Chapter 42: Mathematics and Statistics

Introduction

In this chapter will be discussed two interrelated topics, viz., Mathematics and Statistics, which pervade almost all human activities. Study of mathematics opens up the gateway for many prosperous career opportunities in science and technology including computer science. Test of Mathematical (or Quantitative) Ability and Data Interpretation are two components of many admission and recruitment test. The three proverbial R’s, considered symbolic of functional literacy, besides reading and writing, include arithmetic, a branch of mathematics. Statistics is a branch of mathematics concerned with the collection and analysis of large bodies of data to identify trends and overall patterns. It is a problem-solving and decision-making tool used by a wide cross-section of occupations.

Mathematics

Mathematics is as old as civilisation itself and is one of the most useful and fascinating branches of human knowledge. It encompasses many topics of study and as such it is difficult to define the term "mathematics" which comes from a Greek word meaning "inclined to learn". It may, however, be broadly defined as the scientific study of quantities, including their relationships, operations and measurements expressed by numbers and symbols. The most important skills in mathematics are careful analysis and reasoning, and thus logic is the floor on which the structure of mathematics is built. Starting from widely accepted statements, mathematics use logic to draw conclusions and to develop complete mathematical systems.

Branches of Mathematics

Today, it is customary to divide mathematics into pure mathematics and applied mathematics. However, the boundary between pure and applied mathematics is not always clear. Applied mathematics deals with solutions to practical problems in many areas of science, technology and social sciences. Pure mathematics, on the other hand, deals with the abstract properties of mathematical quantities and systems without regard to application, and seeks to advance mathematical knowledge of its own sake. When a new application is found, a topic which was originally classified as pure mathematics may then become a part of the field applied mathematics. A work in applied mathematics also frequently leads to research in pure mathematics. Computer science, probability statistics, and operations research are often considered part of applied mathematics, whereas abstract algebra, number theory, and topology are usually considered to be in the realm of pure mathematics.

Mathematics may also be divided into various branches, depending on the elements and axioms used. Some of these branches, in turn, can be further subdivided into several sub-branches. Each branch usually consists of definitions, undefined terms, elements, axioms, operations, relations, and theorems. Annexure 1 gives a brief account of several of the main kinds of mathematics. Though not a branch of mathematics, the name "new mathematics" or "modern mathematics" has been given to the contents and methods of teaching mathematics courses initiated in the USA during the 1950s and 1960. The factors which led to the development of new mathematics include general dissatisfaction among mathematicians with both course contents and methods of teaching. However, opinion of mathematicians about the effectiveness of new mathematics is divided.

The Importance of Mathematics

As stated earlier, applied mathematics have many problem solving applications. Mathematics is used by all individuals in everyday life. In science, mathematics is an essential tool for nearly all scientific studies. Scientists use in of designing experiments, analysing data, expressing precisely their findings by mathematical formulas, and making predictions on the basis of these findings. Such physical sciences as astronomy, chemistry, and physics rely heavily on mathematics. Social sciences, economics, psychology, and sociology also depend greatly on statistics and several other branches of mathematics. Economists (notably econometricians) use computers to create mathematical models (or econometric models) of the economic systems.

These models use sets of formulas to predict how a change in one part of economy might affect other parts. Industrial uses of mathematics include designing, developing and testing products, and manufacturing processes. Engineers use mathematics for designing structure, such as buildings, dams, highways, bridges and other engineering projects. Business transactions (buying and selling) and maintaining of other records (stock levels, employees hours and wages) also need mathematics. Banking and financial institutions use mathematics to handle and invest their funds. Mathematics (more precisely Statistics) helps insurance companies calculate risks and commute the premium rates. This task is carried out by actuaries and this branch of knowledge is known as actuarial science (See Chapter 15 - Actuarial Science). And finally, the computer a term derived from the word "compute", is nothing but a mathematical machine that performs calculations at a very high speed.

