eChapters

A simple random sample (SRS) of size n consists of n individuals from the population chosen in such a way that every set of n individuals has an equal chance to be the sample actually selected.

A simple random sample is a subset of a statistical population in which each member of the subset has an equal probability of being chosen. An example of a simple random sample would be the names of 25 employees being chosen out of a hat from a company of 250 employees. In this case, the population is all 250 employees, and the sample is random because each employee has an equal chance of being chosen.

The word ‘statistics’ has been derived from the Latin Word ‘Status’ which means ‘Political State’. The word statistics has been coined by Gottfried Aschenwall, a German Political Scientist. Ronald Aylmer Fisher is regarded as father of modern statistics. The word statistics has meaning both in plural and singular sense. In plural sense, it means quantitative/qualitative figures. For example, the area under food grains in different states of India, the production of guava in different states of India, germination percentage of seeds, etc.. In singular sense, it means as a subject i.e., method of dealing with the quantitative/qualitative information.

Statistics is defined as scientific method which deals with collection, organization presentation, analysis and interpretation of data. It is included under both science and arts. Statistics is used as a tool to obtain knowledge on other subjects of science.

Samples having size, n - 30 are known as large sample. For large values of n (i.e., for n - 30), almost all the distributions are very closely approximated by normal distribution. Thus in this case we apply the normal test, which is based upon the area property of the normal probability curve. Large Sample Test is also known as Z-test or SND (Standard Normal Deviate) test.

This chapter describes fundamental terminology and lays the foundational framework for the course. It offers a general understanding of statistics, its purpose, and its role in addressing diverse problems across various fields of study. Additionally, it explains the rationale behind choosing R software as a powerful tool for data analysis.

Basic Concepts and Terminology

Statistics is an essential tool in agriculture, helping researchers to collect, analyze, interpret, and present numerical data related to plant traits, and agricultural practices. The fundamental goal is to make informed decisions and enhance crop yields, quality, and resilience.

To illustrate, researchers choose to work with a sample due to practical considerations, such as time and cost constraints. For instance, in a field with 10,000 crop plants, a researcher aims to determine the average plant height in the field. To conserve time and resources, they randomly select 200 plants from various locations within the field. These 200 plants constitute a sample from the entire population of 10,000 plants. By analyzing the height observed in the sample, the researcher can conclude the average height of the entire crop field.

This chapter begins by providing an essential overview of different data transformation methods and their importance. It primarily concentrates on exploring three crucial transformation techniques: the square root transformation, log transformation, and arc sine transformation. Additionally, this chapter explores the practical application and analysis of these transformations using the R programming language.

Data Transformation

Data transformation is often needed in statistical analysis and data processing for several key reasons:

• Assumption of homogeneity of variances: ANOVA requires that the variances within each group being compared are equal. When this assumption is violated, i.e., when variances are significantly different across groups, data transformation can help stabilize these variances.

• Normality assumption: ANOVA assumes that the residuals (the differences between observed and predicted values) are normally distributed. However, real-world data often deviates from this normal distribution. Data transformation, such as log or square root transformations, can help in reshaping the distribution of data.

• Handling skewed data: In situations where the data is highly skewed, the results of ANOVA can be misleading. Transformations can reduce skewness, making the distribution more symmetric and suitable for ANOVA.

• Proportional or percentage data: When dealing with data in the form of proportions or percentages, certain transformations (like the arc sine transformation) are necessary. This is because proportions have a bounded scale (0–1 or 0–100%) and their variances are not constant across the range, which can violate ANOVA assumptions

In this chapter, we explore Latin square design (LSD), a specialized form of experimental design renowned for its efficiency in controlling variation. This chapter thoroughly discovers the foundational principles, structure, and application of LSD in experimental research. It details mathematical models for LSD and guides readers through the interpretation of ANOVA tables. Furthermore, the chapter provides a systematic approach to conducting LSD analysis using R software.

Latin Square Design

The LSD is a statistical design used extensively in experiments, especially when specific conditions are met:

• Controlled variation in two directions: LSD is particularly suited for experiments where variation can be controlled along two different factors. This design structures the experimental units into a square matrix, allowing two sources of variability to be accounted for.

• Adherence to experimental design principles: Like other designs, LSD follows the key principles of randomization, replication, and local control. Randomization in LSD is applied within the rows and columns of the square matrix, ensuring unbiased treatment comparisons.

• Efficient use of resources: LSD allows for an effective utilization of resources by controlling two variables simultaneously. This makes it a more efficient design compared to RBD or CRD when there are two sources of nuisance variables.

Partialy Balance Incomplete Block Design (PBIBD)
    
Interblock and Intrablock Information

Q.1.    What is split plot experiment ?

    Answer
    
    If one has to study two type of factors say A and B in the same experiment where A requires larger plot size and B requires smaller plot size then the structure of the experiment will be that of plots within blocks are known as split plots and blocks within replication are known as main (whole) plots. Such a layout of the experiment is known as split plot experiment.

Q.1.    Define strip plot design ?

    Answer
    
    There is a variant of the split plot design in which both factors A and B will be applied to the larger strips by dividing the experimental field into as many rows as the number of levels of one factor A and into as many columns as the number of levels of second factor B. Then the two factors will be applied at random to the rows and columns. Such a layout is known as strip plot design.

2.1. INTRODUCTION

Many biological variables do not meet the assumptions of parametric statistical tests as they are not normally distributed, the variances are not homogeneous, or both. Objects in nature commonly exhibit one of three distributions:

6.1. INTRODUCTION

All ecologist and biology researchers have to find out: (i) how reliable are the observed results? and (ii) what is the probability of difference between observed and expected results? The question of reliability is solved by estimating the confidence interval and the second question is addressed by the application of statistical tests for significance. Both these methods help us to draw conclusions on our observations, hence, may be grouped together as ‘inferential statistics’.

