus, the possible error is = 1/2 x unit of rounding. For example, if the number 23 is rounded to the nearest ten, its value is 20 and possible error is + 5. Hence the value will lie between 20 + 5, ie., between 15 and 25. Q.33 What are different sources of primary data? (iv) Q.32 Ans, Data obtained from original experiments or surveys, ie, the data collected by investigators or enumerators is known as primary data. Also the census data, data released in the Reserve Bank of India bulletins, data published by other authorities in original form are considered as primary data. Q. 34 What are different sources of secondary data? ‘Ans. Published thesis, research papers, project reports, summarised census report, monthly abstracts of CSO and different publications of wade and commerce associations, etc., are the various sources of secondary data. Q.35 What kind of deficiencies of data are checked through editing? Ans, Data are edited to remove mainly four deficiencies which are: (i) Completeness of data; Consistency of data; i) Accuracy of data; and (iv) Homogencity of data.SECTION-B PROGRAMMED STATISTICS Fill in the Blanks Fill in the suitable word(s) or phrase(s) in the blanks: 1. The statement, “Not all numbers are statistical, logarithms for instance, are merely abstract numbers. Statistical data are concrete numbers, which represents objects", was given by and 2. “Great numbers are not counted correctly to a unit, they are estimated”, is the statement of - 3. The defini “Statistics is the science which deals with the collection, classification and tabulation of numerical facts as the basis for explanation, description and comparison of phenomena”, was given by wise decisions in the face of uncertainty” is the definition of statistics given by and 5. St is both, a and an 6. The credit of the statement, “A statistician is a practitioner of the art and science of statistics” goes to 7. “The purpose of statistical methods is to simplify great bodies of numerical data” is the statement given by 8. The definition, “The essence of statistics is not mere counting but comparison”, was given by . 9. The statement, “Statistics enables one to enlarge his horizon”, goes in the name of 10. The statement, “Planning is the order of the day and without statistics planoing i is inconceivable” was given by 1. The author ef the statement, “The science of statistics is a most useful servant, but only of great valuc to those who understand ils proper use", was due to 12. 13. 14. 15. 16. 17. 18, 1, 25. Statistics can prove Use of statistical methods is most st dangerous in the hands of ‘The statement, “On average a a factory labour has become younger in 1991 as compared to 1981", is Statistics deals with only Statistical analysis helps in the of results. Statistics is not applicable 10 observation. Not a matter of statistics. but data are the subject- Statistics are numerical of facts, but all numerical statements are not By statistics we mean quantitative data affected to a marked extent by of causes, (Fle & Kendall) Statistics does not study . Statistics is not a science, its a Statistics are the straw out of which T like every ather economist, have to make (Marshall) the arithmetic of human Statistics Planning on. the basis of inadequate and inaccurate statistics is than no planning at all. (Third-Five-Year Plan, Planning Commission). Statistics is liable to be The data collected from pul known data. Data obtained by conducting a survey is called data. Before analysis, hed reports is the data should beSTATISTICS IN GENERAL 3. uM. 32. Qt units are better than arbitrary units. are used in a mailed enquiry method, Mailed enquiry method cannot by adopted if the respondents arc In personal enquiry method, the response is better than method. Pretesting is essential for preparing a good or The statement, “Statistics is both a science and an an", was given by: (a) R.A. Fisher (b) Tippet () LR, Connor (d) AL. Bowley 47, 7 Assigning number digits to various responses whether quantitative or qualitative is called A figure 16.31 place is. ‘The figure 32,627 rounded to the nearest hundredth place is The figure 32,627 approximated to the thousandth place by the method of discarding figure is 7 rounded to the nearest tenth 48. ‘The figure 45,067 approximated to thousandth 35. Population figures published by the Census Commissione ue. Me place by the method of adding figure is 36. Mistakes and statistical errors are 49. The figure 13.85 rounded to one decimal 37. If a quantity is such that all errors tend to be place is in the same direction, they are called 5 The figure 13.75 rounded to one decimal —_—_ ames place is - 38, The errors caused by the carelessness af the = 54, Government cannot do proper plaoning investigators are called errors, without the help of 39. Formula for the estimation of biased absolite 55 Qbeeryations collected through “sor error 1s : experiments are classified as 40. Formula for the estimation of unbiased 3. Ty know the area under cultivation of absolute error is wheat, the appropriate type of investigation 41, Biased relative error can be estimated by the is . formuta 54, To know the average yield of a crop, an 42. Unbiased relative error can be estimated by appropriate investigation type will be the formula . 43. A survey in which information is collected 55, The compendium which gathers all the facts from each and every individual of the and fascinating memories in a assimilable population is known as. record is known as SECTION-C Multiple Choice Questions Select the correct alternative out of given ones: Q.2 Who stated that statistics is a branch of applied mathematics which specialises in data? (a) Horace Secrist (b) R.A. Fisher (c) Ya-lun-chou (d) LR. ConnorQ3 Qs5 Q. 6. Q. 10 The word ‘statistics’ is used as: (a) Singular ¢b) Plural (¢) Singular and plural both (d) none of the above “Statistics provides tools and techniques for research workers”, was stated by: (a) John 1. Griffin (b) WI. King (c) A.M. Mood (d) ALL. Boddington Out of various definitions given by the following workers, which definition is considered to be mast exact? (a) R.A. Fisher (b) ALL. Bowley (c) M.G. Kendall (d) Cecil H. Meyers. ‘Who stated that there are three kinds of lies: lies, damned lies and statistics. (a) Mark Twin (b) Disracti (c) Darrell Huff (d)_ none of the