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ST-101 Statistics DIRECTORATE OF DISTANCE EDUCATION UNIVERSITY OF JAMMU JAMMU SELF LEARNING MATERIAL B.A. SEMESTER – I SUBJECT : STATISTICS UNIT : I-V COURSE NO. : ST - 101 LESSON NO.: 1-23 COURSE CO-ORDINATOR Proof Reading by : Mr. Stanzin Shakya Dr. Vikas Sharma http:/www.distanceeducationju.in Printed and Published on behalf of the Directorate of Distance Education, University of Jammu, Jammu by the Director, DDE, University of Jammu, Jammu 1 DESCRIPTIVE STATISTICS AND PROBABALITY THEORY © Directorate of Distance Education, University of Jammu, Jammu, 2020 • All rights reserved. No part of this work may be reproduced in any form, by mimeograph or any other means, without permission in writing from the DDE, University of Jammu. • The script writer shall be responsible for the lesson/script submitted to the DDE and any plagiarism shall be his/her entire responsibility. Printed by S K Printers/ 2020 / 100 2 DETAILED SYLLABUS OF B.A. SEMESTER - I STATISTICS PAPER TITLE : DESCRIPTIVE STATISTICS AND PROBABILITY TFEORY Objectives : The objectives of this course is to impart students the basic knowledge of measures of central tendencies and measures of dispersion along with the introduction to concept of probability and its basic theory. UNIT-I Definition, Scope & importance of statistics, General nature of statistical data, qualitative & quantitative data, discrete & continuous data, Primary & Secondary data, classification & Tabulation, frequency distribution & their graphical& diagrammatic representations Histogram, frequency Curves, Bar diagram, Ogive & measure of central tendency (A.M.,G.M., H.M.) Median & mode, their merits& demerits. UNIT- II Measures of Dispersion : Range, Inter-Quartile range, Mean Deviation,Standard Deviation, Variance & Coefficient of Variation, Partition values, Moments (raw & central moments up to order four. Effect of change of origin & scale on moment. Shepherd’s correction (without proof). Skewness & Kurtosis meaning & measures. UNIT-III Bivarite data: Scatter Diagram, product moment correlation coefficient, its properties & simple illustrations. Spearman’s rank correlation coefficient, Intra class corelation Coefficient & correlation ratio. Coefficient of deterimination. UNIT-IV Probability:Random experiment, events, algebra of events, sample space, definitions of probability, simple illustrations for three events, conditional Probability, theorem on Probability of two events & its extension, independent events, simple illustrations, Bayes 3 theorem & its applications. UNIT-V Probability mass function & Probability density function, joint marginal & conditional pmf & pdf , Jacobian Transformation for one and two variables, Independence of random variables, Discrete & continuous random variables, Mathematical expectation, expecta- tion of sum of two random variables & production of two independent random variables, conditional expectation & conditional variance, moment generating function & properties of mgf. NOTE FOR PAPER SETTING The question paper will contain two sections. Section A will contain compulsory ten very short answer type questions of 1 mark each. Section B will contain 7 short answer type question of 5 marks each, atleast one questions from each unit and the student has to attempt any five questions. Section C will contain 10 long answer type questions, two from each unit, of 9 marks each and the student has to attempt five questions selecting one from each unit. BOOKS RECOMMENDED 1. Gupta & Kapoor : Fundamentals of Mathematical Statistics. 2. Kpoor & Saxena : Mathematical Statistics. 3. Goon, Gupta & Dass Gupta : Fundamentals of Statistics Vol-I 4. S.P. Gupta : Statistical Methods. 5. Croxton F.E., Cowden D.J. & Kelin S : Applied General Statistics, Prentice Hall of India. 6. Mood A.M., Boes D.C. & Grybill F.A. : Introduction to the Theory of Statistics. 7. Parazen : Modern Prob. Theory. 8. M.N. Murthy : Introduction to the theory of probability. 9. V.K. Rohatgi : Introduction to the theory of Probability. 4 TITLE OF THE COURSE CONTENTS LESSON TITLE PAGE NO. 1. Definitions, Scope and importance of Statistics 4 2. Classification and Tabulation 17 3. Frequency Distribution 28 4. Measures of Location 44 5. Median and Mode 59 6. Measures of Dispersion 71 7. Standard Deviation and Coefficient of Variance 85 8. Moments 101 9. Skewness and Kurtosis 114 10. Bi Variate Data, Scatter Diagram and Correlation 127 11. Product moment Correlation Coefficient & its Properties 138 12. Spearman’s Rank Correlation Coefficient 148 13. Correlation Ratio 157 14. Random Exeriment 164 15. Axioms of Probability 177 16. Conditional Probability and Independent Events 186 17. Bayes Theorem and its Applications 201 18. Probability Density Function 211 19. Discrete & Continus Random Variables 224 20. Functions of Random Variables 236 21. Mathematical Expectations 246 22. Joint and Marginal Probability Functions 256 23. M.G.F., Conditional Expectation and Variance 268 5 UNIT - I LESSON - 1 DEFINITIONS, SCOPE AND IMPORTANCE OF STATISTICS STRUCTURE: 1.1 Introduction 1.2 Objective 1.3 Origin and development of Statistics 1.4 Definitions of Statistics 1.5 Scope of Statistics 1.6 Importance of Statistics 1.7 Discrete and continuous data 1.8 Primary and Secondary data . 