Advanced Bank Management PDF 2023

ADVANCED BANK 18] INDIAN INSTITUTE OF BANKING & FINANCE ADVANCED BANK MANAGEMENT This book is produced in DAISY text for people with bona fide print disabilities by the National Association for the Blind, Delhi state branch (NAB). This book is created for special distribut...

ADVANCED BANK 18] INDIAN INSTITUTE OF BANKING & FINANCE ADVANCED BANK MANAGEMENT This book is produced in DAISY text for people with bona fide print disabilities by the National Association for the Blind, Delhi state branch (NAB). This book is created for special distribution for the print disabled in accordance with Section 52 (1) (zb) of the "Copyright Act of India 1957 as amended in 2012. This DAISY text only is permitted to be used only by persons with print disabilities. Any further reproduction, distribution to any person without a print disability or any commercial use of this book is strictly prohibited and will be subject to legal action. To know more about NAB please visit our website www.nabdelhi.in  INDIAN INSTITUTE.OF BANKING & FINANCE (ISO 9001:2015 Certified) Kohinoor City, Commercial-II, Tower-1, 2nd & 3rd Floor, Kirol Road, Off-L.B.S. Marg, Kurla-West, Mumbai-400070 Established on 30th April 1928 MISSION To develop professionally qualified and competent bankers and finance professionals primarily through a process of education, training, examination, consultancy/counselling and continuing professional development programs. VISION To be premier Institute for developing and nurturing competent professionals in banking and finance field. OBJECTIVES To facilitate study of theory and practice of banking and finance. To test and certify attainment of competence in the profession of banking and finance. To collect, analyse and provide information needed by professionals in banking and finance. To promote continuous professional development. To promote and undertake research relating to Operations, Products, Instruments, Processes, etc., in banking and finance and to encourage innovation and creativity among finance professionals so that they could face competition and succeed. COMMI ITED TO PROFESSIONAL EXCELLENCE Website: www.iibf.org.in ADVANCED BANK MANAGEMENT Indian Institute of Banking & Finance macmillan education © INDIAN INSTITUTE OF BANKING & FINANCE, MUMBAI, 2023 (This book has been published by Indian Institute of Banking & Finance. Permission of the Institute is essential for reproduction of any portion of this book. The views expressed herein are not necessarily the views of the Institute.) All rights reserved. No part of this publication may be reproduced ortransmitted, in any form or by any means, without permission. Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages. First published, 2023 MACMILLAN EDUCATION INDIA PRIVATE LIMITED Bengaluru Delhi Chennai Kolkata Mumbai Ahmedabad Bhopal Chandigarh Coimbatore Guwahati Hyderabad Lucknow Madurai Nagpur Patna Pune Thiruvananthapuram Visakhapatnam Kochi Bhubaneshwar Noida Sahibabad Hubli ISBN: 978-93-5666-027-4 Published by Macmillan Education India Private Limited (formerly Macmillan Publishers India Private Limited), 21, Patullos Road, Chennai 600002, India Printed at: Capital Offset, Bawana Ind. Area Delhi-39 ADVANCED BANK MANAGEMENT First Edition (2023) Authored, Revised & Updated by: Ms. Anuradha Hait (Basu), Head of Department of Statistics, Thakur College of Science & Commerce (Module A); Mr. Mrityunjay Kumar Gupta, Former Head-HR, BOI (Module B); Mr. P. D. Sankaranarayanan, Former AGM & Faculty, State Bank Academy (Module C); Mr. Shivkumar Sareen, Advisor, IPPB & Former GM & Chief Compliance Officer, BOI (Module D) Vetted by: Mr. Butchi Babu, Former GM, BOI (Module A); Mr. G. S. Bhaskara Rao, Former DGM-Dev. HR, Central Bank of India (Module B); Mr. S. C. Bansal, Ex-faculty IIBF (Module C) “This book is meant for educational and learning purposes. The author(s) of the book has/have taken all reasonable care to ensure that the contents of the book do not violate any copyright or other intellectual property rights of any person in any manner whatsoever. In the event the author(s) has/have been unable to track any source and if any copyright has been inadvertently infringed, please notify the publisher in writing for any corrective action.” FOREWORD Formal education will make you a living; self-education will make you a fortune. -Jim Rohn The banking sector, currently, is experiencing a transformation catalysed by digitalization and information explosion with the customer as the focal point. Besides, competition from NBFCs, FinTechs, changing business models, growing importance of risk and compliance, along with disruptive technologies, have contributed to this radical shift. Such an ever-evolving ecosystem requires strategic agility and constant upgradation of skill levels on the part of the Banking & Finance professionals to chart a clear pathway for their professional development. The mission of the Indian Institute of Banking & Finance is to develop professionally qualified and competent bankers and finance executives primarily through a process of education, training, examination, counseling and continuing professional development programs. In line with the Mission, the Institute has been offering a bouquet of courses and certifications for capacity building of the banking personnel. The flagship courses/examinations offered by the Institute are the JAIIB, CAIIB and the Diploma in Banking & Finance (DB&F) which have gained wide recognition among banks and financial institutions. With banking witnessing tectonic shifts, there was an imperative need to revisit the existing syllabi for the flagship courses. The pivotal point for revising the syllabi was to ensure that, in addition to acquiring basic knowledge, the candidates develop concept-based skills in line with the developments happening in the financial ecosystem and to ensure greater value addition to the flagship courses and to make them more practical and contemporary. This will culminate in creating a rich pool of knowledgeable and competent banking & finance professionals who are capable of contributing to the sustainable growth of their organizations. Keeping in view the above objectives, the Institute had constituted a high-level Syllabi Revision Committee comprising of members from public sector banks, private sector banks, co-operative banks and academicians. On the basis of the feedback received from various banks and changes suggested by the Committee, the syllabi of JAIIB & CAIIB have since been finalized. The revised CAIIB syllabi will now have four compulsory subjects and one elective subject to be chosen from the five elective subjects. The subjects under the revised CAIIB Syllabi are: Compulsory 1. Advanced Bank Management 2. Bank Financial Management 3. Advanced Business & Financial Management 4. Banking Regulations and Business Laws vi I FOREWORD Elective 1. Risk Management 2. Information Technology & Digital Banking 3. Central Banking 4. Human Resources Management 5. Rural Banking A new module on Compliance has been introduced in Advanced Bank Management with Compliance, Corporate Governance and Audit becoming the focal point for a resilient banking system. New units covering Risks in Foreign Trade, GIFT-city etc have been added in the Bank Financial Management. The new subject on Advanced Business & Financial Management will cover the management principles, the advanced concepts of Financial Management and emerging business solutions including Green Finance and Sustainable Financing. The subject Banking Regulations & Business Laws (BRBL) is designed to familiarise the professionals with various laws concerning banking and finance with increased focus on case laws, court judgements covering different areas of banking and finance. The elective subjects on Risk Management, Information Technology & Digital Banking and Rural Banking have also been thoroughly revised and will include new units to make the courses more contemporary. Insofar as the electives on Central Banking and Human Resources Management are concerned, new modules on NBFCs and Emerging Scenarios in HRM have been introduced respectively. As is the practice followed by the Institute, a dedicated courseware for every paper/subject is published. The present courseware on Advanced Bank Management has now been authored in line with the revised syllabus for the subject. The book follows the same modular approach adopted by the Institute in the earlier editions/publications. While the Institute is committed to revise and update the courseware from time to time, the book should, however, not be considered as the only source of information I reading material while preparing for the examinations due to rapid changes being witnessed in all the areas concerning banking & finance. The students have to keep themselves abreast with the current developments by referring to economic newspapers/journals, articles, books and Government / Regulators’ publications I websites etc. Questions will be based on the recent developments related to the syllabus. Considering that the courseware cannot be published frequently, the Institute will continue the practice of keeping candidates informed about the latest developments by placing important updates/Master Circulars/ Master Directions on its website and through publications like IIBF Vision, Bank Quest, etc. The courseware has been updated with the help of Subject Matter Experts (SMEs) drawn from respective fields and vetted by practitioners to ensure accuracy and correctness. The Institute acknowledges with gratitude the valuable contributions rendered by the SMEs in updating/vetting the courseware. We welcome suggestions for improvement of the courseware. Mumbai Biswa Ketan Das 2023 Chief Executive Officer  RECOMMENDED READING The Institute has prepared comprehensive courseware in the form of study kits to facilitate preparation for the examination without intervention of the teacher. An attempt has been made to cover fully the syllabus prescribed for each module/subject and the presentation of topics may not always be in the same sequence as given in the syllabus. Candidates are also expected to take note of all the latest developments relating to the subject covered in the syllabus by referring to Financial Papers, Economic Journals, Latest Books and Publications in the subjects concerned.  