above Who gave the statement, “A well wrapped statistics is better than Hitler's ‘biglie’, it misleads, yet it eannot be pinned on you.” (a) Mark Twin (b) W.A. Neiswanger (c) Darrell Huff (d)_G.W. Snedecor Which of the following represents data? {a) a single value (b) only two values in a set (c) a group of values in a set (d) none of the above Statistics deals with: (a) qualitative information (b) quantitative information {c) bath (a) and (b) (d) none of (a) and (b) Statistical results are, Q14 Q.16 Qa” PROGRAMMED STATISTICS fa) tb) cent per cent correct not absolutely correct (c) always incorrect (d) misleading If ‘a’ isthe.actual value and ‘e* is its estimated value, the absolute error is: (a) a-e {b) la-el (c) ae (d) (a - elle If ‘a’ is the actual value and ‘e” is its estimated value, the formula for relative error is: (a) ale (b) (a - ee (c) la-eVe (d) (a-eVa Data taken from the publication, ‘Agricultural Situation in india’ will be considered as: (a) primary data (b) secondary data (c) primary and secondary data (d) neither primary nor secondary data Mailed questionnaire method of enquiry can be adopted if respondents: (a) live in cities (b) have high income (c) are educated (d) are known ‘The statement, “Designing of an appropriate questionnaire itself wins half the battle”, was given by: (a) AR. Tlersic (b) WL. King (c) H. Huge (d) H. Seerist Statistical data are collected for, (a) collecting data without any purpose (b) a given purpose (©) any purpose (d) none of the above Relative error is always: (a) positiveSTATISTICS IN GENERAL Q 20 Q.22 O23 (b) negative (c) positive and negative both (d) zero Statistical error refers to: (a) Original value - Approx. value (b) Actual value - Estimated value Actual value - Estimated value (c) = Estimated valuc gy Detual value ~ Estimated value @ Actual value Method of complete cnumeration is applicable for: (a) Knowing the production (b) Knowing the quantum of export and import (c) Knowing the population (d) all the above A statistical population may consist of: (a) an infinite number of items (b) a finite number of items (c) either of (a) and (b) {d) none of (a) and (b) Which of the following example does not constitute an infinite population? (a) Population consisting of odd numbers (b) Population of weights of newly born babies (c) Population of heights of 15-year-old children (d) Population of head and tails in tossing a coin successively. Which of the following can be classified as hypothetical population? (a) All labourers of a factory (b) Female population of a country (c) Population of real numbers between 0 and 100 (d) students of the world A study based on complete enumeration is known as; (a) sample survey (b) pilot survey Q.24 Q.26 Q27 Q.28 Q.29 (c) census survey (d) none of the above Tf the actual value of a unit is 415 and its estimated value is 400, the absolute error is: (ay) -15 (b) 15 (c) 0.0375 (d) — 0.0361 If the estimated value of an item is 50 and its actual value is 60, the relative error is: (a) -20 (b) 0.16 (e) 12 (d) 0.20 ‘Who originally gave the formula for the esti- mation of errors? (a) L.R. Connor (b) WL King (©) AL. Bowley (d) A.L. Boddington Boddington gave the formula for the esti- mation of errors of the type: {a) Absolute error biased {b) Absolute error unbiased {c} both (a) and (b) (d) neither (a) ner (b) Boddington's formula for estimation of absolute error (A.E) is: (a) Total ABV Vn (b) Average AE x vn (c) Total AE/n (d) Average AE xn (n= No. of items) Boddington's formula for estimation of relative error is: (a) Total AE @ (b) Average A.BJn Average A.E xn e Average A.E x e () @) (e = estimated value]10 Q.30 O31 Q. 32 Q.33 Qa% Bowley’s formula for absolute unbiased error is: 1 (3) Ty Average (A.B) 2 () 3 (Average AE) 2 average AE) (ce) 3 3 (d) ahh (Average A.E) Statistical results are: (a) absolutely correct. (b) not true (c) true on average (4) universally true The statistical law(s) based on trial and error methods is/are: (a) law of statistical regularity (b) law of inertia of large numbers (c) both laws (a) and (b) {d) none of the laws (a) and (b) The figure 32,64,616.8 approximated to the tenth place by the method of discarding figure ist (a) 32,64,615.8 (b) 32,64,616 (c) 32,64,620 (a) 32,64,610 ‘The figure 32,64,616,8 approximated to the tenth place by adding figure is: (a) 32, 64, 615 (b) 32, 64, 616 (©) 32, 64, 620 (d) 32, 64, 610 The figure 26,476 approximated to the hundredth place by discarding figure is: fa) 26,400 (b) 26,500 (c) 27,000 (d) 25,000 ‘The figure 47, 616 approximated to hundredth place by adding figure is: Q.37 aw PROGRAMMED STATISTICS (a) 47,630 (b) 47,620 (c) 47,700 (d) 47,600 The figure 43,572.6 approximated to the thousandth place by discarding figure is: (a) 43,500 (b) 43,000 (©) 44,000 (d) 44,500 The value 43,572.6 approximated to the thousandth place by adding figure is: (a) 43,500 (b) 43,000 (©) 44,000 (6) 44,600 |. The figure 45,986 approximated to the ten thousandth place by the method of discarding figure is: (a) 40,000 (b) 46,000 (c) 45,500 (d) 45,000 The figure 45,986 approximated to ten thousandth place by the method of adding figure is: (a} 50,000 (b) 46,000 (c) 40,000 (d) none of the above ANSWERS: SECTION-B (1) William A. Spurr and Charles P. Bonini (2) A. L. Bowley (3) Lovitt (4) W.A. Wallis and H.V. Roberts (5) science, art (6) C.H. Meyers (7) A.E. Waugh (8) A.L. Boddington (9) Whipple (10) Tippet (11) WL King (12) anything (13) inexperts (14) acceptable (15) quantitative data (16) interpretation (17) single (18) datum (19) statements; statistics (20) multiplicity (21) individuals (22) scientific method (23) bricks (24) welfare (25) worse (26)STATISTICS IN GENERAL ait misused (27) secondary (28) primary (29) edited Suggested Reading (30) Physical (31) Questionnaires (32) illiterate (33) mailed enquiry (34) schedule; questionnaire (35) primary (36) not same (37) biased (38) unbiased (39) Average A.E x No. of items (40) Average AE x JNo. of items (41) Biased A.E/Estimated value 42) Unbiased A.E 42) Estimated value (45) 16,320 (46) 32,600 (47) 32,000 (48) 46,000 (49) 138 (50) 13.8 (1) statistics. (52) primary data ($3) census method (54) sample method (55) statistical perspective. (43) census survey (44) coding SECTION-C (jb @b Qe Me (a (6b Me (ec Ob (10)b C1)b (12) (3b (aye (Ie (16)b (IT) (18)b (Id Qe Qe (2e C3)c (24)b (25)d (26)d (27)b (28)b (29)d (30) b Ble Ge GId Ga)c (5)a (36)c G7) b) (B8)e Ga (40)a 1. Agarwal, B.L., Basic Statistics, New Age International (P) Ltd. Publishers, New Delhi, 3rd edn., 1996, 2. Gupla, B.N., Siaiistics, Sahitya Bhawan, Agra, 3rd edn., 1978. 