Promotion of Mathematics

Several measures have been taken by the Government of India of the promotion of mathematical education and research. The Department of Atomic Energy (DAE) is primarily involved in this task. The DAE had formed in 1983, the National Board for Higher Mathematics (NBHM) to foster the development of higher mathematics. (See Annexure 2) One of its important activities is to organise Indian National Mathematical Olympiad (INMO) (See Annexure 3) with the twin objectives of identifying mathematical talents at the school level and nurturing them and selecting candidates for the International Mathematical Olympiad (IMO). Introduced in 1959 to encourage school students worldwide to study mathematics, it is held in different countries annually in the month of July.

Indian participation began in 1989 in the 30th IMO, and India hosted the 37th event in 1996 held in Mumbai. In the 39th IMO (1998), in which 405 students from 76 countries participated, India won three gold and three silver medals and secured the seventh rank. Among the aided institutions of the DAE, three are concerned with the mathematics viz., the Tata Institute of Fundamental Research (which has a School of Mathematics), the Mehta Research Institute of Mathematics and Mathematical Physics (Allahabad 211019), and the Institute of Mathematical Sciences (Chennai 600013). The School of Mathematics admits bright young students aspiring for a research career in mathematics as research scholars. Established in 1962, the primary purpose of the Institute of Mathematical Sciences is to foster high quality fundamental research in frontier areas of mathematical sciences.

Educational Opportunities

Mathematics is taught as a compulsory up to Class 10. At the Class 10+2 level, students may opt for the subject, if they so desire. However, those who want to take engineering and technology courses after 10+2 must opt for it 10+2 level. It has been earlier mentioned that Mathematical of Quantitative Ability and Data Interpretation are important components of many admission and recruitment tests. At the undergraduate level, mathematics is offered as one of the combinations both for BA and BSc courses. In some universities, it can be taken up as the principal or honours subject.

The Indian Statistical Institute (Calcutta 700035) has an exclusive course in Mathematics of three-year duration leading to the award of B Math degree. There are stray examples of offering specialised aspects of mathematics as one of the combinations at the graduate level. Some examples are, Numerical Mathematics (Madurai - Kamaraj University). Industrial Mathematics (University of Pune and North Maharashtra University), Mathematical Economics (Devi Ahilya Vishwavidyalaya).

At the Master's degree level, Mathematics is offered by more than 120 universities. The courses are offered by the Departments or Schools of Mathematics. Often there is one Department or School for both mathematics and computer science (See Annexure 4). The nomenclature of the award is both MA and MSc. Some universities make a distinction between Pure Mathematics and Applied Mathematics.

Three universities have introduced five-year integrated Master's degree course open to candidates who have passed the 10+2 examination viz,. Indian Institute of Technology, Kharagpur and Kanpur and Pondicherry University. Besides courses in general, pure, and applied mathematics, a small number of programmes are available in such topics as Computational Mathematics (of Mathematics with Computer Applications), Mathematical Statistics, and Mathematical Economics, Mathematics with Computer Application, Industrial Mathematics, and Functional Mathematics (See Annexure 5). Most of the universities which have Master's degree programmes also have facilities of predoctoral and doctoral research. As mentioned earlier, a number of fellowships are available for pursuing doctoral studies.

Distance Learning: Master's degree programmes in mathematics are available in 14 universities through distance learning mode. (See Annexure 6). Among them is the Alagappa University which offers an MSc course in Industrial Mathematics.

Statistics

Statistics is a set of methods that are used to collect and analyse numerical data. The term "statistics" is used as either a plural noun or a singular noun. As a plural noun, statistics are numerical data; and as a singular noun, statistics is a set of scientific methods used to collect and analyse numerical (or statistical) data. Like mathematics, statistics also has many problem solving applications. Statistical methods help people identify, study, and solve many problems and enable them to take decisions about uncertain situations.

