Application of Mathematical Models and Techniques in the field of Statistics.

Created Feb. 7, 2024 by pawan ray

Application of Mathematical Models and Techniques in the field of Statistics.

 

 

Pawan Kumar Ray

 Assistant Professor

Harkamaya College of Education

6thMile Tadong Gangtok

  Email-ccc4job@gmail.com

 9832359082/7908401075

INTRODUCTION

 

Mathematics is the science of measurement, quantity and magnitude. Mathematics is known as “Ganita” in Hindi which means the science of calculation. Developing children's abilities for mathematics is the main goal of mathematics education. The narrow aim of school mathematics is to develop 'useful' capabilities, particularly those relating to numeracy–numbers, number operations, measurements, decimals and percentages. The higher aim is to develop the child's resources to think and reason mathematically, to pursue assumptions to their logical conclusion and to handle abstraction. It includes a way of doing things, and the ability and the attitude to formulate and solve problems.

Mathematics is the most closely related subject in our daily life. Its knowledge is exact, systematic logical and clear. Mathematics involves the process for intellectual development of mental faculties. It is not that mathematical knowledge is needed only by engineers, doctors or business personals. Even the smallest citizen of society such as laborers and workers need the basic knowledge of mathematics. Besides the mental ability, mathematics develops some quality like concentration, truthfulness, seriousness and reasoning. Thus, in the words of Locke it is rightly said that, “Mathematics is a way to settle in the mind the habit of reasoning”.

The modern age is the age of Science and technology with the advent of technology in all walks of human life, a person feels handicapped if he is unable to utilize them. But even the simplest technological knowhow calls for the basic mathematical knowledge and understanding. Roger Bacon has rightly said that “Mathematics is the gateway and key to all sciences”.In fact mathematics is the language of all sciences. It was only the mathematical interpretation of Newton’s Law of Gravitation that had led the world into the era of Satellite Communication. The emergence of computers, Internet mobiles and modern means of transport has reduced the world into a ‘Global Village’. Thus, Science which is considered as the backbone of technology borrows its exactness and systematic approach by the virtue of mathematics.

 

Mathematics is not just a tool to assist science; it is rather an approach to develop scientific temper, which leads to the highest level of human enquiry. This clearly indicates that “The progress and improvement of mathematics are linked to the prosperity of the state” (Napolean).Despite of the fact that mathematics plays a vital role in our cultural development as well as for our individual progress, it is not the subject of choice for many students. Mathematics is taught like a mechanical subject using no creativity and useful techniques and models. The students are hardly trained to develop mathematical skills of calculation. We are not encouraged to develop Mathematical Thinking Aptitude and Problem Solving approach. The All India Survey of Achievement in mathematics (1970) revealed that achievement of children in mathematics at elementary level is below the expected level.

If the answer books of Secondary School Examination are examined a great number of students would be found failed in mathematics itself. Who should be held responsible, the students, parents or teachers? Is Science and Mathematics phobia an inherited tendency or created? The answer is ambiguous. A common myth prevails in the society that more intelligent a person is, higher will be his aptitude in mathematics. But two people in spite of similar I.Q. might differ in perception, judgment, reasoning, word fluency, vocabulary, spatial orientation, etc. Intelligence on hand is concerned with general mental ability, while aptitude on the other hand is concerned with the specific sensory motor, mechanical, artistic or professional ability. By knowing the Intellectual level on the basis of Intelligence test or School Progress, we may tell the likelihood of the branch of occupation more suitable for the individual.

 

The practice or science of collecting and analysing numerical data in large quantities, especially for the purpose of inferring proportions in a whole from those in a representative sample. Statistics is branch of mathematics concerned with  C2IA  i.e.Collection, Classification, Interpretation &Analysisof numerical facts, for drawinginferences on the basis of their quantifiablelikelihood (probability).It can interpret aggregates of data too large to be intelligible by ordinary observation because such data (unlike individualquantities) tend to behave in regular, predictable manner. It is subdivided into descriptive statistics and inferential statistics.

Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure theory.

Statistical data collection is concerned with the planning of studies, especially with the design of randomized experiments and with the planning of surveys using random sampling. The initial analysis of the data often follows the study protocol specified prior to the study being conducted. The data from a study can also be analyzed to consider secondary hypotheses inspired by the initial results, or to suggest new studies. A secondary analysis of the data from a planned study uses tools from data analysis, and the process of doing this is mathematical statistics.

While the tools of data analysis work best on data from randomized studies, they are also applied to other kinds of data. For example, from natural experiments and observational studies, in which case the inference is dependent on the model chosen by the statistician, and so subjective

Statistics plays a vital role in every fields of human activity. It has important role in determining the existing position of per capita income, unemployment, population growth rate, housing, schooling medical facilities etc in a country. Now statistics holds a central position in almost every field like Industry, Commerce, Trade, Physics, Chemistry, Economics, Mathematics, Biology, Botany, Psychology, Astronomy etc, so application of statistics is very wide.

Modeling and Statistics are two branches of applied mathematics. Modeling involves fitting equations to data, usually just approximately. Statistics is the science of uncertainty. Both of these fields got their starts during the Renaissance, the period during the 14th to the 17th centuries in which European science and culture were "reawakened" from the Dark Ages. Modeling began as physicists attempted to predict the motions of the planets, the phases of the moon, and other natural phenomena. Eventually other disciplines, such as biology, sociology, and business, learned the methods, so that today mathematical modeling is an integral part of research in many fields. Statistics was initially developed to analyze errors in measurements. When different measurements for the same quantity (time between full moons, large distances on the earth's surface) were compared, it was quickly discovered that they formed a pattern, the now-familiar bell curve. Not long afterward, probability was invented to understand games of chance, and then greatly enhanced the understanding and applicability of statistics. Today, the two are intertwined in the area called inferential statistics.