3. Harvey, .M., Sources of Statistics, Clive Bingley, 1969. 4, McCarthy, P-J., Introduction to Statistical Reasoning, McGraw-Hill Book Company, New York, 1957, 5. Monroney, M.J., Facts from Figures, Penguin Books, Baltimore, 1959, 6. Reichman, W.J., Use ard Abuse of Statistics, Penguin Books, Baltimore, 1961. 7. Simpson, G. and Kafka, F,, Basie Statistics, ‘Oxford & IBH, Calcutta, 3rd edn., 1971. 8. Snderson, T. and Sclove, S., An Introduction to the Statistical Analysis of Data, Houghton Mifflin, Boston, 1978.Chapter 2 Classification, Tabulation and Frequency Distribution SECTION-A Short Essay Type Questions. Q.1 What is meant by classification? Ans. Classification is the process of arranging things or items in groups or classes according to their resemblance and affinities and give expression to the units of attributes that may subsist amongst the diversity of individuals. Q.2 > What are the modes of classification? Ans. Different modes of classification are: (i) Geographical classification: classification is according to place, arca or region, (ii) Chronological classification: It is according to the lapse of time, e.g., monthly, yearly, etc, (Gi) Qualitative classification: Data are classified according to the altributes of the subjects or items, ¢.g., sex, qualification, colour, etc. (iv) Quantitative classification: Data are classified according to the magnitude of the numerical values, ¢.g., age, income, height, weight, etc. Q.3 > What are the objectives of classification? Ans. Broadly, there are six objectives of classi- fication: (i) To present the facts in a simple manner. (ii) To highlight items which possess or do not possess certain attributes or qualities. (iii) To provide help in making comparison between items. (iv) To find out mutual relationship between certain measures and their effects. (¥) To present the data in a manner which is suitable for further treatment. (vi) To provide basis for tabulation. Q.4 What do you understand by qualitative classification? Ans. It is the classification on the basis of certain attributes or some qualities of items which cannot be measured quantitatively. Q.5 Describe in brief different kinds of classi- fication. Ans. Different types of classification are: (i) Classification according to attributes. (ii) Simple or two-fold or dichotomous classi- fication (iii) Multiple classification. (iv) Quantitative classification, ie, the classi- fication according to variate values.CLASSIFICATION, TABULATION AND FREQUENCY DISTRIBUTION 13 6 What do you understand by multifactor classification? Ans. Classification criteria based on two or more factors (attributes) is known as multifactor classi- fication. In this type of classification, first the data are classified into two or more classes on the basis of one factor. For each component classification, further classification is done on the basis of second factor and so on, Q.7 What do you understand by open end(s) in group data? Ans. If in grouped classes, the lower limit of the beginning class is not specified and/or the upper limit of the highest (last) class is not specified, it is known as grouped data with open end class(es). Q.8 What are different characteristics of classi- fication? Describe each characteristic in five lines. Ans. Different characteristics of classification are: (i) Exhaustive: The classes should be such that they cover every item of the set, ie, they should also be complete and non-overlapping. For instance, for marital status the classes should be married, unmarried, widow, widower, divorcee, deserted, (i) Stability: Classification should be uniform or standardised so that the results are comparable at different occasions or in different studies. Flexibility: Classification should be amenable according to different situations or require- ments of study. Homogeneity: The. units of measurement of all classes. should be same. Also like units only be accommodated in one class. Suitability: Classification be done according to the objective of the study only. For instance, to study the financial status of people, it will be useless to classify them according to their skin colour or their hair colour, etc, Arithmetic accuracy: The sum of number of units in all classes should be equal to the total number of units. Also in case of observations, the sum of observations in all classes should be equal to the sum of all observations. (iii) Gv) wv) (wi) Q.9 How can one determine the number of classes for a frequency distribution? Ans. In quantitative classification, the number of classes depends upon the class interval. So a formula was suggested by H.A. Sturges to determine the class interval and also the number of classes. The formula is, j-—t 1+3322logign where, i = class interval L = Largest observation S = Smallest observation and m= total number of observations in the set. Also, the denominator, | +3.322 logit is equal to the number of classes. Q. 10 Describe in brief the grouping error. Ans, If the classes are formed in such a way that the frequencies are evenly distributed throughout the class interval, it is justified to assume that the frequencies. are centered at the mid-value of the class. But in cases where such an assumption is not valid, it leads to error which is known as grouping error, Grouping error affects the accuracy of the results. Q. 11 Clarify the difference between exclusive and inclusive class intervals. Ans, Following are the differences between exclusive and inclusive class intervals: (i) In exctusive class intervals, the upper limit of a class is the lower limit of the next class. Also the upper limit of a class is not included in that class. In inclusive class intervals the upper limit of a class instead is not the lower limit of the next class, The lower limit is generally greater by unil measurement. In inclusive method, both the limits of a class are included. To simplify the calculation procedure, inclusive classes are converted into exclusive classes. (o7) (ili) ww)14 (¥) Inclusive classes approach is suitable in case of data given in whole