Math and Statistics

According to Webster's-Online-Dictionary.org:

Statistics is a branch of applied mathematics which includes the planning, summarizing, and interpreting of uncertain observations. Because the aim of statistics is to produce the "best" information from available data, some authors make statistics a branch of decision theory. As a model of randomness or ignorance, probability theory plays a critical role in the development of statistical theory.

According to Wikipedia:

Mathematics is the academic discipline, and its supporting body of knowledge, that involves the study of such concepts as quantity, structure, space and change. The mathematician Benjamin Peirce called it "the science that draws necessary conclusions". Other practitioners of mathematics maintain that mathematics is the science of pattern, and that mathematicians seek out patterns whether found in numbers, space, science, computers, imaginary abstractions, or elsewhere. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.

Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Knowledge and use of basic mathematics have always been an inherent and integral part of individual and group life.

According to Webster's-Online-Dictionary.org:

Mathematics is commonly defined as the study of patterns of structure, change, and space. In the modern formalist view, it is the investigation of axiomatically defined abstract structures using logic and mathematical notation. It is a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement

Science of Mathematics and Science of Statistics

Math and Statistics

Mathematics Statistics Mathematics is an academic subject statistics is a part of applied mathematics Mathematics deals with numbers, patterns statistics is concerned with systematic representation and analysis of data Mathematical concepts are freely used in statistics Mathematics form the basis of our understanding of quantity and measurement statistics makes understanding of data easy  mathematics is mother language of science Both mathematics and statistics find wide usage in different fields

Assumptions and Objective of ANOVA

Assumptions of  ANOVA

1.      Observations and errors are independently distribbuted.

2.      Observations and errors conferm to normal distribution with equal variance.

3.      Treatments and environmental effects are additive

Objective of  ANOVA

1.     It identifies the couses of variation and sort out corresponding components of  variation with associated degrees of freedom.

2.     It provides test of significance based on F-distribution.

Models for ANOVA

Models for ANOVA

There are three types of models used in the analysis of variance,thus are.

  1.Fixed-effects models or Model 1

The fixed-effects model of analysis of variance applies to situations in which the experimenter applies one or more treatments to the subjects of the experiment to see if the response variable values change. This allows the experimenter to estimate the ranges of response variable values that the treatment would generate in the population as a whole.

  2.Random-effects models or Model 2

Random effects models are used when the treatments are not fixed. This occurs when the various factor levels are sampled from a larger population. Because the levels themselves are random variables, some assumptions and the method of contrasting the treatments differ from ANOVA model 1.

  3.Mixed-effects models or Model 3

A mixed-effects model contains experimental factors of both fixed and random-effects types, with appropriately different interpretations and analysis for the two types.

Analysis of variance (ANOVA)

Analysis of variance (ANOVA)

In statistics, analysis of variance (ANOVA) is a collection of statistical models, and their associated procedures, in which the observed variance in a particular variable is partitioned into components attributable to different sources of variation.

In other words, 

Analysis of variance (ANOVA) is the process of partitioning and decomposing total variation contained in the data into various independent components, each of which is attributed to an identifiable cause or sources of variation and a further component due to chance factor.