ADVANCED BANK MANAGEMENT SYLLABUS MODULE A: STATISTICS Definition of Statistics, Importance & Limitations & Data Collection, Classification & Tabulation Importance of Statistics; Functions of Statistics; Limitation or Demerits of Statistics; Definitions; Collection of Data; Classification and Tabulation; Frequency Distribution Sampling Techniques Random Sampling; Sampling Distributions; Sampling from Normal Populations; Sampling from Non­ Normal Populations; Central Limit Theorem; Finite Population Multiplier Measures of Central Tendency & Dispersion, Skewness, Kurtosis Arithmetic Mean; Combined Arithmetic Mean; Geometric Mean; Harmonic Mean; Median and Quartiles; Mode; Introduction to Measures of Dispersion; Range and Coefficient of Range; Quartile Deviation and Coefficient of Quartile Deviation; Standard Deviation and Coefficient of Variation; Skewness and Kurtosis Correlation and Regression Scatter Diagrams; Correlation; Regression; Standard Error of Estimate Time Series Variations in Time Series; Trend Analysis; Cyclical Variation; Seasonal Variation; Irregular Variation; Forecasting Techniques Theory of Probability Mathematical Definition of Probability; Conditional Probability; Random Variable; Probability Distribution of Random Variable; Expectation and Standard Deviation of Random Variable; Binomial Distribution; Poisson Distribution; Normal Distribution; Credit Risk; Value at Risk (VaR); Option Valuation Estimation Estimates; Estimator and Estimates; Point Estimates; Interval Estimates; Interval Estimates and Confidence Intervals; Interval Estimates of the Mean from Large Samples; Interval Estimates of the Proportion from Large Samples Linear Programming Graphic Approach; Simplex Method Simulation Simulation Exercise; Simulation Methodology MODULE B: HUMAN RESOURCE MANAGEMENT Fundamentals of Human Resource Management The Perspective; Relationship between HRM & HRD and their Structure and Functions; Role of HR Professionals; Strategic HRM; Development of HR Functions in India x SYLLABUS Development of Human Resources HRD and its Subsystems; Learning and Development - Role and Impact of Learning; Attitude Development; Career Path Planning; Self-Development; Talent Management; Succession Planning Human Implications of Organisations Human Behaviour and Individual Differences; Employees Behaviour at Work; Diversity at Workplace and Gender Issues; Theories of Motivation and their Practical Implications; ‘Role’: Its Concept & Analysis Employees’ Feedback and Reward System Employees’ Feedback; Reward and Compensation System Performance Management Appraisal Systems; Performance Review and Feedback; Counselling; Competency Mapping and Assessment of Competencies; Assessment Centres; Behavioural Event Interview (BEI) Conflict Management and Negotiation Conflict: Concept & Definition; Characteristics of Conflict; Types of Conflicts; Reasons for Conflict; Different Phases of Conflict; Conflict Resolution; Conflict Management; Negotiation Skills for Resolution of Conflicts HRM and Information Technology Role of Information Technology in HRM; HR Information and Database Management; Human Resource Information System (HRIS); Human Resource Management System (HRMS); e-HRM; HR Research; Knowledge Management; Technology in Training; HR Analytics MODULE C: CREDIT MANAGEMENT Overview of Credit Management Importance of Credit; Historical Background of Credit in India; Principles of Credit; Types of Borrowers; Types of Credit; Components of Credit Management; Role of RBI Guidelines in Bank’s Credit Management Analysis of Financial Statements Which are the Financial Statements; Users of Financial Statements; Basic Concepts Used in Preparation of Financial Statements; Accounting Standards (AS); Legal Position Regarding Financial Statements; Balance Sheet; Profit and Loss Account; Cash Flow Statement; Funds Flow Statement; Projected Financial Statements; Purpose of Analysis of Financial Statements by Bankers; Rearranging the Financial Statements for Analysis; Techniques used in Analysis of Financial Statements; Creative Accounting; Related Party Transactions Working Capital Finance Concept of Working Capital; Working Capital Cycle; Importance of Liquidity Ratios; Methods of Assessment of Bank Finance; Working Capital Finance to Information Technology and Software Industry; Bills/Receivables Finance by the Banks; Guidelines of RBI for Discounting/Rediscounting of Bills by Banks; Trade Receivables Discounting System (TReDS); Non-Fund Based Working Capital Limits; Other Issues Related to Working Capital Finance  SYLLABUS I xi Term Loans Important Points about Term Loans; Deferred Payment Guarantees (DPGs); Difference between Term Loan Appraisal and Project Appraisal; Project Appraisal; Appraisal and Financing of Infrastructure Projects Credit Delivery and Straight Through Processing Documentation; Third-Party Guarantees; Charge over Securities; Possession of Security; Disbursal of Loans; Lending under Consortium/Multiple Banking Arrangements; Syndication of Loans; Straight- Through Loan Processing or Credit Underwriting Engines Credit Control and Monitoring Importance and Purpose; Available Tools for Credit Monitoring/Loan Review Mechanism (LRM) Risk Management and Credit Rating Meaning of Credit Risk; Factors Affecting Credit Risk; Steps taken to Mitigate Credit Risks; Credit Ratings; Internal and External Ratings; Methodology of Credit Rating; Use of Credit Derivatives for Risk Management; RBI guidelines on Credit Risk Management; Credit Information System Restructuring/Rehabilitation and Recovery Credit Default/Stressed Assets/NPAs; Wilful Defaulters; Non-cooperative borrowers; Options Available to Banks for Stressed Assets; RBI Guidelines on Restructuring of Advances by Banks; Available Frameworks for Restructuring of Assets; Sale of Financial Assets Resolution of Stressed Assets under Insolvency and Bankruptcy Code 2016 Definition of Insolvency and Bankruptcy; To Whom the Code is Applicable; Legal Elements of the Code; Paradigm Shift; Corporate Insolvency Resolution Process; Liquidation process; Pre-packed Insolvency Resolution Process for stressed MSMEs MODULE D: COMPLIANCE IN BANKS & CORPORATE GOVERNANCE Compliance Function in Banks Compliance Policy; Compliance Principles, Process and Procedures; Compliance Programme; Scope of Compliance Function; Role & Responsibilities of Chief Compliance Officer (CCO) Compliance Audit Role of Risk Based Internal Audit and Inspection; Reporting Framework and Monitoring Compliance; Disclosure Requirements; Accounting Standards; Disclosures under Listing Regulations of SEBI Compliance Governance Structure Organisational Structure; Responsibility of the Board and Senior Management; Compliance Structure at the Corporate Office; Functional Departments; Compliance Structure at Field Levels; Internal Controls and its Importance Framework for Identification of Compliance Issues and Compliance Risks Compliance Issues; Compliance Risk; Inherent Risk and Control Risk; Independent Testing and Effective Audit Programme; Reporting Framework and Monitoring Compliance; Role of Inspection and Audit; Loan Review Mechanism/Credit Audit; What is Good Compliance xii I SYLLABUS Compliance Culture and GRC Framework How to Create Compliance Culture Across the Organisation; Governance, Risk and Compliance - GRC Framework; Benefits of an Integrated GRC Approach; Whistle-blower Policy; The Components of a Whistle-blower Policy; Reasons for Compliance Failures Compliance Function and Role of Chief Compliance Officer in NBFCs Framework for Scale Based Regulation for Non-Banking Financial Companies; Transition Path; Framework for Compliance Function and Role of Chief Compliance Officer in Non-Banking Financial Companies in Upper Layer and Middle Layer (NBFC-UL & NBFC-ML); Broad Contours of Compliance Framework in NBFCs Fraud and Vigilance in Banks Definition of Fraud; Definition of Forgery; Areas in which Frauds are Committed in Banks; Banking and Cyber Frauds; Fraud Reporting and Monitoring System; Vigilance Function in Banks; RBI Guidelines for Private Sector and Foreign Banks on Internal Vigilance CONTENTS Foreword v MODULE A: STATISTICS 1. Definition of Statistics, Importance & Limitations & Data Collection, Classification & Tabulation 3 2. Sampling Techniques 17 3. Measures of Central Tendency & Dispersion, Skewness, Kurtosis 46 4. Correlation and Regression 71 5. Time Series 86 6. Theory of Probability 105 7. Estimation 134 8. Linear Programming 151 9. Simulation 161 MODULE B: HUMAN RESOURCE MANAGEMENT 10. Fundamentals of Human Resource Management 173 11. Development of Human Resources 197 12. Human Implications of Organisations 249 13. Employees’ Feedback and Reward System 286 14. Performance Management 307 15. Conflict Management and Negotiation 339 16. HRM and Information Technology 362 MODULE C: CREDIT MANAGEMENT 17. Overview of Credit Management 393 18. Analysis of Financial Statements 422 19. Working Capital Finance 447 20. Term Loans 468 21. Credit Delivery and Straight Through Processing 482 22. Credit Control and Monitoring 494 23. Risk Management and Credit Rating 501 24. Restructuring/Rehabilitation and Recovery 511 25. Resolution of Stressed Assets under Insolvency and Bankruptcy Code 2016 530 xiv I CONTENTS MODULE D: COMPLIANCE IN BANKS & CORPORATE GOVERNANCE 26. Compliance Function in Banks 555 27. Compliance Audit 569 28. Compliance Governance Structure 592 29. Framework for Identification of Compliance Issues and Compliance Risks 613 30. Compliance Culture and GRC Framework 634 31. Compliance Function and Role of Chief Compliance Officer in NBFCs 656 32. Fraud and Vigilance in Banks 669  STATISTICS Unit 1. Definition of Statistics, Importance & Limitations & Data Collection, Classification & Tabulation Unit 2. Sampling Techniques Unit 3. Measures of Central Tendency & Dispersion, Skewness, Kurtosis Unit 4. Correlation and Regression Unit5. Time Series Unit 6. Theory of Probability Unit 7. Estimation Unit 8. Linear Programming Unit 9. Simulation 4 I ADVANCED BANK MANAGEMENT 1.0 OBJECTIVES After studying this unit, you will be able to: Develop an understanding to use the proper methods to collect the data, employ the correct analyses, and effectively present the results. Familiarise with Data Management techniques by using Classification, Tabulation and Presentation. As Statistics is a crucial process behind how we make discoveries in science, make decisions based on data, and make predictions, collecting correct data and present it properly and making it ready for analysis and interpretation is very important. From this chapter, you will understand correctly getting the data ready to be analysed and interpreted. 