eChapters

What is dispersion?

In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range. For instance, when the variance of data in a set is large, the data is widely scattered. On the other hand, when the variance is small, the data in the set is clustered.

Dispersion is contrasted with location or central tendency, and together they are the most used properties of distributions.

Measure of dispersion

(a) Range

The range is defined as the difference between the largest and smallest observations. Range(X)= Max(X)- Min (X)

Example

The range in statistics for a given data set is the difference between the highest and lowest values. For example, if the given data set is {2,5,8,10,3}, then the range will be 10 – 2 = 8. Thus, the range could also be defined as the difference between the highest observation and lowest observation.

In R there is facility of reshaping the data. Reshaping of data includes joining of two or more vectors to create a data frame. Also two or more data frames can be merged. Appropriate functions are available for these jobs. While merging the data frames, we can join columns or rows.

Illustration of joining two vectors by using cbind() function.

15.1. Introduction

Meta-analysis is a statistical technique used to combine the results of multiple studies to produce a single, more precise estimate of the effect size of a particular intervention or association. It is a powerful tool in research, particularly in f ields such as medicine, psychology, education, and social sciences, where it is often necessary to synthesize evidence from numerous studies to draw meaningful conclusions. This chapter provides a comprehensive overview of meta-analysis, including its history, methodology, advantages, limitations, and applications.

15.2. Historical Background

The concept of meta-analysis dates back to the early 20th century, but it was not until the 1970s that the term was coined by Gene V. Glass. Glass defined meta-analysis as “the statistical analysis of a large collection of analysis results from individual studies for the purpose of integrating the findings” (Glass, 1976). Since then, meta-analysis has evolved significantly, with advancements in statistical methods and software making it more accessible and robust.

15.3. The Need for Meta-Analysis

15.3.1. Accumulation of Evidence

In many fields, particularly in medicine and psychology, a large number of studies are conducted on the same or similar research questions. However, individual studies may have small sample sizes, leading to low statistical power and inconclusive results. Meta-analysis allows researchers to pool data from multiple studies, increasing the overall sample size and statistical power, thereby providing a more reliable estimate of the effect size.

At country level, there is major paradiam shift from cropping systems towards farming system research in XI Plan period. The mandate of Project Directorate for Cropping Systems Research, Modipuram stands changed with focus on Farming Systems and re-named as Project Directorate for Farming Systems Research. w.e.f. 1.4.2009. The All India Co-ordinated Research Project on Cropping Systems with its 69 on-station and on-farm centres representing different agro-climatic zones of the country also stands re-named as AICRP on Integrated Farming Systems. As such, the much needed methodologies and procedures for planning, monitoring and interpretation of farming systems related data are not readily available. Therefore, some methodologies and procedures which can be adopted to execute Farming Systems Research for drawing meaningful conclusions are discussed here.

2.1  Write True/False

Q.1    From theoretical point of view, the most general method of estimation is the “Maximum Likelihood Estimation Method.”
Q.2    The method of “maximum Likelihood Estimation” was initially formulated by the statistician named as “C.F. Gauss.”

2.1 Simple Random Sampling

The simplest and common most method of sampling is simple random sampling. In this procedure, the sample is drawn unit by unit with equal probability of selection for each unit at each draw. It is sometimes referred to as unrestricted random sampling.

2.1.1 Definition

“A sampling procedure which gives equal chance to all possible samples each of size n which can be drawn from a population of size N, is called simple random sampling.”

13.1 Introduction

In the previous chapters, we have considered different tests of significance based on normal distribution, F distribution, student’s t distribution and Chi-square (x²) distribution which are frequently applied by the users of different disciplines of applied sciences and social sciences. In this chapter some more tests of significance are considered which have not been covered in those chapters. These tests are also based on t, F, Fisher’s z transformation,  and some other test statistics are also discussed one by one in the following sections.

13.2 t Test for Testing the Significance of an Observed Sample Correlation Coefficient r ^xy

Let r be the observed sample correlation coefficient based on a small sample of size n drawn from a bi-variate normal population with population correlation coefficient r. The standard error of r is given as

1. Which of the following states has nutrient use of < 100 kg/ ha?

(a) Madhya Pradesh                (b) Karnataka

(c) West Bengal                       (d) Uttar Pradesh

2. Which of the following is the leading state in maize production

(a) Bihar                                   (b) Uttar Pradesh

(c) Punjab                                (d) Gujarat

Abiotic Stress Tolerance: The ability of plants to withstand and adapt to nonliving environmental stressors, such as drought, salinity, or extreme temperatures. Afforestation and Reforestation: Afforestation involves planting new forests in previously non-forested areas, while reforestation involves restoring degraded or deforested lands with trees to sequester carbon. Agrarian Society: A society characterized by agriculture as the primary means of subsistence and economic activity. Agrarian: Pertaining to the cultivation of land, agriculture, or rural life. Agribusiness: The commercial activities, including production, processing, and distribution, related to agriculture and farming. Agricultural Chemistry: A branch of chemistry that focuses on chemical processes and substances related to agriculture, including soil chemistry, fertilizer composition, and pesticide development. Agricultural Diversification: The strategy of expanding the range of crops, livestock, or other agricultural activities in a rural area to increase income and food security. Agricultural Economics: A branch of economics that focuses on the production, distribution, and consumption of agricultural goods and services, as well as the economic factors and policies that influence agricultural markets. Agricultural Extension: The application of scientific research and new technologies to agricultural practices through education and advisory services. Agricultural Innovation: The process of developing and adopting new ideas, practices, and technologies in agriculture. Agricultural Machinery: Various mechanical devices and equipment designed for use in farming, such as tractors, plows, and harvesters. Agricultural Subsidies: Financial incentives provided by governments to farmers or agricultural producers to support their production, stabilize prices, or achieve other economic or social objectives. Agriculture: The science and practice of cultivating crops, raising livestock, and managing resources for food production and other agricultural products. Agroecology: The scientific discipline that applies ecological principles to the design and management of sustainable agricultural systems, focusing on biodiversity, nutrient cycling, and reduced environmental impact.

