eChapters

In 1935, Prof. Ronald A. Fisher laid the foundation for the subject in his monumental work entitled “The Designs of Experiments”. Experimental designs concern the arranging of treatments in such a manner that the inferences and conclusions regarding the effects of these treatments can be easily done and their reliability measured. Experiments are made with a view to find the validity of a particular hypothesis and to have an idea about the extent of the reliability that can be placed on a particular conclusion arrived. The selection of the design will have a very great bearing on the accuracy of the ultimate results. By a random selection of experimental units it is possible to remove the ambiguity about the casual interpretation of the observed associations. Random sampling is the most essential ingredient of all experimental designs. Besides, there are many devices for increasing the precision of the inference and the calculations.

To understand design of experiments, let us first get acquainted with some important terms and concepts associated with design of experiments:

• Experiment: Experiment is a scientifically planned method. The experiment is conducted draw a valid conclusion about a particular problem. The conclusion is based on statistically observation.

• Treatment: The object or the procedures under comparison in an experiment are known as treatment. For Example: Fertilizers, varieties, cultivation practices, irrigation level in case of agricultural experiments and drugs, breeds, farms, etc. in case of animal experiments.

• Yield: The response of the treatment is measured by some indicator such as crop production, milk production, body temperature, etc. Such an indicator is called yield. The treatments are applied to some units such as field plots, sample of cows, sample of patients, etc. and the effect on the yield is observed.

• Experimental material: The experimental material is nothing but a set of experimental units. In other words, the material which is used in experiment is known as experimental material. For Example: A piece of land i.e. Agricultural field in agricultural experiments and a group of cattle i.e. animals in animal experiments.

Statistical Inference

It is the process of drawing valid inference or conclusion about the population on the basis of a sample drawn from the population. In other words, Statistical inference is the process of using a sample to infer the properties of a population. It requires studies pertaining to three aspects:

• Point Estimation • Interval Estimation

• Testing of Hypothesis • Point Estimation: It is to find a single value, which is used as an estimator of unknown parameter. For example, sample mean may be used as an estimate of the population mean µ.

• Interval Estimation: It is to find confidence limits based on sample values, within which the unknown parameter lies with confidence coefficient described in probability term. In other words, we estimate an unknown parameter using an interval of values that is likely to contain the true value of that parameter and state how confident we are that this interval indeed captures the true value of the parameter.

• Testing of Hypothesis: The tests used to ascertain whether the differences between estimator and the parameter or between two estimators are real or due to chance are called tests of hypothesis or tests of significance. In other words, the procedures, which enable us to decide whether to accept or reject hypothesis, are called tests of hypothesis or tests of significance. Two types of problems mainly arise in tests of significance:

• Testing the significance of the difference between estimator and the pa rameter.

An index number is a statistical measure designed to show changes in a variable or group of related variables with respect to time, geographical location or other characteristics like: income, profession, etc. A collection of index numbers for different years, months, locations, etc. is called an index series. Since index number are usually used to measure the level of business and economic activities over a period of time, so they are rightly called barometers of economic activities or simple called economic barometers. The various problems faced in the construction of an index number are to known the specific purpose of index number, selection of items, data for index numbers, choice of base periods, choice of an average, selection of weights and choice of a suitable formula.

Methods of Constructing Index Numbers

A number of formulae have been developed for constructing index numbers, which may be grouped into the following categories:

• Unweighted index numbers

• Weighted index numbers

• Unweighted index numbers: In this case, weights are not expressly assigned. 

Time Series

A series of observations recorded over time is known as a time series. In other words, a time series is a set of observations taken at specified times, usually at equal intervals. For example: Annual retail sales, rainfall measurements, heartbeats per minute, etc. Such studies are to be based on the analysis of time series data collected over time. Thus, the analysis of time series plays an important role in an investigation of economic, social, commercial and even biological phenomenon.

Components of time series

In a long time series, generally, there are following four components:

• Secular trend or long term movements

• Seasonal variations

• Cyclic variations • Random or irregular movements

• Secular trend or long term movements: It is that characteristic of a time series which extends consistently throughout the entire period of time under consideration. It shows a long term tendency of an activity to grow or to decline. The term long period of time in a relative phenomenon and cannot be defined exactly. For some cases a period as small as a weak may be fairly long while in other cases, a period as long as two years may not be assumed long.

Theoretical Probability Distribution

Many random variables associated with statistical experiments have similar properties, and as such can be described by the same probability functions. The distribution obtained from a theoretical listing of outcomes and probabilities, which can be described by a mathematical formula representing same phenomenon of interest is called a theoretical probability distribution. The distributions of discrete and continuous random variables are accordingly called discrete or continuous probability distributions.

