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

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

**Pawan
Kumar Ray**

** Assistant Professor**

**Harkamaya
College of Education**

**6 ^{th}Mile
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
C^{2}IA 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 14^{th} to the 17^{th}
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
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(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