 

Concept of the Mathematical techniques& Modeling:

A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering), as well as in the social sciences (such as economics, psychology, sociology, political science). Mathematical models can take many forms, including dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include logical models. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed. There are three kinds of mathematical models: linear, exponential, and power. The main idea is to collect data that relates two variables, say x and y, graph the data, and use the shape of the graph to pick a likely family of possible equations. Each family has infinitely many members (think of all the possible lines there are: different slopes, different y-intercepts), and one of these members will be the best fit for the data. The fit might not be exact; in fact, it might be very poor (not all data is remotely linear). Using mathematical analysis, we can judge just how useful that particular equation is. After we decide on the appropriate equation, the model, we can use it to make predictions about the variables.

The Importance & Uses of Mathematical techniques and Modeling in the field of Statistics:

Mathematical Models

There are several situations in which mathematical models can be used very effectively in introductory education.

  • Mathematical models can help students understand and explore the meaning of equations or functional relationships.

  • Mathematical modeling software such as Excel, Stella II , or on-line JAVA /Macromedia type programs make it relatively easy to create a learning environment in which introductory students can be interactively engaged in guided inquiry, heads-on and hands-on activities.

  • After developing a conceptual model of a physical system it is natural to develop a mathematical model that will allow one to estimate the quantitative behavior of the system.

  • Quantitative results from mathematical models can easily be compared with observational data to identify a model's strengths and weaknesses.

  • Mathematical models are an important component of the final "complete model" of a system which is actually a collection of conceptual, physical, mathematical, visualization, and possibly statistical sub-models.

Statistical Models

A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of some sample data and similar data from a larger population. A statistical model represents, often in considerably idealized form, the data-generating process.

The assumptions embodied by a statistical model describe a set of probability distributions, some of which are assumed to adequately approximate the distribution from which a particular data set is sampled. The probability distributions inherent in statistical models are what distinguishes statistical models from other, non-statistical, mathematical models. A statistical model is usually specified by mathematical equations that relate one or more random variables and possibly other non-random variables. As such, a statistical model is "a formal representation of a theory. A solid statistical background is very important in the sciences. But the extent to which statistical ideas are appropriate in an introductory course depends on specific course objectives and the degree or institutional structure. Here we list several examples showing why and when statistical models are useful.

Statistical models or basic statistics can be used:

  • To characterize numerical data to help one to concisely describe the measurements and to help in the development of conceptual models of a system or process;

  • To help estimate uncertainties in observational data and uncertainties in calculation based on observational data;

  • To characterize numerical output from mathematical models to help understand the model behavior and to assess the model's ability to simulate important features of the natural system(model validation). Feeding this information back into the model development process will enhance model performance;

  • To estimate probabilistic future behavior of a system based on past statistical information, a statistical prediction model. This is often a method use in climate prediction. A statement like 'Southern California will be wet this winter because of a strong El Nino' is based on a statistical prediction model.

  • To extrapolation or interpolation of data based on a linear fit (or some other mathematical fit) are also good examples of statistical prediction models.

  • To estimate input parameters for more complex mathematical models.

  • To obtain frequency spectra of observations and model output

Conclusion:

 

The mathematical modeling and techniques are used in mathematical statistics, are studied. First comes the discussion of the necessity of mathematical modeling of mathematical statistics application then the application of mathematical modeling in mathematical statistics is discussed including by mathematical models and mathematical statistics course the whole process of integrating mathematical modeling in mathematical statistics including strengthening the construction of teaching material encouraged and tends students to participates in mathematical modeling contests and reform assessments methods. All these ways can help to achieve good results in mathematical statistics contests, simulate students interest in learning and inspire students creative thinking.

 

References:

 

  • Adèr, H.J. (2008), "Modelling", in Adèr, H.J.; Mellenbergh, G.J., Advising on Research Methods: a consultant's companion, Huizen, The Netherlands: Johannes van Kessel Publishing, pp. 271–304.

  • Burnham, K. P.; Anderson, D. R. (2002), Model Selection and Multimodel Inference (2nd ed.), Springer-Verlag, ISBN 0-387-95364-7.

  • Cox, D.R. (2006), Principles of Statistical Inference, Cambridge University Press.

  • Konishi, S.; Kitagawa, G. (2008), Information Criteria and Statistical Modeling, Springer.

  • McCullagh, P. (2002), "What is a statistical model?"(PDF), Annals of Statistics, 30: 1225–1310, doi:10.1214/aos/1035844977.

  • Lakshmikantham,, ed. by D. Kannan,... V. (2002). Handbook of stochastic analysis and applications. New York: M. Dekker. ISBN 0824706609.

  • Schervish, Mark J. (1995). Theory of statistics (Corr. 2nd print. ed.). New York: Springer. ISBN 0387945466.

  • Freedman, D.A. (2005) Statistical Models: Theory and Practice, Cambridge University Press. ISBN 978-0-521-67105-7

  • Hogg, R. V., A. Craig, and J. W. McKean. "Intro to Mathematical Statistics." (2005).

  • Larsen, Richard J. and Marx, Morris L. "An Introduction to Mathematical Statistics and Its Applications" (2012). Prentice Hall.

·         https://www2.stetson.edu/~mhale/stat/index.htm retrieved on August 19, 2018

·         https://en.wikipedia.org/wiki/Mathematical_statisticsretrieved on August 18, 2018

·         https://serc.carleton.edu/introgeo/mathstatmodels/why.htmlretrieved on August 16, 2018

 

 

Return to top