1.1 INTRODUCTION The word6Statistics’ has been derived from the Latin word ‘statisticum’, Italian word ‘statistia’ and German word ‘ statistik’, each of which means a group of numbers or figures that represent some information of human interest. It was first used by professor Achenwell in 1749 to refer to the subject-matter as a whole. Achenwell defined statistics as the political science of many countries. In the early years statistics is to be used only by the kings to collect facts about the state, revenue of the state or the people in the state of administrative or political purpose. Gradually the use of statistics which means data or information has increased and widened. It is now used in almost in all the fields of human knowledge and skills like Business, Commerce, Economics, Social Sciences, Politics, Planning, Medicine and other sciences, physical as well as natural. In many practical situations in life, we come across different types of data which are needed to be understood, analysed, compared and interpreted correctly. For example, in a college we need to analyse the data of marks obtained, in a hospital we need to analyse the data of number of patients having different diseases, rate of mortality, Different types of data need to be analysed in Economics, Government and Private organisations, Sports and in many other fields. Data mean information, which can be of two types - Qualitative and Quantitative. Statistics means quantitative or numerical data, which can be used for further calculations. Statistical analysis of data can be comprised of four distinct phases: 1. Collection of data: In this first stage of investigation, numerical data is collected from different published or unpublished sources, primary or secondary. 2. Classification and Tabulation of data: The raw data collected is to be represented properly for further calculations. The raw data is divided into different groups or classes and represented in a form of a table. 3. Analysis of data: Classified and Tabulated data is analysed using different formulas and methods according to purpose of the study or investigation. 4. Interpretation of data: At the final stage, relevant conclusions are drawn after the data is thoroughly analysed  DEFINITION OF STATISTICS. IMPORTANCE & LIMITATIONS... I 5 1.2 IMPORTANCE OF STATISTICS Statistics is the subject that teaches how to deal with data, so statistical knowledge helps to use proper methods for collection of data, properly represent the data, use appropriate formula and methods to analyse correctly and effectively get the results and interpret the data. Applications of Statistics is important in every sphere of field - Business and economics, Medical, Sports, Weather forecast, Stock Market, Quality Testing, Government decisions and policies, Banks, Different educational and research organisations, etc. 1.2.1 Business and Economics In Business, the decision maker takes suitable policies and strategies based on information on production, sale, profit, purchase, finance, etc. By using the techniques of time series analysis, the businessman can predict the effect of a large number of variables with a fair degree of accuracy. By using ‘Bayesian Decision Theory’, the businessmen can select the optimal decisions to directly evaluate the payoff for each alternative course of action. In Economics, Statistics is used to analyse demand, cost, price, quantity, different laws of demand like elasticity of demand and consumer’s maximum satisfaction which is determined on the basis of data pertaining to income and expenditure. 1.2.2 Medical Statistics have extensive application in clinical research and medical field. Clinical research involves investigating proposed medical treatments, assessing the relative benefits of competing therapies, and establishing optimal treatment combinations. 1.2.3 Weather Forecast Statistical methods, like Regression techniques and Time series analysis, are used in weather forecasting. 1.2.4 Stock Market Statistical methods, like Correlation and Regression techniques, Time series analysis are used in forecasting stock prices. Return and Risk Analysis is used in calculation of Market and Personal Portfolios and Mutual Funds. 1.2.5 Bank In banking industry, credit policies are decided based on statistical analysis of profitability, demand deposits, time deposits, credit ratio, number of customers and many other ratios. The credit policies are based on the application of probability theory. 6 I ADVANCED BANK MANAGEMENT 1.2,6 Sports Players use statistics to identify or rectify their mistakes. A proper understanding of the statistics determines the success of a team or a single athlete. 1.3 FUNCTION OF STATISTICS 1. Statistics present the facts in definite form. 2. Statistics simplify complex data. 3. It provides a techniques of comparison. 4. Statistics study the relationship between two or more variables. 5. It helps in formulating policies. 6. It helps in forecasting outcomes. 1.4 LIMITATIONS OR DEMERITS OF STATISTICS 1. Statistics do not deal with Individuals Statistical methods can’t be applied for individual values of the observations as for individual observation, there is no point of comparing anything or analysing anything. Statistics is the study of mass data or a group of observations and deals with aggregates of facts. 2. Statistics does not study Qualitative Data Statistical methods can’t be applied for qualitative or non-numerical data. Statistics is the study of only of those facts which are capable of being stated in number or quantity. 3. Statistics give Result only on an Average Statistical methods are not exact. Generally, when we have large number of observations, it becomes difficult to handle it. A part of the data (sample) is collected for study and draw conclusion from, as a representative for the whole. As a result, the result obtained are not exactly same, had we analysed the whole data. The results are true only on an average in the long run. 4. The results can be biased: The data collection may sometime be biased which will make the whole investigation useless. Generally, this situation arises when data is handled by inexperienced or dishonest person. 1.5 DEFINITIONS Population It is the entire collection of observations (person, animal, plant or things which is actually studied by a researcher) from which we may collect data. It is the entire group we are interested in and from which we need to draw conclusions. Example 1. If we are studying the weight of adult men in India, the population is the set of weights of all men in India.  DEFINITION OF STATISTICS, IMPORTANCE & LIMITATIONS... I 7 2. If we are studying the grade point average of students of Mumbai University, the population is the set of GPA’s of all students of Mumbai University. Sample: Sometimes the population from which we need to draw conclusion is too large to study. At times collecting data from too large a population becomes time-consuming and expensive. To save time and money, generally a part of population is selected for study. A sample is a part (a group of units) of population which is representative of the actual population. By studying the sample, it is expected that valid conclusions are drawn about the whole group. Example: The population for a study of infant health might be all children bom in India in one particular year. The sample might be all babies bom on one particular day in that year. Data can be classified into two types, based on their characteristics. They are: 1. Variates 2. Attributes A characteristic that varies from one individual to another and can be expressed in numerical terms is called variate. Example: Prices of a given commodity, wages of workers, heights and weights of students in a class, marks of students, etc. A characteristic that varies from one individual to another but can’t be expressed in numerical terms is called an attribute. Example: Colour of the ball (black, blue, green, etc.), religion of human, etc. Quantitative or Numerical variables can be further classified as discrete and continuous. A variate which takes discrete or distinct value or in other words can take only a countable and usually finite number of values is called Discrete Variable. Example: Number of members in a family, Number of accidents, Age in years. A variate that can take any value within a range (integral/fractional) is called Continuous Variable. Example: Percentage of marks, Height, Weight. A parameter is a numerical value or function of the observations of the entire population being studied. A parameter is usually an unknown value that is fixed. Example: Population mean, population median, population standard deviation, etc. Since parameter is unknown, it has to be calculated or estimated from a sample. Statistic is used to estimate parameter. A statistic is a quantity or function of the observations of the sample of data. It is used to give information about unknown values in the corresponding population. For example, the sample mean is used to estimate the parameter population mean. Statistic is also called Estimator. 