numbers. In rest of the cases exclusive class approach is suitable. Q. 12 If mid-values of the classes are known, how can the classes be formed? Ans. Find the difference between two consecutive mid-values. Subtract half of the difference from the mid-value and again add it to the mid-value. The values obtained on subtracting and adding half of the difference are the lower and upper limits of the class of which the mid-valuc has been used. If m is the mid-value of a class and j is difference between two consecutive mid-values, the lower and upper class limits are (m~ ‘) and G + 3) respectively. Q.13 Ilustrate exclusive and inclusive class intervals, Ans, In exclusive class intervals uppers limit of the class is not included, c.g., in the class 10-20 those values are included which are 10 or more and tess than 20. Similarly in the inclusive class intervals, both the limits of a class are included. The classes may be of the type, 5 - 5.99, 10 - 14.99, etc. Q. 14 Distinguish between real limits and apparent class limits of a distribution in grouped data. Ans. If the distribution is for a discrete variable, the real and apparent class limits are same, e.g., 5-10, 11-16, 17-22, ... since there is no recorded value between 10 & 11, 16 & 17, etc. But if the class intervals are exclusive, in that case either the upper limit or the lower limit is to be excluded since the value can be included in one class only. For instance, let the classes be 5-10, 10-15, 15-20, etc. Suppose the upper limits are excluded. In that case, the real limits are 5-9, 10-14, 15-19, etc. Of course there is no point in between 9 & 10, 14 & 15, etc. If the lower limits are excluded, the real limits are 6-10, 11-15, 16-20, ete. Q. 15 What do you understand by tabulation? ‘Ans, It is the process of presenting data collected through survey, experiment or record in rows and columns so that it can more casily be understood and can be used for further statistical analysis. Q. 16 What are different parts of a standard table? PROGRAMMED STATISTICS Ans. There are five parts of a table. (i) Title, (ii) captions and stubs, (iii) Body; (iv) Prefatorial, and (v) Source note. ‘Q..17 | What are the objectives of tabulation of data? Ans. The objectives of tabulation are: (i) To clarify the object of investigation. (ii) To reduce complexity of data (iii) To economise space (iv) To depict the relation among data if it exists. (v) To facilitate analysis of data. Q. 18 What are the requisites of a standard table? Ans. Requisites of a standard table are: (i) It should be suitable for the purpose. Gi) Clarity and completeness of table is necessary. (iii) Table should be of adequate size. (iv) Units of measurements should be specified. (v) Logical arrangement of items. (vi) Totals and sub-totals be given. Q. 19 What are the main purposes of tabulation? Ans. The main purposes of tabulation are: i) To present the haphazard data in simple and concised manner. (ii) To save: space. (ii) To show the trend of data, if any. (iv) To facilitate comparison of data. (v) To detect errors and omissions of data, if any. (vi) To facilitate the process of statistical-analysis. (vii) ‘To know the source of data. Q. 20 What is the difference between classification and tabulation? Ans. Classification is meant for arranging the data into characteristics or groups where each group has the number of item attached to it. In case of variables, it is given in the form of frequency distribution. Tabulation is the logical and systematic arrange- ment of data in rows and columns. In a table, data may be presented in modified form as well, e.g., in per cent, proportion, total or average values, etc.CLASSIFICATION, TABULATION AND FREQUENCY DISTRIBUTION 15 Q. 21 What is an original table (classification table)? Ans. In an original table, the data are presented in the same formy in which they are collected. Q. 22 What is a derivative table? Ans. In.aderivative table the data are not presented in its original form but the values based on original observations are presented. For instance, totals for different classifications, the means, percentage, ratios or proportions, etc., are presented ina derivative table, Q. 23 Distinguish between frequency and cumu- lative frequency. Ans. Frequency: Number of times a variate value is repeated is called its frequency. Cumulative frequency: This is the number of observations corresponding to less than (more than) or equal to a specified value. Q. 24 Differentiate between a time series and a spatial series. Ans. (i) Time series is an ordered data arranged in sequence of time period. Time periods may be weekly, monthly, yearly, quinquennially, decadal, etc, Time series is also known as historic series. (ii) Spatial series is one in which the data are arranged according to the place or space. The place or space may be localities, cities, states, countries, etc. Q. 25 Explain briefly the stem and leaf display of data, Ans. Itisa method of presentation of data in which each value is divided into two parts. One part consists of one or more leading digits as stem and remaining of the digits as leaf, SECTION-B Fill in the Blanks Fill in the suitable word(s) or phrase(s) in the blanks: 1, Classification is the of facts that are distinguished by some significant 2. Fora good classification, the class should be and . 3. Classification can be done according to 4. Quantitative classification leads to 5. Yearwise recording of data of food production will be called classification. 6. The census data published for citywise popu- lation in India will be known as classification. 7. The data recorded according to standard of education like illiterate, primary, secondary, graduate, technical, etc., will be known as classification. 8. Distribution of families according to their size will be classified as classification. 9, The difference between the upper and lower limit of a class is called 10. The average of the upper and lower limits of aclass is known as 11. There is a general assumption that the class frequency is centered at the of the class. 12. Departure from the assumption that the frequencies are evenly distributed over the class interval leads to error. 