Elementary statistics

statistics The Nature of Statistics as a subject: What Statistics is not? “Statistics should not be taught as a subject correlative with physics, chemistry, economics and sociology. Statistics is not a science; it is a scientific method.” (Croxton and Coroden: Applied General Statistics) In order to get a clear conception of what statistics is, let us first see what statistics is not. 1. Statistics is not a subject of study in the sense that economics, chemistry, physics, psychology, history etc. are. 2. Statistics is not a science like physics or chemistry or biology. 3. Statistics is not Mathematics, although the statistical methods are basically mathematical. What Statistics is? Having seen what statistics is not we now proceed to see what statistics is not we now proceed to see what statistics is. Let us proceed step by step. 1. Phenomena are many and various, such as chemical, physical, biological, graphical, social, psychological, economical, business etc. 2. Each phenomenon has its own peculiarities, characteristics and problems. 3. For studying any phenomenon, for one practical purpose, some method is required 4. Method of study or research available are several namely: a. The Experimental Method b. The Empirical Method c. The Historical Method d. The Case Method e. The Deductive Method f. The Statistical Method It is, therefore, clear that statistics is methodology. It is a method of studying problems relating to any phenomenon. That is, Statistics is a technique of research. A given phenomenon or an aspect of a phenomenon cannot be studied by any one method: The choice of method depends upon the nature of the phenomenon to be studied,” It must not be assumed that the statistical method is the only method to use in research, just as the carpenter needs to use more than one tool in completing a piece of work, so that the research worker must often make use of not

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formulas

TABLE OF CONTENTS
Page
4 Core Mathematics C1
4 Mensuration
4  Arithmetic series                  
5 Core Mathematics C2
5 Cosine rule
5 Binomial series
5 Logarithms and exponentials
5 Geometric series
5 Numerical integration
6 Core Mathematics C3
6 Logarithms and exponentials
6 Trigonometric identities
6 Differentiation
7 Core Mathematics C4
7 Integration
8  Further Pure Mathematics FP1
8  Summations
8 Numerical solution of equations
8 Conics
8 Matrix transformations
9 Further Pure Mathematics FP2
9 Area of sector
9 Maclaurin’s and Taylor’s Series
10 Further Pure Mathematics FP3
10 Vectors
11 Hyperbolics
12 Differentiation
12 Integration
13 Arc length
13 Surface area of revolution

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Chapter 01: Real numbers, limits and continuity

Contents and summary

• Articles of Exercise 1.1
  • Rational numbers, Irrational numbers, Real numbers, Complex numbers
  • Properties of real numbers, Order properties of R
  • Absolute value or modules of a R
  • The completeness property of R
  • Upper bound, lower bound
  • Real line, Interval
  • Working rule for the solution of inequality
  • Binary relation (B.R), Domain of B.R, Range of B.R
  • Function, Onto or surjective funiton, (1-1) Function, Bijective Function
  • Real valued function, Image of a function, Bracket function
• Articles of Exercise 1.2
  • Finite limit at a finite point
  • Left hand limit
  • Right hand limit
• Articles of Exercise 1.3
  • Continuity

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Mathematics is not just a study of numbers, nor is it simply about calculations.

It is not about applying formulas, either. It can perhaps be better described as “a field of creation through accurate and logical thinking." Mathematics has a long, rich history and continues to grow rapidly. New findings are regularly presented at conferences in and outside of Japan and articles based on such findings are published in mathematics journals in countries around the world.

Mathematics is a very diverse filed. The appendix shows a list of subcategories classified under mathematics by the American Mathematical Society (the list is an excerpt from the AMS's Mathematical Reviews journal). Many of you must be surprised to see so many subfields of mathematics. "Algebra" and "Geometry", for instance, may be familiar subjects of mathematics in high school, but they are divided further into more specified categories in the list, which also includes combinations of subfields, such as "algebraic geometry."

Mathematics is an academic discipline of great depth, with a number of unsolved problems. It has progressed on cumulative contributions from countless mathematicians in the world who have tackled those problems while creating new areas of inquiry.

Some of the subfields in the appendix, such as fluid mechanics, quantum theory, information and telecommunication, and biology, may seem irrelevant to mathematics at first glance. These fields, however, take mathematical approaches to describing and analyzing phenomena under study.

They illustrate how a number of subfields of mathematics have benefited from and evolved through interactions with other disciplines. Mathematics, on the other hand, has made considerable contributions to the advancement of other academic fields. In fact, mathematics is often described as the foundation of scientific studies.

One of the major characteristics of mathematics is its general applicability.

One equation, for instance, can represent a particular phenomenon in physics as well as a certain logic in economics. This general nature of mathematical equations enables unified treatment of diverse phenomena in various academic fields. Furthermore, mathematical theorems are no respecter of age or seniority: any theorem has to be proven through appropriate mathematical procedures whether you are a novice student researcher or an eminent professor of mathematics. And once proven true, mathematical theorems will never be reversed. This "universality" of mathematics is another important feature that allows the discipline to transcend time and space.

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