1.6 COLLECTION OF DATA Researchers or investigators need to collect data from respondents. There are two types of data. 8 I ADVANCED BANK MANAGEMENT 1.6.1 Primary Data Primary data is the data which is collected directly or first time by the investigator or researcher from the respondents. Primary data is collected by using the following methods: Direct Interview Method: A face to face contact is made with the informants or respondents (persons from whom the information is to be obtained) under this method of collecting data. The interviewer asks them questions pertaining to the survey and collects the desired information. Questionnaires: Questionnaires are survey instruments containing short closed-ended questions (multiple choice) or broad open-ended questions. Questionnaires are used to collect data from a large group of subjects on a specific topic. Currently, many questionnaires are developed and administered online. Census and sample survey In a census, data about all individual units (e.g., people or households) are collected in the population. In a survey, data are only collected for a sub-part of the population; this part is called a sample. These data are then used to estimate the characteristics of the whole population. In this case, it has to be ensured that the sample is representative of the population in question. For example, the proportion of people below the age of 18 or the proportion of women and men in the selected sample of households has to reflect the reality in the total population. 1.6.2 Secondary Data Secondary data are the Second hand information. The data which have already been collected and processed by some agency or persons and is collected for the second time are termed as secondary data. According to M. M. Blair, “Secondary data are those already in existence and which have been collected for some other purpose.” Secondary data may be collected from existing records, different published or unpublished sources, like WHO, UNESCO, LIC, etc., various research and educational organisations, banks and financial places, magazines, internet, etc. Distinction between primary and secondary data 1. The data collected for the first time is called Primary data and data collected through some published or unpublished sources is called Secondary data. 2. The primary data in the hands of one person can become secondary for all others. For example, the population census report is primary for the Registrar General of India and the information from the report is secondary for others. 3. Primary data are original as they are collected first time from the respondents directly or by preparing questionnaires. So they are more accurate than the secondary data. But the collection of primary data requires more money, time and energy than the secondary data. A proper choice between the two forms of information should be made in an enquiry.  DEFINITION OF STATISTICS, IMPORTANCE & LIMITATIONS... I 9 1.7 CLASSIFICATION AND TABULATION So, we learned about the different methods of collecting primary and secondary data. The raw data, collected in real situations are arranged randomly, haphazardly and sometimes the data size is very large. Thus, the raw data do not give any clear picture and interpreting and drawing any conclusion becomes very difficult. To make the data understandable, comparable and to locate similarities, the next step is classification of data. The method of arranging data into homogeneous group or classes according to some common characteristics present in the data is called Classification. Example: The process of sorting letters in a post office, the letters are classified according to the cities and further arranged according to the streets. Classification condenses the data by removing unimportant details. It enables us to accommodate large number of observations into few classes and study the relationship between several characteristics. Classified data is presented in a more organised way so it is easier to interpret and compare them, which is known as Tabulation. There are four important bases of classifications: 1. Qualitative Base: Here the data is classified according to some quality or attribute such as sex, religion, literacy, intelligence, etc. 2. Quantitative Base: Here the data is classified according to some quantitative characteristic like height, weight, age, income, marks, etc. 3. Geographical Base: Here the data is classified by geographical regions or location, like states, cities, countries, etc. like population in different states of India. 4. Chronological or Temporal Base: Here the data is classified or arranged by their time of occurrence, such as years, months, weeks, days, etc. This classification is also called Time Series data. Example: Sales of a company for different years. Types of Classification 1. If we classify observed data for a single characteristic, it is known as One-way Classification. Ex: Population can be classified by Religion - Hindu, Muslim, Christians, etc. 2. If we consider two characteristics at a time to classify the observed data, it is known as a Two-way classification. Ex: Population can be classified according to Religion and sex. 3. If we consider more than two characteristics at a time in order to classify the observed data, it is known as Multi-way Classification. Ex: Population can be classified by Religion, sex and literacy. 1.8 FREQUENCY DISTRIBUTION Frequency: If the value of a variable (discrete or continuous) e.g., height, weight, income, etc. occurs twice or more in a given series of observations, then the number of occurrences of the value is termed as the “frequency” of that value. The way of representing a data in a form of a table consisting of the values of the variable with the corresponding frequencies is called “frequency distribution”. So, in other words, Frequency distribution 12 I ADVANCED BANK MANAGEMENT 2. Continuous Frequency Distribution: Variable takes values which are expressed in class intervals within certain limits. Problem 2: Marks obtained by 20 students in an exam for 50 marks are given below-convert the data into continuous frequency distribution form. 18, 23, 28, 29, 44, 28, 48, 33, 32, 43, 24, 29, 32, 39, 49, 42, 27, 33, 28, 29. Solution Problem 3: Following data reveals information about the number of children per family for 25 families. Prepare frequency distribution of number of children (say variable x, taking distinct values 0, 1,2, 3,4). 3 2 112 4 0 12 3 1 2 0 4 2 2 12 3 2 1 3 4 0 1 Solution: Frequency distribution of number of children in 25 families Number of children Number of families Problem 4: For the following frequency distribution, prepare cumulative frequency distribution of less than and greater than type  DEFINITION OF STATISTICS, IMPORTANCE & LIMITATIONS... I 13 1 2 3 4 5 Solution X Frequency Less than cf Greater than cf 1 5 5 43 + 5 = 48 2 7 5 + 7 = 12 36 + 7 = 43 3 15 12 + 15 = 27 21 +15 = 36 4 11 27 + 11 = 38 10 + 11 =21 5 6 38 + 6 = 44 4 + 6 = 10 6 4 44 + 4 = 48 4 Total 48 Problem 5: Following is the marks of 50 students. Prepare cumulative frequency distribution of both the types. Also find Relative Frequencies. Marks No. of Students Solution Problem 6: For the following frequency distribution, obtain cumulative frequencies, relative frequencies and relative cumulative frequencies. Class Interval 30-50 50-70 70—90 90-110 110-130 130-150 | Frequency 8 15 25 16 I 7 I 4 14 I ADVANCED BANK MANAGEMENT Solution Class interval Freq Cumu. Frequency Relative Relative Cumu. frequency Less than More than Frequency Less than More than 30-50 8 8 75 0.11 0.11 1.00 50-70 15 23 67 0.20 0.31 0.89 70-90 25 48 52 0.33 0.64 0.69 90-110 16 64 27 0.22 0.86 0.36 110-130 7 71 11 0.09 0.95 0.14 130-150 4 75 4 0.05 1 ,.00 0.05 Total 75 1.00 Problem 7: Represent the information given below with a suitable table. 1400 candidate were medically examined in afitness test, out ofwhich 21% were girls. From the doctor’s report, it was found that 396 males and 104 females were unfit. 40 % of the remaining males and 60% of the remaining females were in good health. The rest were declared as temporarily unfit. Solution: Distribution of candidates according to Male/Female and Fitness Fitness X Sex Male Female Total Good health 284 114 398 Temp unfit 426 76 502 Unfit 396 104 500 Total 1106 294 1400 Problem 8: Tabulate the information given below. In 2020, out of total of 3000 workers in a factory, 2300 were skilled workers. The number of woman employees were 300 out of which 250 were unskilled. In 2021, the number of skilled workers was 2750 of which 2500 were men. The number of unskilled workers was 760 of which 300 were women. Solution: Classification of workers according to sex and skill for the year 2020, 2021 Sex/ Skill Men Women Total Year Skilled Unskilled Total Skilled Unskilled Total Skilled Unskilled Total 2020 2250 450 2750 50 250 300 2300 700 3000 2021 2500 460 2960 250 300 550 2750 760 3510 Problem 9: Out of total number of 2000 candidates interviewed for employment in a company, 628 were from Pune and the rest from Nashik. Amongst the graduate from Pune, 350 were experienced and 80 were unexperienced. While the corresponding figures for undergraduates from Nashik were 615 and  DEFINITION OF STATISTICS, IMPORTANCE & LIMITATIONS... I 15 52 respectively. The total number of in experienced candidates from Pune and Nashik were 175 and 192 respectively. Present the above information in a suitable tabular form. Solution: Distribution of candidate from Pune and Nashik according to Education and Experience Education Graduate Undergraduate Total Experience Exp Inexp Total Exp Inexp Total Exp Inexp Total City Pune 350 80 430 103 95 198 453 175 628 Nashik 565 140 705 615 52 667 1180 192 1372 Total 915 220 1135 718 147 865 1633 367 2000 Questions 1. The data of some worker’s salary are given as 2300, 2400, 2500, 2100, 2000, 2000, 2300, 2800, 3000, 2300, 2700, 2400, 2500. If desired number of class intervals is 10, class width is (a) 100 (b) 200 (c) 300 (d) 400 2. The largest and smallest values of a data are 60 and 40 respectively. If desired number of class intervals is 5, class width is (a) 25 (b) 20 (c) 4 (d) 5 3. The class intervals where upper and lower limits are also in the class interval are called (a) Exclusive type (b) Inclusive type (c) Discrete type (d) Continuous type 4. The type of cumulative frequencies where the frequencies are added starting from the highest class to the lowest class are called______________ (a) Relative Frequency (b) Percentage Frequency (c) Less than Cumulative Frequency (d) Greater than Cumulative Frequency 16 I ADVANCED BANK MANAGEMENT 5. The data