Discrete Probability Distribution: It can be defined as a probability distribution of discrete random variables. Such a distribution will represent data that has a f inite countable number of outcomes. If the random variable X is a discrete random variable, the probability function P(X) is called probability mass function and its distribution is called discrete probability distribution. There are two conditions that a discrete probability distribution must satisfy. These are given as follows:

Types of Discrete Probability Distributions

There are several important discrete probability distributions-

• Uniform distribution

• Binomial distribution

• Negative Binomial distribution

• Geometric distribution

• Poisson distribution

• Hyper-Geometric distribution

The theory of probability has its origin in the games of chance related to gambling such as throwing a die, tossing a coin, drawing cards from a pack of cards etc. Jerame Cardon (1501-1576), an Italian mathematician was the first man to write a book on the subject entitled “Book on Games and Chance”, which was published after his death in 1663. The systematic and scientific foundation of the mathematical theory of probability was laid in mid-seventeenth century by two French mathematicians B. Pascal (1623-1662) and Pierre de Fermat (1601 1665).

Important terms and concepts

To understand probability, let us first get acquainted with some important terms and concepts associated with probability.

• Trial: An experiment is any natural or planned activity for which it is not possible to predict the particular result called a trial.

• Events: The possible outcomes of experiments of a trial are called events. For example: Tossing of a coin in an experiment and one toss of the coin is called trial. In tossing of a coin there are only two outcomes or events, i.e. head or tail.

Types of events

Events can be of following different types:

• Exhaustive events

• Equally likely events

• Mutually exclusive events

• Dependent events

• Independent events

• Null or impossible events

• Exhaustive events: It is the total number of all the possible outcomes of an experiment.

Correlation

In a statistical analysis, when two variables are studied simultaneously, then it is called “Bivariate study’. Now, in a bivariate study, if it is found that change in the values of one variable then in sympathy with the change in the values of other variable, then these two variables are said to be associated and the association is termed as correlation. In other words, the term correlation indicates the relationship between two such variables in which with changes in the values of one variable, the values of the other variable also change. In short, the tendency of simultaneous variation between two variables is called correlation.

Types of Correlation

Correlation can be classified by-

• Positive and Negative correlation: The correlation may be classified according to the direction of change in the two variables. In this regard, correlation may be either positive or negative. Positive correlation refers to the movement of variables in the same direction. Negative correlation efers to the movement of variables in the opposite direction.

• Simple, Partial and Multiple correlations: When only two variables are studied it is a problem of simple correlation. When three or more variables are studied simultaneously, it is a problem of multiple correlations. On the other hand, in partial correlation we recognize more than two variables, but consider only two variables to be influencing each other the effect of other influencing variables being kept constant.

• Linear and non-linear correlations: In linear correlation for every unit change in the values of one variable, there is constant change in the value of other variable. The perfect positive and negative correlations are also linear correlations. In non-linear correlation the change in the values of one variable does not have a constant ratio to the change in the other variable.

Dispersion

Measure of central tendency tells about the central position of the series. They do not throw like on the formation of series because in some cases, two series may have a same measure of central tendency but differ in their composition. In order to study the formation, composition, structure and scatteredness of the items of the series, measure of second order are necessary. The measure of second order will give idea about the formation of the series. As the representation of a series on the basis of its measure of central tendency is not sufficient but when a series is represented on the basis of its measure of central tendency as well as its measure of dispersion, then it will give the complete adequate representation of the series. Therefore, it is defined as the extent or degree of which data tend to spread around an average is called the dispersion. It is also known as variability or measure of second order.

Characteristics of an ideal Measure of Dispersion

An ideal measure of dispersion should possess the following properties:

1) It should be rigidly defined i.e. it has a fixed and finite value

2) It should be easy to understand

3) It should be easy to calculate

4) Its calculation should be based on all the observations

5) It should be capable for further algebraic treatment

6) It should not be affected by extreme values of the observations

7) It should be least affected by fluctuations of sampling

Need of an average

Through frequency distribution are the condensed form of raw data, but these are not sufficient for making comparison between two or more data. Therefore further condensation of frequency distribution into a single figure is made. This single f igure is always representative of the frequency distribution and such a figure is known as average.

Measure of Central Tendency

According to Croxten and Cowden “An average is a single value within the range of the data that is used to represent all the values in the series”. A statistical measure used for representing the centre of central value of a set of observations is known as “Measure of Central Tendency”. It is also called as “Measure of f irst order”.