13. Formula for determining the number of classes was given by . HA. Sturges formula for Getermining the number of classes is Number of classes depend on H.A. Sturges formula for finding out the class interval is ‘The number of classes and class interval for the distribution of marks from 0 to 100 of 50 students of a class should be and respectively. 15. 16, 1.18. 19. S RRES B 8 ai. 3B. Class boundaries are also sometimes called limits. Mid-values of the classes are also called Frequency density of a class is the frequency of class. In the mid-valuc of a class interval is 20 and the difference between two consecutive mid- values in 5, the class limits are and ‘An arrangement of data in rows and columns is known as Tables help in of data, Tabulation makes the data easily Tabulation follows - —___ facts cannot be presented in the form of a table. A general purpose table is also known as or table. A general table is a of a large amount of data, ‘The table which do not present the data but the results of analysis are called tables. Headings of the columns of a table are known as . Headings of the rows of a table are called The portion of the table in which data are presented is designated as ofa table. The manner in which the frequencies are distributed according to variate values is known as Frequency distributions are often constructed with the help of Relative frequency is the ratio of a frequency to the ‘of the distribution. PROGRAMMED STATISTICS 36. Percentage frequency is the multi- plied by 100, Frequencies added successively in an ordered series giving the number of items up to that value are called ____. A frequency distribution with upper limits of classes and corresponding cumulative frequencies is known as type distribution. A frequency distribution with lower limits of classes and corresponding cumulative fre- quencies is known as type distri- bution. ‘The graphs of less than type and more than type distributions intersect at A grouped series, in which either the lower limit of the first group or the upper limit of the last group is missing or both, is called an 3. 41. A series arranged in accordance with cach and every observation is known as__. ‘The distribution of frequencies according to individual variate values is called distribution. A series of data with exclusive classes along with the corresponding frequencies is called distribution, Given the following frequency distribution, fill in the missing frequencies: Cumulative FrequencyQi CLASSIFICATION, TABULATION ANO FREQUENCY DISTRIBUTION aw SECTION-C Multiple Choice Questions Select the correct alternative out of given ones: —Q.7 If the number of students in a school is 200 . - and maximum and minimum marks earned Numerical ddatn prescoted in descriptive forma are 90 and 10 respectively, for the distribution (a) classified tation of mats the class interval (rounded) is: (b) tabular presentation ® 9 (©) graphical presentation (©) 12 (d) textual presentation (d) none of above Whether classification is done first or tabu- {Given logy2 = 0.3010, log,,3 = 0.4771] on? " se ai - Q. 8 If the lower and upper limits of a class are 10 (a) Classification follows tabulation, and 40 respectively, the mid-points of the (b) Classification precedes tabulation, class is: (©) Both are done simultaneously. (a) 25.0 (a) No criterion. ® 2s For the mid-values given below, (c) 15.0 25, 34, 43, 53, 61, 70 (d) 30.0 ‘The first class of the distribution is: Q.9 In a grouped data, the number of classes (a) 245-345 preferred are: {b) 25-34 (a) minimum possible (e) 20-30 (b) adequate (a) 205-295 (c) maximum possible In an exclusive type distribution, the limits (4) any arbitrarily chosen number. excluded are: Q. 10 Class interval is measured as: {@) lower limits (a) The sum of the upper and lower limit (b) upper limits | (b) half of the sum of lower and upper limit (c) either of the lower or upper limit (c) half of the difference between upper and (d) lower limit and upper limits both lower limit Asscrics showing the sets of all distinct values (4) the difference between upper and lower individually with their frequencies is known limit. as: Q. 11 The class interval of the continuous grouped (a) grouped frequency distribution (b) simple frequency distribution (c) cumulative frequency distribution (d) none of the above A series showing the sets of all values in classes with their corresponding frequencies. is known as: (a) grouped frequency distribution (b) simple frequency distribution (c) cumulative frequency distribution (d)_ none of the above data: 10-19 20-29 30-39 40-69 50-59 is: @) 9 (b) 10 (©) 145 @45 Q. 12 A grouped frequency distribution with uncertain first or last classes is known as:18 Qi O16 Q.i7 Q18 (a) exclusive class distribution (b) inclusive class distribution (c) open end distribution (d) discrete frequency distribution ‘The distribution, Values Frequency Less than 5 5 Less than 10 12 Less than 15 21 Less than 20 27 Less than 25 31 Less than 30 33 is of the type: (a) inclusive class type: (b) exclusive class type (c) discrete type (@) none of the above Data can be well displayed or presented by way of: (a) stem and leaf display (b) cross classification {c) two or more dimensional table (d) all the above A simple table represents: {a) only one factor or variable (b) always two factors or variables (c) two or more number of factors or variables (d) all the above A complex table represents: (a) only one factor or variable (b) always two factors or variables (c) two or more factors or variables (d) all the above ‘The headings of the rows given in the first column of a table are called: (a) stubs (b) captions (c) titles (d) prefatory notes. ‘The column heading of a table are known as: (a) sub-titles (b) stubs Qs Qn PROGRAMMED STATISTICS (c) reference notes (d) captions The series, Place No. of accidents per day Delhi 10 Kolkata 5 Mumbai 48 ‘Chennai 7 Indore: 7 is of the type: (a) Spatial {(b) Geographical (c) Industrial (d) Time series ‘The series, Year Production of food (M. Tonnes) 1987 160 1988 168 1989 170 1990 172 1991 174 1992 176 is categorised as: (a) individual series. (b) continuous series (c) discrete series (d) time series ‘The income of five persons is as follows: Person. Income (Rs/Month) MrA 1,700 MrB 2,300. Mrc 7,000 Mr D- 8,500 MrE 5,400 ‘The above series is of the type: (a) Individual series (b) Discrete series (c) Continuous series (d) Time seriesCLASSIFICATION, TABULATION AND FREQUENCY DISTRIBUTION Q. 22 The