classification, which is based on variables, like, demand, supply, height and weight is considered as (a) Qualitative data (b) Quantitative data (c) Time series data (d) Discrete data 6. The data which is classified or arranged by their time of occurrence, such as years, months, weeks, days, etc., is called (a) Time series data (b) Geographical data (c) Historical data (d) Both (a) and (c) Answers (a); l. 2. (c); 3. (b); 4. (d); 5. (b); 6. (d)  Sampling Techniques STRUCTURE 2.0. Objectives 2.1. Introduction 2.2. Random Sampling 2.3. Sampling Distributions 2.4. Sampling from Normal Populations 2.5. Sampling from Non-Normal Populations 2.6. Central Limit Theorem 2.7. Finite Population Multiplier Keywords/Gloss ary Questions Annexure - Probability Table 18 I ADVANCED BANK MANAGEMENT 2.0 OBJECTIVES The objectives of this unit are as follows: Learn to take a sample from an entire population and use it to describe the population Make sure the samples represent the population. Introduce the concepts of sampling distributions Understand the tradeoff between costs of larger samples and accuracy Introduce experimental design - sampling procedures. Estimation - data analysis and interpretation of sample data Testing of hypotheses - one-sample data Testing of hypotheses - two-sample data 2.1 INTRODUCTION As you know, statistics is a tool used in business and finance. Statistics is an appreciated and maligned tool, depending on how it is used. We need statistical methods to reduce risk and uncertainty and improve our decision-making skills. Decision making in a situation involves collecting data and then using it for the future strategy. Usually. We cannot use the entire data because of the sheer size or numbers. Therefore, we take a sample and test it. For example, if a milk plant processes 1 lakh litres of milk every day, one cannot break open each packet and test the milk for quality. Here we take samples from each batch. If we want to do a market survey, we cannot interview each and every household. We take a representative sample. Thus, sampling becomes an integral tool of the quantitative methods we use. We take a sample, collect data from the sample, and attempt to generalise the whole data results. Tea tasters at tea auctions are very highly paid employees of tea companies. They sample a small portion of the tea produced from the plantation before the auction. Food products are often tasted before being sold. Before buying Diwali sweets, you may take a small bite before buying it. Obviously, everything cannot be opened and tasted or tested as there would be nothing left to sell. We have to select a sample and test that only. If you want to write a report on why many people are migrating from India to Canada or Australia, contacting each Indian who migrated would be time consuming and expensive. So you would choose a sample and make a report accordingly. Thus, time and size are decisive factors which make it necessary to take business decisions on a sample. If your sample is properly chosen, your report would truly reflect the reasons of the entire population for migration. Of course, the best results would be available in any situation if we collected data from the entire population. Such complete enumeration or census is used in the population census carried out in our country every ten years. Statisticians use the word population to refer not only to people but to all items that are to be studied. A sample is a part or subset of the population selected to represent the entire group. The word sample is used to describe a portion chosen from the population, in other words. We can describe samples and populations using mean, median, mode, and standard deviation measures. When these terms describe a sample, they are called startsricand are not from the population but estimated from the sample. When these terms describe a population, they are called parameters. A statistic is a characteristic of a sample; a parameter is a population characteristic.  SAMPLING TECHNIQUES I 19 Conventionally, statisticians use lower case Roman letters to denote sample statistics and Greek or Capital letters to denote population parameters. Table 2.1 lists these symbols. TABLE 2.1 Important Statistic Notation and Symbol Population Sample Definition Collection of all items Part of the population Characteristics Parameters Statistics Symbols Size - N Size - n Mean - p Mean - X Standard Standard Deviation - a Deviation - q Types of sampling The process of selecting respondents is known as "sampling.9 The units under study are called sampling units, and the number of units in a sample is called a sample size. There are two methods of selecting samples from populations: non-random or judgement sampling and random or probability sampling. In probability sampling, all the items in the population have a chance of being chosen in the sample. In judgement sampling, personal knowledge or opinions are used to identify the items from the population that are to be included in the sample. A sample selected by judgement sampling is based on someone’s experience with the population. An oil drilling company would ask an experienced geologist to test different terrains or land beneath the sea before deciding where to explore for oil. Sometimes a judgement sample is used as a pilot or trial sample to decide how to take a random sample later. If we want to launch a new city newspaper, a pilot test of the paper can be launched at a judgement sample to see the response. But, the rigorous statistical analysis, which can be done with random probability samples, cannot be done with judgement samples. On the other hand, they are more convenient and can be used successfully even if we cannot test their validity. But, if a study uses judgement sampling and loses a significant degree of representativeness, it will have purchased convenience at a high price. Biased samples Suppose the Parliament is debating on the women’s bill. You are asked to conduct an opinion survey. Because women are the most affected by the women’s bill, you interviewed many women in different cities, towns and rural areas of India. Then you report that an overwhelming 95 per cent are in favour of reservation for women in Parliament. Sometime later, the government has to take up the issue of Foreign Direct Investment (FDI) in print media. Since newspaper publishers are the most affected, you contact all of them, both national and regional, in India and report that the majority is not in favour of FDI in print media. 20 I ADVANCED BANK MANAGEMENT In both these cases, you picked a biased sample by choosing people who would have strong feelings on this issue. You have to have sound samples. A report based on the data collected from such a biased sample would not truly reflect public opinion. If we follow random sampling, it is possible to statistically determine the reliability of the estimates obtained from the sample to avoid such errors. 2.2 RANDOM SAMPLING There are four main types of random sampling. 1. Simple Random Sampling 2. Systematic Sampling 3. Stratified Sampling 4. Cluster Sampling Simple Random Sampling Simple Random Sampling selects samples by methods that allow each possible sample to have an equal probability of being picked and each item in the entire population to be included in the sample. Suppose we have four teenagers participating in a talk show. We want a sample of two teenagers at a time to participate with the chat show host. The following table illustrates the possible combinations of samples of teenagers, the probability of each sample being picked and the probability that each teenager will be in a sample. Teenagers A, B, C, D Possible samples of two teenagers: AB, AC, AD, BC, BD, CD Probability of drawing these samples of two people is the same as below: P(AB)= 1/6 P(AC) = 1/6 P(AD) = 1/6 P(BC) = 1/6 P(BD) = 1/6 P(CD) = 1/6 You would observe that any one student appears in 3 of the 6 possible samples. Therefore, probability of a particular student in the sample is: P(A) = 1/2 P(B)= 1/2 P(C) = 1/2 P(D) = 1/2  SAMPLING TECHNIQUES I 21 The example illustrates a finite population of four teenagers. If we write A, B, C, and D on four identical slips of paper, fold the papers, and randomly pick any two, we get a sample. While picking up two paper slips, we may pick up one, keep it away, and pick another from the remaining three. This type is called sampling without replacement There is another way of doing it. Suppose after picking the first slip, we note the name on it and put the slip back in the lot, i.e replace the paper slip. Then we draw the second slip. There is a chance that we may draw the same student again. This is called sampling with replacement. Theoretically, it is possible to have an infinite population. For example, the population of all prime numbers is infinite. Although many populations seem exceedingly large, no truly infinite population of physical objects actually exists. After all, given unlimited resources and time, one can enumerate any finite population. As a practical matter, we will use the term infinite population when we are talking about a population that could not be enumerated in a reasonable period of time. Thus, we use a theoretical concept of infinite population as an approximation of a large finite population. How to do Random Sampling? Suppose there are 100 employees in a company, and we wish to interview a randomly chosen sample of 10. We write the name of each employee on a slip of paper and deposit the slips in a box. After mixing them thoroughly, we draw 10 slips at random. The employees whose names are on these 10 slips, are our random sample. This method of drawing a sample works well with small groups of people but presents problems with large populations. Also, add to this the problem of the slips of paper not being mixed well. We can also select a random sample by using random numbers. These numbers can be generated either by a computer programmed to scramble numbers, or by a table of random digits. The table below shows some random digits. These numbers have been generated by a completely random process. The probability that any one digit from 0 through 9 will appear is the same for each digit, and the probability of one sequence of digits occurring is the same as for any other sequence of the same length. 