Characteristics of an ideal Measure of Central Tendency

An ideal measure of central tendency should possess the following properties:

1) It should be rigidly defined i.e. it has a fixed and finite value

2) It should be easy to understand 3) It should be easy to calculate

4) Its calculation should be based on all the observations

5) It should be capable for further algebraic treatment

6) It should not be affected by extreme values of the observations

7) It should be least affected by fluctuations of sampling

Diagrammatic Presentation of Data

The quantitative data when classify are converted into averages and if the number of average is large then it is not possible to draw a conclusion from their large numbers of averages. Moreover quantitative data are dull, confusing, boring for many of us. Therefore, a statistical device to make such confusing data more attractive was developed i.e. if such complex data are presented by geometric f igures, pictures, graphs or curves then it will become more attractive and more impressive, is called as diagrammatic representation of data. These diagrams are nothing but the use of geometrical figures to improve the overall presentation and offer visual assistance for the reader.

Types of Diagrams

Generally, diagrams are made for a special series or a time series and they are divided into five different categories:

1) One dimensional diagrams

2) Two dimensional diagrams

3) Three dimensional diagrams

4) Pictograms

5) Cartograms

Population: The word population or universe in statistics means a set of observations relating to a phenomenon under investigation.

Sample: It is a fraction or a subset of population drawn through a valid statistical procedure so that it can be regarded as representative of the entire population.

Sampling: The statistical procedure of drawing a sample from the population is called Sampling.

Census enquiry or complete enumeration: When each and every unit of the population is investigated for the characteristic under study, known as census enquiry or complete enquiry.

Sample enquiry or sample enumeration: It is a part of the population is investigated for the characteristic under study, known as sample enquiry or sample enumeration.

Necessity of sampling: Since, the process of collecting information from all the elements of a large population may be, in general, expensive, time consuming and difficult. We help to sampling and generalise the properties of the sample to the entire population. The statistical procedures which are used for drawing inferences or conclusions about the population from sample data are covered under Inferential Statistics or Statistical inference. Thus, sampling theory is the basis of statistical inference in which we wish to obtain maximum information about the population with minimum effort and maximum precision.

Statistical Series: When observations of any characteristic present in the individual of a group are recorded and arranged one after the other in a systematic order, they form a series, known as statistical series. For example: milk yield of 20 cows, height of the students in a class in a systematic order, etc. These are some important types of series:

• Discrete series: In this series, observations can take up only exact values and not any fractional values, is called discrete series. That means, this type of series has whole number of counting. For example: number of the students in a class, members of a family, number of births in a certain year, etc

• Continuous series: A series which can take up any numerical value (integral/ fractional) within a certain range is called continuous series. For example: milk yield of 20 cows, height, weight, temperature, etc.

• Array series: When we arranged the observations of a series either in ascending order of their magnitude or descending order of their magnitude, then it becomes an array series.

• Spatial series: When items of a series are given according to space or place or time, then such type of series is known as spatial series. For example: month wise milk yield of a herd of cows, month wise income or expenditure of an industry, population of a country according to states, etc.

Introduction to Statistics

The word statistics has been derived from Latin word “status”, the German word “statistik”, the Italian word “statistica” or the French word “statistique”. During early periods, it was used by emperors and kings only for collecting the information about state and other information which was needed about their people, their number, land, status of people, revenue of the state etc. Gradually, it meaning and uses extended and there onwards its nature changed.

Some scientist/ researchers were given definition of statistics as per time and place.

• An ancient date, A.L. Bowley gives some definitions according their use and place. He seems to be inadequate because these do not include all aspects of statistics. As per time, place or requirement, he gave three definitions of statistics:

• Statistics may be called the science of counting

• Statistics may rightly be called the science of average

• Statistics as the science of counting and average

 

4.1 Principles of Economics

• ‘Economics’ word derived from two Greek words, oikos (a house) and

nemein (to manage) means ‘managing an household’

• Microeconomics: Study of the behavior of individual, small, isolated

and disaggregated units.

• Macroeconomics: Study of groups and broad aggregates of the economy.

• Adam Smith wrote the book- “An Enquire into the Nature and Causes of

Wealth of

• Nations” or in short “Wealth of Nations” in 1776.

• Adam Smith is known as father of modern economics.

4.2 Wealth Definition - Adam smith

• Adam smith (1723 - 1790), in his book “An Inquiry into Nature and

Causes of Wealth of Nations” (1776)

4.3 Welfare Definition - Alfred Marshall

• Alfred Marshall (1842 - 1924) wrote a book “Principles of Economics”

(1890)

• Alfred Marshall defined “Political Economy” or Economics is a study

of mankind in the ordinary business of life; it examines that part of

individual and social action which is most closely connected with the

attainment and with the use of the material requisites of well being”.