series, Marks No. of Students 20-30 5 30-40 14 40-50 2 50-60 12 60-70 9 0-80 2 is of the type: (a) Discrete series (b) Continuous series (c) Individual series (d) none of the above A frequency distribution can be: {a) discrete (b) continuous (c) both (a) and (b) (d)_ none of (a) and (b) Q. 24, In an individual series, each variate value: (a) has same frequency (b) has frequency one (c) has varied frequency (d) has frequency two Q. 25 Which of the following statements is true? (a) An individual series is a particular case of discrete series 0) of continuous series c discrete and continuous serics (d) There is nothing like individual series Frequency of a variable is always: (a) in percentage {b) a fraction (c) an integer (d) none of the above be called as: (a) a continuous series (b) a discrete series (c) an individual series (d} time series The following frequency distribution, An individual series is a particular case An individual series is a special case of ‘The data given as, 5,7, 12, 17,79, 84, 91 will Q31 Q.32 19 x: 92, 17, 24, 36, 45, 48, 52 fe 25 32 8 9 6 1 is classified as: {a) continuous distribution. (b) discrete distribution. (c) cumulative frequency distribution (d) none of the above In an ordered series, the data are: (a) in ascending order (b) in descending order {c) cither (a) or (b) (d) neither (a) or (b) ‘The following frequency distribution, Classes Frequency 0-10 3 0-20 8 0-30 14 0-40 20 0-50 25 is known as: (a) continuous frequency distribution (b) discrete frequency distribution (c) cumulative distribution in more than type (d) cumulative distribution in less than type The following frequency distribution, Classes Frequency 0-15 7 0-10 8 0-5 3 is classificd as: {a) cumulative distribution in less than type ({b) cumulative distribution in more than type (c) discrete frequency distribution (d) cumulative frequency distribution Classification is applicable in case of: (@) quantitative characters (b) qualitative characters (c) both (a) and (b) (d) none of the aboveANSWERS Section-B (1) grouping; characteristics (attributes) (2) exhaustive; mutually exclusive (3) attributes (4) frequency distribution (5) Chronological (6) geographical (7) qualitative (8) quantitative (9) class interval (10) mid-value (11) mid-value (12) grouping (13) HLA. Sturges (14) 1 + 3.322 log,, (15) class interval (16) (L - S) (1 + 3.322 logy, n) where L = largest value, S = smallest value and n = No. of values (17) 7 and 15 (18) mathematical (19) class marks (20) per unit (21) 17.5 and 22.5 (22) tabulation (23) analysis (24) understandable (25) classification (26) qualitative (27) primary or reference (28) repository (29) specific purpose (30) captions (31) stubs (32) body (33) frequency distribution (34) tally marks (35) total frequency (36) relative frequency (37) cumulative frequency 8) less than (39) more than (40) median (41) open end series (42) individual series (43) discrete frequency (44) continuous frequency (45) freq = 14; as, cuefreg-27, 41, 48. SECTION-C Gd @)yb Gd Bc Gb Ha Mb Ba Ob (1d (Ib (He PROGRAMMED STATISTICS (13) b (19) b (25) a GI)b aad (20) 4 (26) ¢ 2) c (5) a Qla Ce (16) (22)b (28)b (7a (23) ¢ 29) ¢ (8)d 4b (30) d Suggested Reading 1. Agarwal, B.L., Basic Statistics, New Age International (P) Ltd. Publishers, New Delhi, 3rd edi, 1995. 2. Devore, JL., Probability and Statistics for Engineering and the Sciences, Brooks/Cole Publishing Co., California, 1982. 3. Gupta C.B., Am Introduction to Statistical Methods, Vikas Publishing House, Delhi, 8th edn., 1978, 4. Kenny, J.F, and Keeping, E.S., Mathematics of Starissics, Part 1, D. Won Nostrand Co., New York, 1951, 3. Sancheti, D.C. and Kapoor, V.K., Statistics, Sultan Chand & Sons, New Dethi, 7th edn., 1991. 6. Shukla, M.C. and Gulshan, S.S., Statistics, Sultan Chand & Co., New Delhi, 1970. 7. Simpson, G. and Kafka, F., Basic Stasistics, Oxford & IBH, Calcutta, 3rd Indian print, 1971,Chapter 3 Diagramatic and Graphical Representation SECTION-A Short Essay Type Questions Q.1 What are the advantages of diagramatic representation of data? Ans. Following are the advantages of diagramati: representation of data: (i) Diagrams give a bird's-eye view of complex data. They have long lasting impression. Easy to understand even by a common man. They save time and labour. {v) They facilitate comparison. Q.2 > Give the names of diagrams which are one- dimensional. iti) (iv) Ans. Bar diagrams and line diagrams are one- dimensional. Different types of bar diagrams are: (i) simple bar diagram (ii) multiple bar diagram (iii) sub-divided or component bar diagram (iv) percentage bar diagram. Q.3 Name diagrams which are two-dimensional. Ans. Rectangles, circles and pie diagrams are two- dimensional diagrams. Q. 4 What arc the diagrams categorised under three- dimensional diagrams? ‘Ans, Cubes, cylinders and spheres are categorised as three-dimensional diagrams. Q.5 What types of diagram are known as non- dimensional diagrams? Ans. Pictograms. are known as non-dimensional diagrams, Q.6 What do you understand by bar diagram and a sub-divided bar diagram? ‘Ans. A bar diagram represents the magnitude of a single factor according to time periods, places, items, ete. But when the magnitude of the factor is given with its sub-factors, cach bar is further sub-divided into components in proportion to the magnitude of the sub-factors. Yearwise Sales ‘Sales (Grove Fis.) 1994 1993 13992 1991 1999 ‘Year o 1 2 3 4.5 Fig. 3.1, Bar diagram22 Such a diagram is known as sub-divided bar diagram. Countrywise Tourist in Various Cities No. of Tourist, Madras Udaipur et a a . Srinagar rT thi a a Fig. 3.2. Sub-divided bar diagram Q.7 When do you prefer a multiple bar diagram (compound bar diagram). Ans. To depict a number of related factors for ‘comparison in various years or at a number of places, multiple bar diagrams are: preferable. Q.8 | What is a multiple bar diagram? Ans, Ina multiple bar diagram, adjoining bars are drawn according to the number of factors and their heights in proportion to the values of the factors in the same order for each period or place. Each bar of a group is shown by different patterns or colours to make them easily distinguishable and this pattern is retained in all the groups. A constant distance is Display of Proft and Expenses o——____— — Prost ~ BE penses, 1990 1991 1992 993 1994 Fig. 