15,819 20,685 82,621 83,748 69,662 09,281 72,950 85,961 48,980 06,840 41,120 48,326 18,824 54,466 94,470 74,574 63,491 00,923 04,142 51,336 00,995 02,727 96,703 12,671 31,976 59,987 93,247 72,301 34,290 69,051 69,787 72,950 56,709 54,709 70,945 71,642 54,358 40,316 88,897 99,907 35,939 34,406 11,987 23,691 70,582 67,494 30,909 20,395 50,973 74,338 22 I ADVANCED BANK MANAGEMENT 96,974 22,756 34,574 81,235 60,089 20,079 07,000 50,890 37,689 39,622 45,847 76,661 55,448 14,318 74,840 38,401 64,888 08,409 90,352 56,923 63,151 91,208 72,286 78,612 98,120 01,904 02,727 18,928 53,446 01,778 55,700 48,000 38,595 60,089 26,117 26,926 36,478 33,822 45,786 36,984 31,406 67,652 15,134 53,872 87,520 70,645 65,145 96,286 59,837 79,304 83,832 97,712 31,209 83,209 09,007 11,865 68,401 98,654 43,211 11,559 16,264 19,856 40,972 75,623 09,406 52,203 64,287 05,963 23,673 32,329 49,420 22,936 42,003 32,367 15,130 02,875 33,739 27,092 29,805 75,614 88,094 92,125 72,709 42,514 40,618 86,905 64,893 74,804 69,873 40,372 31,196 26,299 50,839 67,173 82,465 Let us see how to use this table. We assign each employee a number from 00 to 99, look up the table above and pick a systematic method of selecting two-digit numbers, like the first two digits. So, we have 15, 09, and so on till we get our 10 numbers. Systematic Sampling In systematic sampling, elements are selected from the population at a consistent level that is measured in time, order, or space. If we wanted to interview every twentieth student on a college campus, we would choose a random starting point in the first twenty names in the student directory and then pick every twentieth name after that. Systematic sampling differs from simple random sampling in that each element has an equal chance of being selected, but each sample does not have an equal chance of being selected. This would have been the case if, in our earlier example, we had assigned numbers between 00 and 99 to our employees and then began to choose our sample of 10 by picking every tenth number beginning 1, 11,21,31, and so forth. Employees numbered 2, 3, 4 and 5 would have no chance of being selected together. In systematic sampling, there is a probability of introducing an error into the sampling process. The system chosen may cause a problem. If we want to check the chances of people eating out on different days of the week, choose Friday. There is a higher likelihood of Friday as it is the beginning of the weekend, and we get a higher result.  SAMPLING TECHNIQUES I 23 Systematic sampling has some advantages. Even though systematic sampling may be inappropriate when the elements lie in a sequential pattern, this method may require less time and sometimes results in lower costs than the simple random sample method. Stratified Sampling To use stratified sampling, we divide the population into relatively homogenous groups, called strata. Then we use one of the following two approaches. Either we select at random from each stratum a specified number of elements corresponding to the proportion of that stratum in the population as a whole or, we draw an equal number of elements from each stratum and give weight to the results according to the stratum’s proportion of the total population. With either approach, stratified sampling guarantees that every element in the population has a chance of being selected. When to use Stratified Sampling Stratified sampling is appropriate when the population is already divided into groups of different sizes, and we wish to acknowledge this fact. Example - middle class, upper class, lower middle class, etc. or according to age, race, sex or any other stratification. A jeans company may want to study which age group prefers to wear jeans the maximum. Thus the age groups may be 13 to 20,20 to 30, 30 to 40, or 40 plus. The advantages of stratified samples are that when they are properly designed, they more accurately reflect the characteristics of the population from which they are chosen than do other kinds of samples. Cluster Sampling A well-designed cluster sampling procedure can produce a more precise sample at considerably less cost than simple random sampling. In cluster sampling, we divide the population into groups or clusters and then select a random sample of these clusters. We assume that these individual clusters are representative of the population as a whole. Suppose a market Research team is attempting to determine by sampling the average number of television sets per household in a large city. They could use a city map and divide the territory into blocks and then choose a certain number of blocks (clusters) for interviewing. Every household in each of these blocks would be interviewed. Comparison of Stratified and Cluster Sampling The population is divided into well-defined groups with both stratified and cluster sampling. We use stratified sampling when each group has a small variation within itself, but there is wide variation between the groups. We use cluster sampling in the opposite case - when there is considerable variation within each group, but the groups are essentially similar. Basis of Statistical Inference: Simple Random Sampling Systematic sampling, stratified sampling and cluster sampling attempt to approximate simple random sampling. All are methods that have been developed for their precision, economy or physical ease. How­ ever, as we do problems, we shall assume that the entire sample we are talking about are data based on simple random sampling. The process of making statistical inferences is based on the principles of 24 ADVANCED BANK MANAGEMENT random sampling. Once you understand the basics of random sampling, the same can be extended to other samples with some amendments which are best left to professional statisticians. It is important that you get a grasp of the concepts concerned. 2.3 SAMPLING DISTRIBUTIONS In this section, we presume you are familiar with mathematical concepts such as mean, mode, median, standard deviation, etc. Each sample you draw from a population would have its own mean or measure of central tendency and standard deviation. Thus, the statistics we compute for each sample would vary and be different for each random sample taken. Let us take an example. We take a finite population of 5 young boys; A, B, C, D, and E, and collect data about their heights in centimetres. The data is shown in Table 2.2. TABLE 2.2 Sampling Distribution Table | Boy A B c D height (cm) 160 162 164 170 156 Now, if we take samples of size 3 (that is, select 3 boys in each sample), we will get 10 different samples. We list these samples, the corresponding data and their mean in Table 2.3. TABLE 2.3 Samples, Their Data and Mean No 1 2 3 4 5 6 7 8 9 10 Sample ABC ABD ABE BCD BCE ACD ACE ADE BDE CDE r data 160 160 160 162 162 160 160 160 162 164 162 162 162 164 164 164 164 170 170 170 164 170 156 170 156 170 156 156 156 156 mean 162 164 159.33 165.33 160.66 164.66 160 162 162.66 163.33 From Table 2.3, you can see that sample mean for each sample is different. This collection of different values of the sample mean for samples of size 3, forms a distribution of sample means. This distribution has a mean. If we add all sample means in Table 2.3, and divide the sum by the number of samples, i.e., 10, we get 162.397 (say, 162.40) Normally, we will be dealing with large populations. Hence the number of samples of a particular size is also very large. Suppose we have to determine the proportion of sugar plants in a plantation affected by pest disease in samples of 100 plants taken from a very large plantation. We have taken a large number of these 100 item  SAMPLING TECHNIQUES I 25 samples. If we plot a probability distribution of the proportions of infested plants in all these samples, we would see a distribution of the sample proportion. (The term proportion here refers to the proportion that is infected.) We could also have a sampling distribution of a proportion. Sampling distribution is the distribution of all possible values of a statistic from all possible samples of a particular size drawn from the population. Describing Sampling Distributions Any probability distribution (and, therefore, any sampling distribution) can be partially described by its mean and standard deviation. Table 2.4 describes how different sampling distributions can be described. TABLE 2.4 Different Sampling Distribution S. No. Population Sample Sample Statistic Sampling Distribution 1. Water in a River 10-one litre con­ Mean number of parts of Sampling distribution of tainers of water mercury per million parts the mean of water 2. All professional Groups of 5 Median height Sampling distribution of basketball teams players the median 3. All parts pro­ 50 of each part Proportion defective Sampling distribution of duced in a manu­ the proportion facturing process Each of the above sampling distributions can be partially described by its mean and standard deviation. Concept of Standard Error The standard deviation of the distribution of the sample means is called the standard error of the mean. Similarly, standard error of the proportion is the standard deviation of the distribution of the sample proportions. The term standard error is used because it has a very specific connotation. For example, we take various samples to find the average heights of college girls across India and calculate the mean height for each sample. Obviously, there would be some variability in the observed mean. This variability in sampling statistics results from the sampling error due to chance. Thus, the difference between the sample and population means is due to the choice of samples. Thus, the standard deviation of the sampling distribution of means measures the extent to which the means vary because of a chance error in the sampling process. Thus, the standard deviation of the distribution of a sample statistic is known as the standard error of the statistic. Thus, a standard error indicates the size of the chance error and the accuracy we are likely to get if we use the sample statistic to estimate a population statistic. Thus, a mean with a smaller standard deviation is a better estimator than one with a higher standard deviation. 