Studying a problem using statistics typically involves four basic steps, viz., (1) defining the problem; (2) collecting data; (3) analysing data; and (4) reporting the results. These results help in taking appropriate decisions. As such, statistical methods are used by a wide variety of occupations, such as doctors, economists, policy makers in governments, scientists, meteorologists, demographers, to name only a few. However, the most visible application of statistics is by the governments. The measurement of inflation indicative of soaring price levels, and rate of economic growth, expressed in terms of GDP reported regularly in the mass media can hardly escape anybody's attention.

Notwithstanding denigrating comments of many adversaries, some often in lighter vein (e.g. How to Lie With Statistics, by D huff and I Gies. NY Norton, 1954), the importance of statistics in governance was highlighted by the United Nations more that 45 years back thus: "The demand for statistics of the highest quality by governments, business, economists, social workers and others, has increased enormously. This is accounted for partly by the growing complexity of the modern world, but even more by the fact that many government today have accepted wide responsibility for the welfare of the citizens and have embarked upon a wide variety of economic planning, regulation, and social security measures... Governments which do not have well developed statistical organisations are severely handicapped" (Handbook of Statistics NY, UN, 1954. p. 55).

National Statistical System

India has one of the most developed statistical systems in the world. An idea about the statistical data available on a wide variety of socio-economic dimensions may be had from the annual compendium Statistical Abstracts, India, published by the Central Statistical Organisation (Department of Statistics). The 35th edition covering the data for 1998 has 252 statistical tables grouped under 39 broad headings. Besides the Central Statistical Organisation (CSO) and the National Sample Survey Organisation (NSSO) in the Department of Statistics (Ministry of Statistics and Programme Implementation), two major statistical agencies are the Office of the Register General and Census Commissioner and the Directorate General of Commercial Intelligence and Statistics.

While the former is responsible for conducting the decennial population census (next one due in 2001), the latter handles foreign trade data. Most of the departments in the Central Government have their own statistical establishments and all State Governments and Union Territory Administrations have Directorates, Departments or Bureaus of Statistics. There is a large number of periodical publications brought our by these statistical agencies for disseminating latest statistical data.

The Department of Statistics in the Ministry of Statistics and Programme Implementation overviews the national statistical activities in conjunction with the State Statistical Directorates. The Ministry has introduced several programmes for the promotion of statistical research which are executed by the Central Statistical Organisation. Annexure 7 gives a brief account of the activities of the Department of Statistics. The Government of India constituted the Indian Statistical Service (ISS) in 1961 by encadering statistical function posts then operating in different ministries a competitive examination conducted by the Union Public Service Commission every rear (See Annexure 8).

In order to ensure high quality of statistical data, committees and commissions are appointed from time to time to review the situation, identify the shortcomings, and recommend measures for improving the existing systems. The Committee to Review the National Statistical System constituted in 1979-80 made recommendations of far-reaching consequences. In view of the drastic changes in the national economic situation following the liberalization of the economy, it became necessary to appoint a high power National Statistical Commission in early 2000, under the Chairmanship of Dr. C Rangarajan, a distinguished economist (now Governor of Andhra Pradesh), to reassess the situation and recommend measures for a systematic revamping of the national statistical system in the changed context.

Educational Opportunities

Internationally acclaimed statistical scientist, Prasanta Chandra Mahalanobis (1893-1972) may be said to be the father of statistical education in India. Among the many achievements of Mahalanobis, the most monumental was the establishment of the Indian Statistical Institute (ISI) in 1931. In course of time ISI got recognition as a premier centre of scientific research, not only in India but also the world over. The Institute which made a jumble beginning in the small Statistical Laboratory in the Presidency College (Calcutta), where Mahalanobis was a professor of Physics, not only had produced a generation of statisticians of world stature but has also played a significant role in making statistics the highly developed precise science that it is today.

Beginning with imparting training mainly to government officials, the Institute introduced in 1939 a professional course of one year duration open to holders of master’s degree. The ISI was instrumental in introducing MA/MSc degree course course in Statistics in the Calcutta University in 1941 and provided the entire faculty support for quite some time. At present, ISI offers as many 10 courses in statistics, mathematics, computer science, quality control, as well as in a host of other subjects including library and information science. (See Annexure 9).