3.3. Multiple bar diagram PROGRAMMED STATISTICS maintained between groups of bars drawn for periods or places, Such a diagram is known as multiple or compound bar diagram (Fig. 3.3). Q.9 What do you understand by deviation bar diagram? Ans. Deviation bar diagrams are suitable to show the net deviations during various years or according to different countries or places, etc. In the deviation bar diagrams, positive deviations are shown to the right side of the base line and negative deviations are shown to the left side of the same base line. For instance, the gaps between imports and exports, profit and loss in different years or from different countries can be very well displayed through deviation bar diagrams (Fig. 3.4). Net Results of a Company Loss Profit 1985, 1986: 1987 1988 1989 1990 1991 Fig, 3.4. Deviation bar diagram Q. 10 Write a short note on duo-directional bar diagrams. Ans. Duo-directional bar diagrams are used to exhibit the two aspects of a single factor at a glance given for different periods or places. In this type of diagram, one part of the bar remains above the base line and the other below the base line. The heights of the bars below and above the line are in proportion to the values of the two aspect separately whereas the bar as a whole represents the factor, For example, ‘we want to show the price of certain item in different years. The price consists of two parts, the cost and[DIAGRAMATIC AND GRAPHICAL REPRESENTATION the profit. So profit may be taken above the line and cost below the line. It is a sort of sub-divided bar diagram (Fig. 3.5). Price of terns. Price (Crore Ais.) Fig. 3.5. Duo-directional bar diagram Q. 11 Write a brief note on paired bar diagram. Ans. When two related factors having different units of measurements are to be displayed for comparison in various periods or places, paired bar diagrams are suitable. In this diagram usually, the periods or places are shawn in a strip and horizontal ‘bars for each factor are drawn to the right and left of ‘the vertical strip or vertical bars are drawn below and above the horizontal strip. For instance, area and production of paddy in different years in India can be very well displayed through a paired bar diagram (Fig. 3.6). Q. 12. What are sliding bar charts? Explain in brief. Ans, Sliding chart is a bilateral chart in which two ‘components of a factor are represented by two parts of the bar. One part is on the left and the other is on the right of the base line, The scale may be the absolute numbers or in percentages. Such a chart is suitable in situations such as a numbers arrested in a criminal case or patients operated for different diseases. What percentage or number of suspects have been cleared and what are still under trial. How many patients are cured and how many of them still 23 Area and Production of Paddy Area, Production 1990-91 1980-81 1970-71 1960-61 1950-514 80 60 49 20 0 20 40 60 80 Fig. 3.6. Paired bar diagram suffer. The two components can be better displayed by a sliding bar diagram. The base line may be Tepresenting the type of operations, kind of court cases, etc. The percentage or numbers in the two components, cured and not cured, cases cleared and not cleared may be changing from time to time. For each type of operation crime (factor), a separate sliding bar will be drawn. Patients Operated Not cured Uicer 85% | 9 70 200 20 70 95 Fig. 3.7. Stiding bar chart Q. 13° What is a broken bar diagram? ‘Ans, Often an investigator comes across cases where some figures are very large as compared toothers. In this situation, if the scale is chosen for proper portray of small values by bars, the bars for large values will expand to a unpalatable size. Again, if the scale is chosen for proper display of large values by bars, the bars for small value will become non-existent. Hence, to remove this discrepancy, broken bars are constructed. First, a small scale is taken and bars are erected at all periods or places up to the highest small values and/or a round off value, Then with a gap another base line and a new scale for large values is chosen. Bars are constructed for remaining value on the new base line. Such a bar diagram is known as broken bar diagram. These diagrams be interpreted very carefully to avoid any wrong conclusions. Admissions in Various Subjects: Phy Chem Gomp ME Fig. 3.8, Broken bar diagram Q. 14 What is the basis of comparison in bar charts? Ans. The basis of comparison in bar charts is linear or one directional. Q. 15 What is the difference between bar diagrams (charts) and column charts (diagrams)? Ams. In the bar diagrams, the bars are arranged horizontally on a vertical base line, whereas in the column charts, bar are arranged vertically on a horizontal base line. Q.16 What are the differences between a bar diagram and a rectangular diagram? PROGRAMMED STATISTICS Ans. The differences are: (In bar diagram all bars are of equal width. (ii) The width of bars in bar diagram is chosen arbitrarily for qualitative characters like places, states, countrics, etc., or according to the equal class intervals, In a rectangular diagram, there is always a variable and its magnitude. The width of the rectangles are according to variate values and heights of the rectangles are according to their respective magnitudes. In bar diagram, comparisons are based on the heights of bars only whereas in rectangular diagrams, comparisons are based on the area of the rectangles. Q. 17 Explain a line diagram in two lines. ‘Ans. A line diagram is a one-dimensional diagram in which the height of the line represents the frequency corresponding to the value of the item or a factor. (iii) (iv) No. of Accidents on Week Days Man: Tues ‘Wed Thur Fei ‘Sat ‘Sun Fig. 2.9. Line diagram Q. 18 Explain in brief a pie-chart. Ans. A pie-chart is a circular diagram which is usually used for depicting the components of a single: factor. The circle is divided into segments which are: in proportion to the size of the components. They are shown by different patterns or colours to make them attractive (Fig. 3.10).DIAGRAMATIC AND GRAPHICAL REPRESENTATION India's imports from various sources (per cent) 1385-86 Fig. 3.10. Pie chart Q.19 Explain a histogram. Ans. A histogram isa bar diagram which is suitable for frequency distributions with continuous classes. The width of all