30 I ADVANCED BANK MANAGEMENT The areas also show the probabilities under the probability curve shown in Fig. 2.2. 2.5 SAMPLING FROM NON-NORMAL POPULATIONS In the preceding section, we stated that when the population is normally distributed, the sampling distribution of the mean is also normal. But, we come across many populations that are not normally distributed. How does the sampling distribution of the mean behave when the population from which the samples are drawn is not normal? An illustration will help us answer this question. Consider the data in Table 2.5, concerning five motorcycle owners and the life of the Tyres. Because only five people are involved, the population is too small to be approximated by a normal distribution. Let us take all of the possible samples of the owners in groups of three, compute the sample means (x) list them, and compute the mean of the sampling distribution (|i_). We have done this in Table 2.5. These calculations show that even in a case in which the population is not normally distributed, (p_), the mean of the sampling distribution is still equal to the population mean, p. TABLE 2.5 Experience of Five Motorcycle Owners with Life of Tyres Owner Chetan (C) Dinesh (D) Eswar (E) Feroz(F) George (G) Tyre Life in months 3 3 6 9 15 Total life = 36 months Mean = 36/5 = 7.2 months  SAMPLING TECHNIQUES I 31 TABLE 2.6 Calculation of Sample Mean Tyre Life with n = 3 Samples of Three Sample Data Sample Mean EFG 6 + 9 + 15 10 DFG 3 + 9 + 15 9 DEG 3 + 6+15 8 DEF 3+6+9 6 CFG 6+6+9 7 CEG 3 + 6 + 15 8 CEF 3+6+9 6 CDF 3+3+9 5 CDE 3+3+9 5 CDG 3 + 3 + 15 8 Total = 72 months Mean = 7.2 months Now, look at Fig. 2.3(a), which shows the population distribution of tyre lives for the five motorcycles owners, a distribution that is anything but normal in shape. In Fig. 2.3(b), we show the sampling distribution of the mean for a sample size of three, taking the information from Table 2.6. Notice the difference between the probability distributions in Figs. 2.3(a) and 2.3(b). In Figure 2.3(b), the distribution looks a little more like the bell shape of the normal distribution. If we repeat this exercise and enlarge the population size to 40, we could take samples of different sizes. Then plot the sampling distributions of the mean that would occur for the different sizes. This will show quite dramatically how quickly the sampling distribution of the mean approaches normality, regardless of the shape of the population distribution. 0.45 “ 0.4 - ♦ 0.35 - 0.3 - 0.25 - 0.2 - 0.15 - 0.1 - 0.05 - 0-I---------- 1------ ------ r 1------ ------ 1------ ♦------ 1------ —i 0 2 4 8 10 12 14 16 JIT aDD FIGURE 2.3 (a) Problem Distribution 32 I ADVANCED BANK MANAGEMENT FIGURE 2.3 (b) Distribution of Mean Tyre Life 2.6 CENTRAL LIMIT THEOREM The example in the above Table and the probability distribution in the above graphs tell us many things. First, the mean of the sampling distribution of the mean will equal the population mean regardless of the sample size, even if the population is not normal. As the sample size increases, the sampling distribution of the mean will approach normality, regardless of the shape of the population distribution. The Central Limit Theorem is the relationship between the shape of the population distribution and the shape of the sampling distribution of the mean. The central limit theorem is perhaps the most important in all statistical inference. It assures us that the sampling distribution of the mean approaches normal as the sample size increases. 1. Actually, a sample does not have to be very large for the sampling distribution of the mean to approach normal. 2. Statisticians use the normal distribution as an approximation to the sampling distribution whenever the sample size is at least 30, but the sampling distribution of the mean can be nearly normal with samples of even half the size. 3. The significance of the central limit theorem is that it permits us to use sample statistics to make inferences about population parameters without knowing anything about the shape of the frequency distribution of that population Let’s illustrate the use of the central limit theorem. The distribution of annual earnings of all bank tellers with five years’ experience is as shown below in Fig. 2.4. This distribution has a mean of Rs. 19,000 and a standard deviation of Rs. 2,000. If we draw a random sample of 30 tellers, what is the probability that their earnings will average more than Rs. 19,750 annually? In Fig. 2.4 we show the sampling distribution of the means that would result and highlight the area representing ‘earnings over Rs. 19,750.’ Our first task is to calculate the standard error of the mean from the population standard deviation as follows:  SAMPLING TECHNIQUES I 33 FIGURE 2.4 CT Standard error of the mean 2000 2000 = Rs. 365.16 5.477 Because we are dealing with a sampling distribution, we must now use the equation for z value and the Standard Normal Probability Distribution (App. Table z = (x - p)/ ct- For x = Rs. 19750; (19750-19,000) 365.16 750 ~ 365.16 = 2.05 Annexure Table gives us the probability of 0.4798 for az value of 2.05. We show the corresponding area in Fig. 2.4 as the area between the mean and Rs. 19,750. Since half or 0.5000 of the area under the curve lies between the mean and the right-hand tail, the shaded area must be 0.5000 (Area between the mean and the right-hand tail) - 0.4798 (Area between the mean and 19,750) 0.0202 (Area between the right-hand tail and 19,750) 34 I ADVANCED BANK MANAGEMENT Thus, we have determined that there is slightly more than a 2 per cent chance of average earnings being more than Rs. 19,750 annually in a group of 30 tellers. The central limit theorem is one of the most powerful concepts in statistics, which states that the distribution of sample means tends to be a normal distribution. This is true regardless of the shape of the population distribution from which the samples were taken. Result 1: If X ~ Bin(n, p), then Z = ( ? tends to standard Normal Deviation as n —> a/W X-X Result 2: If X ~ PX), then Z = __ tends to standard Normal Deviation as sample size —> «> vX Examples 1. A sample of 25 observations from a normal distribution has a mean of 98.6 and a standard deviation of 17.2. (a) What is P(92 < x < 102)? (b) Find the corresponding probability given a sample of 36. Solution: (a) N= 25, p = 98.6,o= 17.2, ax = o I yl~n~ = 17.2 IV25 = 3.44 P (92 < x < 102) = p[(92 - 98.6)/3.44 < (x - m)/s/ (102 - 98.6)/3.44] = P (-1.72 < z < 0.99) = 0.4573 + 0.3389 = 0.7962 (b) n = 36, 0^=0/7^ = 17.2/736 = 2.87 P (92 < x < 102) = z?[(92 - 98.6)/2.87 < (x - p,)/o. < (102 - 98.6)/2.87] = P(-2.30 < z < 1.18) =.4893 + 0.3810 = 0.8703 2. Kamala, an auditor for a large credit card company, knows that, on average, the monthly balance of any given customer is Rs. 112, and the standard deviation is Rs. 56. If Kamala audits 50 randomly selected accounts, what is the probability that the sample average monthly balance is (a) Below Rs. 100? (b) Between Rs. 100 and Rs. 130? Solution: The sample size of 50 is large enough to use the central limit theorem p = 112, o = 56, n = 50, cr? = 56/ V50 = 7.920 (a) P(x < 100) = P[(x -p) /crv< (100-112)/ 7.920] = P(z < -1.52) = 0.5 - 0.4357 = 0.0643 (b) P(100 < x < 130) = P[(l 00 -112) / 7.920 < (x -p) /o- < (130 -112) / 7.920] = P(-1.52 < z < 2.27) = 0.4357 + 0.4884 = 0.9241  SAMPLING TECHNIQUES I 35 3. It has been found that 2% of the tools produced by a certain machine are defective. What is the probability that in a shipment of 400 tools, 3% or more defective? The sample size of 400 is large enough to use the central limit theorem. —A Solution: X ~ P (2 = np = 0.02*400 = 8), then Z = tends to standard Normal Deviation 3% of 400= 12 P(X > 12) = P[(x - 8)/V8 > (12 - 8))/V8 = P([(x - 8)/2.82 > 1.43) = 0.5 - 0.4236 = 0.0764 4. A coin is tossed 700 times. Using Normal approximation find the probability of getting number of Heads between 280 and 375. The sample size of 700 is large enough to use the central limit theorem. z x X-np X-350 Solution: X~ Bin (n = 700, p = 0.5), then Z = ,---- =---------- tends to standard Normal Deviation yjnpq 13.22 P(280 < X < 375) = P[(280 - 350)/13.22 < (X - 350)/l3.22 < (375 - 350)/l3.22)] = P[-5.29 < Z < 1.89] = 0.5 + 0.4706 = 0.9706 An Important Consideration in Sampling: The Relationship between Sample Size and Standard Error We saw earlier in this chapter that the standard error, is a measure of the dispersion of the sample means around the population mean. If the dispersion decreases (if av becomes smaller), then the values taken by the sample mean tend to cluster more closely around m. Conversely, if the dispersion increases (if becomes larger), the values taken by the sample mean tend to cluster less closely around m. We can think of this relationship this way: As the standard error decreases, the value of any sample mean will probably be closer to the value of the population mean. As the standard error decreases, the precision with which the sample mean can be used to estimate the population mean, increases. If we refer to Equation, we can see that as n increases, s* decreases. This happens because, in an Equation, a larger denominator on the right side would produce smaller s*. on the left side. Two examples will show this relationship; both assume the same population standard deviation s of 100. When n = 10, x = 23 Arithmetic Mean of Grouped data If a variate X take values xp x2,..., x with corresponding frequencies f ,f...,f respectively, then the ~ n - ~ n arithmetic mean of these values is given by X = ---- , xz are the class marks of the class intervals for grouped continuous data. Example 2: Find the Arithmetic mean for following: 1 2 3 4 5 6 7 8 Solution: A.M. = M = 173/43 = 4.02 Example 3: Find Arithmetic Mean of the following data: Class Interval 20-30 30-40 40-50 50-60 60-70 70-80 80-90 Frequency 5 12 10 14 5 8 4 X= =.3030 = 52 24 ZZ 58 Example 4: If the mean of the following distribution is 54, find the value of x. Class Interval 0—20 20-40 40-60 60-80 80-100 Frequency 7 X 10 9 13 => 54 = (2370 + 30x)/(39 + x) => x = 11  MEASURES OF CENTRAL TENDENCY... I 49 3.3 COMBINED ARITHMETIC MEAN If X} and X2 are the arithmetic mean of two samples of size w and n2 respectively then, the Combined arithmetic mean X of the distribution combining the two can be calculated as - _ n{Xx + n2X2 n}+n2 Example 5: The average marks of a group of 100 students in Mathematics are 60 and for other group of 50 students, the average marks are 90. Find the average marks combined group of 150 students. _niXl+ n2 X2 _ 100x60 + 50x90 _7q nx+n2 150 Example 6: In private health club, there are 200 members, 100 men, 80 women and 20 children. The average weight of men, women and children are 60 kgs, 50 kgs and 35 kgs respectively. Find the average weight of the combined group. Ans: = 100, n2 = 80, n3 = 20 Xj = 60, x2 = 50, x3 = 35 Combined mean = x= — + — - + = 10700/200 = 53.5 «i + «2 + n3 Example 7: An average daily wage of all the 90 workers in a factory is Rs. 60. An average daily wage of female workers is Rs. 45. Calculate an average daily wage of male workers if one third workers are male. Solution: = No. of male workers = 90 x 1/3 = 30 n2 = No. of female workers = 90 x 2/3 = 60 X2 = mean daily wages of female workers = 45 X= ^60=30-r, + 60*45 + n2 90 1 Example 8: Mean of 25 observations was found to be 78.4. But later on it was found that 96 was misread as 69. Find the corrected mean. Solution: We have n = 25 As X = 2x /n, Wrong sum of observations = Lx = nX = 25 x 75.4 = 1,960 Correct sum of observations = Ex - Wrong Observation + Correct Observation = 1,960-69 + 96= 1,987 So, Corrected Mean = 1987/25 = 79.48 Example 9: The mean salary of 50 employees of a company was computed as Rs. 680 per month. Later it was found that salary of Mr X was taken as Rs. 270 instead of 720. What will be the correct mean salary? 50 I ADVANCED BANK MANAGEMENT Ans: x = Rs. 680 7V= 50, = ft x The sum of observation = 50*680 = 34000 Corrected sum = the original sum - wrong value + correct value = 34000-270 + 720 = 34450 i Corrected sum Corrected mean =---------------------- = 34450/50 = 689 No. of observation Merits of Arithmetic Mean 1. It is rigidly defined 2. It is easy to calculate and simple to follow 3. It is based on all the observations 4. It is determined for almost every kind of data 5. It is finite and indefinite 6. It is readily put to algebraic treatment 7. It is least affected by fluctuations of sampling. Demerits of Arithmetic Mean 1. It is highly affected by extreme values. 2. It cannot average the ratios and percentages properly. 3. It is not an appropriate average for highly skewed distribution. 4. It cannot be computed accurately if any item is missing. 5. The mean sometimes does not coincide with any of the observed value. 6. Mean cannot be calculated when open-end class intervals are present in the data 3.4 GEOMETRIC MEAN The Geometric Mean (GM) is the average value or mean which measures the central tendency of the set of numbers by taking the root of the product of their values. Geometric mean takes into account the compounding effect of the data that occurs from period to period. Geometric mean is always less than Arithmetic Mean and is calculated only for positive values. Applications It is used in stock indexes. It is used to calculate the annual return on the portfolio. It is used in finance to find the average growth rates which are also referred to the compounded annual growth rate. It is also used in studies like cell division and bacterial growth, etc.  MEASURES OF CENTRAL TENDENCY... I 51 Geometric Mean of Ungrouped or Raw Data _______ 1 G.M. = ”]xxx2...xn = (x{x2... xn)" where Xj ,x2..xn are n observations of x. Example 10: Find the G.M. of the values 10, 24, 15, and 32. Solution: Given 10, 24, 15, 32 ____________ _[ We know that G.M. = ^xxx2....xn = (xjX2....xw)« = (10x24x15x32)^ = (115200)^ = 18.423 Geometric Mean of Grouped or Raw Data i where x1,x2...,xn are n observations of x with frequencies fx,f2 ^.,fn respectively and n = /j + /2 + + A Example 11: Find the G.M. for the following data. X i Solution: G.M. = (x/’x/2...x/")" = (l5 x 26 x 35 x 410j(5+6+5+10) = 2.4705 Merits of Geometric Mean 1. It is useful in the construction of index numbers. 2. It is not much affected by the fluctuations of sampling. 3. It is based on all the observations. Demerits of Geometric Mean (i) It cannot be easily understood. (ii) It is relatively difficult to compute as it requires some special knowledge of logarithms. (iii) It cannot be calculated when any item or value is zero or negative. 3.5 HARMONIC MEAN Harmonic Mean is defined as the reciprocal of the arithmetic mean of reciprocals of the observations. Arithmetic mean is appropriate measure of central tendency when the values have the same units whereas 52 I ADVANCED BANK MANAGEMENT the Harmonic mean is appropriate measure of central tendency when the values are the ratios of two variables and have different measures. So, generally Harmonic mean is used to calculate the average of ratios or rates. Applications It is used in finance to find average of different rates. It can be used to calculate quantities such as speed. This is because speed is expressed as a ratio of two measuring units such as km/hr. Harmonic Mean of Ungrouped or Raw data: n H.M. =-------- — where r r are n observations of x. Example 12: Find the HM of the values 10, 24, 15, and 32 Solution: Given 10, 24, 15, 32 4 4 We know that H.M. = = 16.6945 1111 0.2396 ------ 1-------- 1-------- 1------ 10 24 15 32 Harmonic Mean of Grouped or Raw data: where X\,x2,....... xn are n observations of x with frequencies......fn respectively and Example 13: Find the H.M. for the following data. 1 2 3 < I n 26 26 Solution: H.M. =-------- 7- = -7—7—-—777 =-------------- y" A 12.1667 A=ix. 1 2 3 4 Comparison between Arithmetic, Geometric and Harmonic Mean The arithmetic mean is appropriate if the values have the same units, whereas the geometric mean is appropriate if the values have different units and harmonic mean is appropriate if the data values are ratios of two variables with different measures, called rates.  MEASURES OF CENTRAL TENDENCY... I 53 Arithmetic Mean > Harmonic Mean > Geometric Mean A.M. x H.M. = (G.M.)2 Example 14: Find the Harmonic mean of two numbers a and b, if their Arithmetic mean is 16 and Geometric mean is 8. Solution: Let a and b are the two numbers. A.M. x H.M. = (G.M.)2 Solution Given: A.M. = 16 and G.M. = 8 The relation between A.M. G.M. H.M. is given as: A.M. x H.M. = G.M.2 16 x H.M. = 82 16 x H.M. = 64 H.M.= 64/6 = 4 3.6 MEDIAN AND QUARTILES The median is the middle value of a distribution, i.e., median of a distribution is the value of the variable which divides it into two equal parts. It is the value of the variable such that the number of observations above it is equal to the number of observations below it. Observations are arranged either in ascending order or descending order of their magnitude. Median is a position average whereas the arithmetic mean is a calculated average. Quartiles A quartile represents the division of data into four equal parts. First, second intervals are based on the data values and third their relationship to the total set of observations. By dividing the distribution into four groups, the quartile calculates the range of values above and below the mean. A quartile divides data into three points - the lower quartile QI, the median Q2, and the upper quartile Q3, to create four dataset groupings. The interquartile range is a measure of variability around the median, which is calculated using the quartiles are denoted by QI, Q2 and Q3. 54 I ADVANCED BANK MANAGEMENT Interquartile Range = Q3- Q1 Courtesy: https://stats.stackexchange.com/ Median of Ungrouped or Raw data The formula to calculate the median of the data is different for odd and even number of observations. Median of odd Number of Observations If the total number of given observations is odd, then the formula to calculate the median for a number of n observations is: Median = n~— th observation 2 Median of even Number of Observations If the total number of given observations is even, then the median formula to calculate the median for n number of observations is:  MEASURES OF CENTRAL TENDENCY... I 55 Median = {( y )th observation + ( )th observation)} / 2 Example 15: Find Median of 34, 32, 48, 38, 24, 30, 27, 21, 35. Solution: Arranging the data in ascending order, we have 21,24,27, 30, 32, 34, 35, 38,48. n = 10 ; hence = 5 and —— = 6 2 2 Median is ( 5th observation + 6th th observation ) / 2 = (15 + 17)/2 = 16. Example 16: Find the median of the daily wages of ten workers from the following data: 20, 25, 17, 18,8, 15,22, 11,9, 14. Solution: Arranging the wages in ascending order, we have 8, 9, 11, 14, 15, 17, 18, 20, 22, 25 Z7 + 1 Median = —— th observation = 5.5th observation = (15 + 17)/2 = 16. Median of Grouped data: If a variate X take values ,x2 , x^with corresponding frequencies /J ,/2..fn respectively, then the median of these values is given by Median =/. + Median class is the class in which the corresponding value of less than cumulative frequency just exceeds the value of N/2. Where, l} = lower limit of the median class, l2 = upper limit of the median class f= frequency of the median class, cf = cumulative frequency of the class preceding the median class, N = total frequency. Example 17: Find Median for the following data. Class interval 20-30 30-40 40-50 50-60 60-70 Frequency 8 26 30 20 16 56 I ADVANCED BANK MANAGEMENT Solution: c/ Frequency Class Mark(x) Less than cf 20-30 8 25 8 30-40 26 35 34 40-50 30 45 64 50-60 20 55 84 60-70 16 65 100 Total N=100 N/2 = 50, (40-50) is the Median class. „ (50-40)(50-34) f 30 Example 18: Find the Median for the following: 3 4 5 6 7 8 9 Solution 7V=E/ = 200 , —=100 2 128 is first c/just exceeding 100. The value ofx, corresponding to c/128 is 6 which gives median.  MEASURES OF CENTRAL TENDENCY. I 57 Example 19: Find Median for the following data. Class interval 5-9 10-14 15-19 20—24 25—29 30-34 35-39 Frequency 8 18 27 V I 12 I 8 1 Solution Cl New Cl Frequency Less than cf 5-9 4.5-9.5 8 8 10-14 9.5-14.5 18 26 15-19 14.5-19.5 27 53 20-24 19.5-24.5 21 74 25-29 24.5-29.5 10 84 30-34 29.5-34.5 8 92 35-39 34.5-39.5 7 99 Total