1.9 Sources of secondary data. 1.10 Self assessment questions 1.1 INTRODUCTION:- The word statistics , when referring to the scientific discipline, is singular, as in “Statistics is an art. This should not be confused with the word statistic , referring to a quantity calculated from a set of data, whose plural is statistics. Some scholars pinpoint the origin of statistics to 1663, with the publication of Natural and Political Observations upon the Bills of Mortality by Jhon .Early applications of statistical thinking revolved around the needs of states to base policy on demographic and economic data. The scope of the discipline of statistics broadened in the 6 early 19th century to include the collection and analysis of data in general. Today, statistics is widely employed in government, business, and the natural and social sciences. Its mathematical foundations were laid in the 17th century with the development of probability theory by Blaise pascal and Pierre de fermat. Probability theory arose from the study of games of chance. The method of least squares was first described by Carl Friedrich Gauss around 1794. The use of modern computers has expedited large-scale statistical computation, and has also made possible new methods that are impractical to perform manually. 1.2 OBJECTIVE: Almost every aspect of natural phenomena and human and other activity is now subjected to measurement and interpretation in terms of statistics. The figures have become so important in human affairs that they form the basis of rational thinking and rational decisions in many spheres of human activities and events are proving that the decisions based on figures field better results. The reason being that the knowledge obtained with the help of satisfied methods and supported by numerical facts is accurate and precise. Through application of appropriate statistical methods, current performance may be measured, significant relationships may be studied, past experience may be analyzed and probable future trends appraised. To acquire knowledge and to express it precisely, statistical methods and statistics, have a vital role to play 1.3 ORIGIN AND DEVELOPMENT OF STATISTICS: The term ‘STATISTICS’ has been derived from the Latin word ‘STATUS’ which means political state. More directly, the Italian word ‘STATUS’ seems to be the origin of the term ‘statistics’. The term was used in the 15 th century for state. Germans have used this word in the same sense and they spelled it as ‘STATISTIK’. These words mean ‘political state’ or the ‘statesman’s art’. In 1770, Baron J.F.Von Bielfied defined statistics in his book. ‘The elements of universal Erudition’, as “The science that teaches what is the political arrangement of all the modern states of the known world”. In Germany, systematic collection of statistics by the state started during the end of the 18 th century. In England it was started during Napoleonic wars because to raise revenue to meet the wars expenses it was thought proper to bring precisions in the statistics of public revenue and expenditure. 7 In India, too, the collection of Statistics is an old age tradition. In the ancient works like ‘Manusmirity’,’ Shukraniti’ etc, there is description of methods of organization for the collection of Statistics for running the state. Magasthenese had given an account of method of collection of data regarding revenue and expenditure, births and deaths, military, land etc, during Chandragupta’s time. Kautalya’s Arthashatra too describes all these. During Mughal period, Statistics used to be collected and the system of collection was described in ‘Tuzuk-i-Babri’ and ‘ Ain-i- Akbari’. During Akbar’s time, Raja Toder mal collected land Statistics for determining revenue. In India, Prof. P.C. Mahalonobis , has contributed a lot in the theoretical and applied field of Statistics .In the applied field the names of C.R.Rao, R.C. Desai, Dr. P.V.Sukhatme etc are also noteworthy. Statistics has now become a universal applicable science. Statistics are required as a basis of action in all fields whenever information can be measured . 1.4 DEFINITION OF STATISTICS : Statistics has been defined differently by different authors from time to time. To a layman, it simply means a mass of figures of collection of data. It is true that Statistics deals with aggregates
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