4.4 Scarcity Definition - Lionel Robbins

• Lionel Robbins published a book “An Essay on the Nature and

Significance of Economic Science” in 1932.

• Robbins defined, “economics is a science which studies human behaviour

as a relationship between ends and scarce means which have alternative

uses”.

Computers: Input and Output Devices

3.1 Input Device

Input devices are the pieces of hardware that allow to enter or input data into

a computer.

Keyboard

• A computer keyboard serves as a data input device.

• A keyboard is a device containing keys, each of which denotes an

alphabet, number, or number instruction that may be served to a computer

to do multiple tasks.

• The keyboard is a significant input device, both computers and laptops

use keyboards to enter commands into the computer.

Mouse

• Mouse is a pointing device.

• The mouse is used to open various files and programs by directly clicking

on the various icons on the system.

• A mouse has three buttons on top and one trackball on the bottom, which

are used to select and move the mouse.

• A touchpad is provided as an alternative for the mouse on laptops, which

aids in mouse pointer movement.

Number line

Every real number occupies a position on the number line and conversely,

every point of the line can be matched with a real number.

Absolute value

The absolute value of modules of a number is equivalent to the value of that

number ignoring the sign. For example, the absolute value for -1 is 1, and the

absolute value of +1 is also 1.The absolute value is defined by the symbol

?x?.

Statistics has originated as a science of statehood and found applications slowly and steadily in Agriculture, Economics, Commerce, Biology, Medicine, Industry, planning, education and so on. Sir Ronald A. Fisher, (1890 – 1962), who is called “the Father of Statistics”, drew many solid conclusions

1.2 Definition

Statistics is the science which deals with the collection of data, organisation of data (or) classification of data, presentation of data, analysis of data and interpretation of data, which are known as the statistical methods.

“Statistics is the science, technology and art of creating new knowledge.”

–RA Fisher.

“By statistics we mean, aggregate of facts affected to a marked

extent by multiplicity of causes numerically expressed enumerated or

estimated according to reasonable standard of accuracy collected in a

systematic manner for a predetermined purpose and placed in relation

to each other.”

– Professor Horace Secrist

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.

In this chapter, we will explore multiple comparison tests, with a particular emphasis on three significant tests: least significant difference (LSD), Bonferroni, and the techniques to visualize their outcomes. We aim to facilitate a comprehensive understanding of how to conduct and interpret the results of these comparison tests visualization using the R software.

Treatment Comparison

The “doebioresearch” package is commonly used for testing the significance of treatment groups in various experimental designs such as randomized block design (RBD), completely randomized design (CRD), and least significant difference (LSD). This package excels in conducting multiple comparison tests like LSD, Duncan’s multiple range test (DMRT), and Tukey’s honestly significant difference (HSD). It efficiently calculates all comparisons in the background, resulting in the generation of only group labels, as previously demonstrated in the output of the “doebioresearch” package. Additionally, the “agricolae” package is another useful tool that provides detailed values for multiple comparison tests. These three tests—LSD, DMRT, and HSD—are particularly popular in agricultural experiments and form the focus of our discussion.

I. Least significant difference (LSD) test

• To conduct the least significant difference test for comparing treatment means, the “agricolae” package is essential.

• To install the “agricolae” package, use the following code:

• install.packages (“agricolae”)

• After installation, activate the package with this command:

• library (“agricolae”)

• The data frame required for analysis in this context remains consistent with what is needed for randomized block design (RBD), completely randomized design (CRD), LSD, and variability analysis, among others. As an illustration, consider an example involving variability analysis.

This chapter focuses on split plot and strip plot designs, essential tools in experimental design known for effectively handling complex experiments. It covers the key principles and applications of these designs, emphasizing how they can be used to address specific research challenges. The chapter also provides practical guidance on conducting analyses of split and strip plot designs using R software, making it a valuable resource for researchers seeking to apply these methods in their work.

Split Plot Design

The split-plot design is a statistical approach widely used in experiments where specific conditions apply:

• Hierarchical arrangement of treatments: This design is ideal for experiments involving two levels of experimental units, typically “whole plots” and “subplots.” The main treatments are applied to the whole plots, while the sub-treatments are applied within these to the subplots.

• Adherence to experimental design principles: The split-plot design follows the principles of randomization, replication, and local control, but with a hierarchical twist. The main treatments are randomized across the whole plots, and the sub-treatments are randomized within these main plots. • Resource management in unequal experimental conditions: This design is particularly useful when some factors require larger experimental units or when changing levels of one factor is more resource-intensive than changing levels of another factor.