bars is equal to class interval and heights of the bars are in proportion to the frequen- cies of the reSpective classes. In this diagram bans touch each other but one bar never overlaps the other (Fig. 3.11). Distribution of Wages No. of Workers 3 o SRS8582 38 Wages/Month (Fs.) Fig. 3.11, Histogram Q. 20 Describe a frequency polygon. Ans. When the mid-points of the tops of the adja- cent bars of a histogram are joined in order, then the graph of lines so obtained is called a frequency polygon. Frequency Distribution of Wages Fig. 3.12. Frequency polygon Q. 21 Discuss a frequency curve in brief. ‘Ans. A frequency curve is a graphical representa tion of frequencies corresponding to their variate values by a smooth curve. A smoothened frequency polygon represents a frequency curve. Distribution of Marks ‘No. of Students, Fig. 3.13. Frequency curve Q. 22 What is a false line in reference to a graph? ‘Ans, When the variation in the magnitudes of a variable is small as compared to the variate values,‘the yertical scale chosen as zero at the origin fails to depict the fluctuation prominently as desired by the investigator. Hence a fine parallel to abscissa (X- axis) is drawn a little above it and the point joining the ordinate (Y-axis) is taken as origin which usually represents the minimum valuc (an approximate value by discarding figure) from amongst the magnitudes to be taken on the vertical scale. Then proper scale is chosen measuring distances from this false base line. Such a process makes the fluctuations conspi- cuous. The graph so obtained is called gee whiz graph. False line on the graph is usually shown by a saw- tooth line. Q. 23. What is the range curve (chart)? ‘Ans. To depict the spread of highest and lowest values of a time series, the band formed by the line gtaphs of highest and lowest values is known as range curve. Range curve can also be represented through bar segments drawn at each period which are of magnitudes of the differences between highest and lowest values, These bars start from the lowest price and go up to the highest value. Q. 24 How do you draw a histogram when the widths of all classes are not equal? Ans. When widths of all classes of a frequency distribution are not equal, heights of the bars are taken in proportion to the frequency density (frequency per unit interval). Q. 25 What is a ratio chart or a semi-logarithmic graph? Ans. It is a line graph obtained by plotting the points (x, log y) in such a way that the a-values are taken along X-axis on a natural scale and valucs log y are taken along Y-axis. Hence, ratio chart is a graph of the points (x, log y) plotted on an ordinary graph paper. It is also called semi-logarithmic graph. in the sense that log-values are used only for y (Fig. 3.14). Q. 26. What do you understand by a graph? Ans. A graph is a display of points and lines. In a graph of paired values (x, y), so called the co- ordinates of a point, are plotted on a graph paper by suitably choosing the scales along X-axis (abscissa) and ¥-axis (ordinate). The plotted points are joined PROGRAMMED STATISTICS Yield Per Hectare of Kharif Pulses & &@ 410 82888 Fig. 3.14, Ratio chart by straight lines in their sequence of occurrence. ‘The figure so obtained is called a graph. The graph depicts the trend, fluctuations, variability, ctc., very prominently (Fig. 3.15). ‘Two ot more graphs made on the same graph paper having a common scale along X-axis and Y-axis facilitate the comparison of data tremen- dously, Population and Availability of Pulses Pulses Per Capita (gms) 64 66 «6687072 Population (Crores) Fig. 3.15. Graph Q.27 Describe an ogive curve in brief. Ans, It is graph plotted for the variate values and their corresponding cumulative frequencies of a frequency distribution. Its shape is just like clongatedDIAGRAMATIC AND GRAPHICAL REPRESENTATION ‘5. An ogive curve is prepared either for more than type or less than type distribution. 50 40 8 8 No. of Students —> 3 0 10 2 8630 40 50 60 Marks ——> Fig. 2.16. Ogive curve Q. 28. What are the uses of ogive curve? Ans. Ogive curveis useful in finding out quartiles, deciles, percentiles, ete. Q. 29. At what point the ogives for more than type and less than type distribution intersect? Ans. The ogives for more than type and less than type distributions intersect at the median. Q. 30 Explain briefly a Lorenz curve. Ans, Lorenz Curve is a special type of cumulative frequency curve which is used to portray the data to indicate whether a factor is equally distributed in relation to the other factor for certain segment of the population, It was originally developed by M.O. Lorenz and is named after him. Q. 31 How can one draw a Lorenz curve? Ans. Following steps are involved while drawing a Lorenz curve: |. The variate values giving information about the segment of population are ignored. 2. Cumulative totals for the magnitudes or frequencies of the two other factors are found ‘out separately. 3. Cumulative totals are expressed as percentage of their respective grand totals. aT 4. Paired cumulative percentages are plotted on a graph paper choosing same scale along axes from 0 to 100. 5. Plotted points are joined by a smooth free hand curve. This curve always starts from the origin and terminates at the end point (100, 100) (Fig. 3.17). Parcent of Income ° 20 6400 608 Percent of Persons 100 Fig. 3.17. Lorenz curve Q. 32 What is the importance of the Lorenz curve? Ans. Incase of equal distribution of both the factors in the segment of population, the graph will be a straight line. But the Lorenz curve farther from straight line shows the inequality of distribution of two factors. Q.33 How are the data portrayed by pictograms? Ans. In pictograms, the data are displayed by the pictures of the items to which the data pertain. A single picture represents a fixed number. Forexample, the population is shown by man, milk production by milk cans, fleets of aeroplane by the pictures of aeroplanes, ete. Population of India in 1993 is 89.6 crore. This can be shown by a pictogram having the pictures of nine men, each man representing 10 crore. Picto- grams are the least satisfactory type of diagrams. ‘They are inaccurate 100. Even then they are preferred by novice and diletiante people. Display of data through pictograms was initiated by Dr, Otto Newrath in 1923 (Fig. 3.18).