Statistics as a subject is offered at the first degree, postgraduate, diploma, pre-doctoral, and doctoral levels. At the first degree level, it is offered as one of the combinations. A small number of universities have also introduced topics in specialized areas of statistics as one of the combinations. Some examples are Mathematical Statistics (BA Hons) (University of Delhi), Statistical Methods (University of Pune), Statistical Techniques (North Maharashtra University), Elements of Economic Statistics (Madurai – Kamaraj University), Business Statistics and Biostatistics (Avinashilingam Institute for Home Science and Higher Education for Women), Applied Statistics (University of Madras).

Indian Statistical Institute and Visva-Bharathi offer stand-alone BStat courses of three-year duration. At the postgraduate level statistics courses are offered by more that 70 universities. Like mathematics, the nomenclature of the awards is also MA and MSc. Annexure 10 gives an illustrative list of universities offering MA and MSc courses. Several universities have also introduced specialized courses in the subject, e.g., Statistics with Computer Applications (Annamalai University), Industrial Statistics (North Maharashtra University), Statistics with Computer Programming (Bundelkhand University), It may be mentioned that the population studies programmes of International Institute for Population Sciences (Mumbai 400080) cover Statistical Methods for Population Studies and Biostatistics. (See Chapter 52 Mixed Bag).

Two other most important areas in statistics, viz., Statistical Quality Control (SQC) and Operations Research (OR) have been discussed in subsequent sections. Diploma and Certificate courses are very small in number and mostly involve special areas of the subject. Some examples are, Diploma in Health Statistics (DHS) – All India Institute of Hygiene and Public Health (Calcutta 700073), PG Diploma in Statistics (Dr. B R Ambedkar University) PG Diploma In Statistics (Punjab University), PG Diploma in Statistics of two-years duration (Bananas Hindu University), Certificate in Statistics (Andhra University), Certificate in Statistical Computing (Karnataka University). Many universities also offer pre-doctoral and doctoral courses.

Statistical Quality Control: Quality control is a general term, widely used in industrial engineering, for the processes of testing an industrial output (and often input) for conformity to design specifications, including defining testing procedures, minimum quality standards, and the like. Statistical Quality Control (SQC) is the procedure that employs statistical extrapolation from a test sample to determine the quality of entire lots. Quality Control should not be confused with Quality Assurance, Quality Management or Total Quality Management. Quality Assurance, a management function, is the system of documented control within the business and industry that minimizes the opportunities for the errors affecting both the business and industry and their customers.

ISO (International Standards Organisation)-9000 provides the framework for the implementation of the Quality Assurance System. SQC is widely used in industries involved in the production of commodities from consumer goods to industrial products. The Indian Statistical Institute has done pioneering work in the areas of SQC education and implementation of SQC Process in a large number of industries. It offers three courses in the subject: (i) two-year MTech course in Quality Reliability and Operations Research; (ii) two-year Specialist Development Programme in SQC and OR; and (iii) one-year Certificate/two-year Diploma Programme in SQC & OR (See Annexure 9)

Operations Research: Operation Research (OR), as defined in the technical dictionaries, is the mathematical analysis of optimisation problems especially of dynamic systems subject to a given set of constraints as well as any application of such analysis within a system in order to evaluate its effectiveness in achieving identified goals, The topic is a component of MSc course in applied mathematics. Exclusive Master’s degree programmes are offered by seven universities. Besides, another seven universities have introduced PG Diploma courses in the subject (See Annexure 11).

Career Opportunities

Teaching and research offer reasonable good career opportunities for persons holding doctoral degrees in mathematics and statistics. Mention has been made about the new opportunity in the field of actuarial science in view of the privatisation of the insurance industry and the opening the up of insurance sector for the foreign companies. The Life Insurance Corporation of India has now introduced and Actuarial Apprenticeship scheme to meet its extended growing manpower demand.