Referring back to the chapter on sampling, we may recall that it is usually not possible to calculate the ‘population parameter’ directly. Instead, we have to rely on estimates. Two types of estimates can be used, a point estimate and an interval estimate. A point estimate is a single numerical value of a sample statistic that is used to estimate the corresponding population parameter. For example, the sample mean, x , is used as an estimate of the population mean, μ Point estimates are not used widely because the value of some statistic, say mean, varies from sample to sample and we can’ t conclude with any degree of certainty that a single point estimate equals the population parameter. Therefore, an interval estimate is typically favoured. An interval estimate is a range of values within which the population parameter is likely to lie.

Age old practice of photography was taken with still camera in parallel direction, not aerially scan or snapping which really present wider prospective of any object or places in rearranging or planning skilfully in two dimensional working. Earlier aeroplane played a important role in over head condition to capture snaps but again accuracy and angle of camera limited the works because these photographs were diagonally or trigonally not matched the reality of ground.

Most of the scientific research studies –except few descriptive survey research studies–will have one or more hypotheses to be tested as a part of study. Hypothesis is a prediction intended to be tested in a research study (Aron, Coups, and Aron, 2012). Hypothesis is the guide which directs the purpose and process of research by indicating the dependent and independent variables to be observed, extraneous variables to be controlled and the active variables to be manipulated.

There can be no scientific studies which do not involve the process of measuring. Measurement process is the integral part of research activity. Measurement can be defined as the assignment of numerals to the objects or events according to set of rules.  The emphasis in this definition is on phrase ‘set of rules’. If the rules can be set up based on some rational or empirical basis, then theoretically anything can be measured. It is very much easy to understand the concept of measurement in case of physical sciences (Example weight, length, mass etc.) but it is harder understand the concept of measurement in case of psychological objects/phenomenon (intelligence, aggressiveness, attitude etc.) without understanding the predetermined rules of measurement. A rule is a guide, command that tells you what to do/how to do.

1.1 Introduction
The concept of invention is the fundamental requirement for design thinking. Since the birth of mankind, people started inventing many concepts, which are mainly observations of facts and innovation of ideas for producing products that are useful to the society. The design thinking requires unconventional thinking to design processes for maximal operational efficiency of manufacturing the products, which give maximum satisfaction to customers with more value for money. Hence, the team that is involved in the design thinking projects may gain the required skills by knowing several inventions of the past. Hence, a review of inventions is presented in this chapter.
1.2 Tracing Inventions
The content at https://www.explainthatstuff.com/timeline.html gives a complete account of inventions starting from 4–5 billion years ago up to 2022. The article on solar cell energy gives an account of the sun started producing energy. About 1.2 million years ago, humans invented fire, which has been further innovated for biofuels, candles, IC engines, aviation engines, etc. In the 7th century B.C., ants were killed by burning in the heat produced by concentrated sun rays using magnifying lens. Greeks and Romans utilised burning mirrors to light candles during religious ceremonies in the third century B.C. About 3.5 million years ago, stone, wood, antlers, and bones were used by humans to make first kind of tools. These were further improved to make tools for machines. About 4000 BCE back, iron was used to make decorative ornaments. About 3500 BCE, people invented the concept of wheel, which brought quantum revolution in transportation.

Reliability aims at assessing the accuracy and consistency of whatsoever the measuring instrument measures irrespective of whether we are measuring what we are supposed to measure or not. Whereas the validity emphasises on the question “are we measuring what we were supposed to measure?”

To clarify the difference between reliability and validity, let us take an example of instruments that measure colour intensity. Assume, an expert develops two separate instruments X and Y to measure the intensity of green and red colour, respectively. But, the commercial manufacturer wrongly tags the red colour measuring instrument as X. When someone uses this wrongly tagged instrument for measuring the green colour intensity, whatever intensity values the instrument yields are reliable but not valid. It is so because the instrument is measuring red colour intensity accurately (hence data is reliable) but it is not measuring what it was supposed to measure –the green colour intensity (hence data is not valid).

Quantitative studies are preferred over qualitative studies for their relatively higher objectivity. But many of the times, complexities involved in the occurrence of phenomenon, its causes, and consequences demand for in-depth analysis of a single unit of study. Whenever we want to study a single unit in depth both qualitatively and quantitatively then case study is the best method of research. Case study is a best tool to use, when we are interested in exploring a new phenomenon on which there is very less scientific information available and also when we are interested in formulating new hypothesis for causal study.

Interview method: It is a two way interactive (either face-to-face or through multimedia technology), method of data collection wherein the researchers elicit required response from the respondent by presenting stimulus in the form of series of questions pertinent to it.

Interview schedule: It is a data collection tool consisting of properly worded, sequentially arranged questions, tests relevant to objective of research.

Existing positive as well as negative interactions among the members of the society determine the communication structure, leadership and technology diffusion pattern within the given social system. Sociometry is a method for gathering and analyzing data on communication choice and interaction pattern of individuals, within a group in order to identify communication structure and pattern.

Frequency distribution table is a statistical table which shows the sets of all distinct values of the variables arranged in order of their magnitude either individually or in groups with their corresponding frequencies side by side.

Need for frequency distribution table

1. To condense the data
2. To provides quick reference for the entire data
3. To facilitate quick & easy statistical analysis of data
4. To reveal the pattern within the data