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.

Frequency Distribution

Frequency Distribution: A frequency table is used to summarize categorical, nominal, and ordinal data. It may also be used to summarize continuous data when the data set has been divided  into meaningful groups.

Count the number of observations that fall into each category. The number associated with each category is called the frequency and the collection of frequencies over all categories gives the frequency distribution of that variable.

Table 1 Frequency Distribution of Time

Time	Count
110	1
115	2
120	4
125	3
130	5
135	3
140	4
145	2
150	1

The relative frequency is a number which describes the proportion of observations falling in a given category. Instead of counts, we report relative frequencies or percentages.

Quantitative variable

Quantitative variable

·         Quantitative or numerical data arise when the observations are frequencies or measurements.

·         The data are said to be discrete if the measurements are integers (e.g. number of  employees of a company, number of incorrect answers on a test, number of participants in a program…)

·         The data are said to be continuous if the measurements can take on any value, usually within some range (e.g. weight).   Age and income are continuous quantitative variables. For continuous variables, arithmetic operations such as differences and averages make sense.

Analysis can take almost any form:

Þ      Create groups or categories and generate frequency tables.

Þ      All descriptive statistics can be applied.

Þ      Effective graphs include: Histograms, Stem-and-Leaf plots, Dot Plots, Box plots, and XY Scatter Plots (2 variables).

·         Some quantitative variables can be treated only as ranks; they have a natural order, but  these values are not strictly measured.  Examples are:  1) age group (taking the values child, teen, adult, senior), and 2) Likert Scale data (responses such as strongly agree, agree, neutral, disagree, strongly disagree).  For these variables, the distinction between adjacent points on the scale is not necessarily the same, and the ratio of values is not meaningful.

Analyze using:

Þ      Frequency tables

Þ      Mode, Median, Quartiles