Moments

In statistics, moments are measures of the shape and variability of a data set. They are used to describe the location and dispersion of the data. There are several types of moments that can be calculated, each providing different information about the data set.

Moments is a term which refers to a measure of force with respect to its tendency which provides rotation. The strength of the tendency depends upon the amount of force and distance from the point of origin where force is exerted.

So total moments of all the forces about origin, if total moment is divided by the total force, which is called moment actually arithmetic mean for frequency distribution.

Hence, arithmetic mean is called first moment about the origin

• Arithmetic mean of various powers expressed as deviation from mean in any distribution is called the moment of the distribution.

5.1  Write True/False

Q.1    The null hypothesis H₀ is said to be composite if it does not specify the joint probability of x₁, x₂, ......, x_n completely.
Q.2    We may write null hypothesis as :
    H₀ : θΣ Ωθ_0-

Mosaic is the combination of two or more raster images into single image. Sometimes, single image captured by remote sensing satellite doesn’t completely covers the desired study area mostly. Therefore, users resort for mosaicking two or more images captured by satellite sensors from different paths so that the whole study area is merged into a single raster image to facilitate the image processing and analysis. The pixel size of the raster images should be same for mosaicking.

In most of the cases, there is spatial overlapping among the raster images from which we want to create mosaic. These overlapping areas are handled in multiple ways which is being discussed further using R – code example. For mosaicking raster images in R, mosaic function of the raster package is used.

In many situations, experiments may not known in advance which contrasts they wish to compare, or they may be interested in more than (m – 1) possible comparisons.  In many exploratory experiments, the comparisons of interest are fixed only after preliminary examinations of the data.
 
13.1 Scheffe’s Method for Comparing All Contrasts

Scheffe (1953) has proposed a method for comparing any all possible contrasts between treatment means.  In the Scheffe method, the type I error is at most a for any of the possible comparisons.

Q.1 Cluster sampling restricts the spread of sample over the population which results generally in increased sampling variance of estimators under considerataion.

Q.2 In adopting cluster sampling, few selected clusters are enumerated partially.

7.1 Introduction

In chapter 5, the cluster sampling has been discussed in which clusters were considered as sampling units and all the elements in the selected clusters were enumerated completely. It has been stated there that the cluster sampling is economical under certain circumstance but the method restricts the spread of sample over the population which results generally in increased sampling variance of the estimator under consideration. It is, therefore, logical to expect that the efficiency of the estimator will be increased by distributing the elements over a large numbers of clusters. Hence instead of observing all the units in the selected clusters, one may like to observe only few elements of the selected clusters. This will involve the sampling work to be done also within the selected clusters. If we do so, the process of selection will involve sampling work to be done at two stage, one for selecting the clusters where sampling units are the clusters and the second for selecting the units (elements) within the selected clusters where sampling units are the elements.

Multivariate analysis is essentially the statistical process of simultaneously analyzing multiple independent variables with multiple dependent (outcome or criterion) variables using matrix algebra. During the last two or three decades, multivariate statistical analysis has become increasingly popular. The theory has made great progress, and with the rapid advances in computer technology, routine applications of multivariate statistical methods are implemented in several statistical software packages. The traditional approach to the teaching of multivariate statistical analysis, as exemplified by Anderson (1958), relies heavily on advanced matrix mathematics. On the other hand, Hury and Riedwyl (1988) suggest that it is possible to understand most of the basic ideas underlying multivariate statistical analysis without a mastery of such mathematics, provided that these are conveyed with the help of real data sets. In this chapter we propose a non-mathematical data-driven approach for teaching multivariate statistical methods to students. Despite this, we are mindful of the need for students to know some basic linear algebra and univariate statistical concepts. Such basic knowledge provides students with the foundation necessary for the application of the appropriate multivariate statistical procedures and for the interpretation of results.

True/False

Q.1.    The most fundamental assumption for “Parametric inference” is that the random sample drawn is from “Normal” population.
Q.2.    The most fundamental assumption of “Non-parametric” test is that the random sample drawn is from “Continuous distribution”
Q.3.    The standard parametric methods are applicable for “Estimating the parameter” as well as for “Testing the hypothesis” about the parameters.
Q.4.    Non-parametric methods, are applicable only for testing the hypothesis about  distribution but not applicable for estimation purpose of parameter.

Q.1 The most fundamental assumption to be followed by random variable for using the standard method (parametric statistic) or parametric inference is that the random variable follows normal distribution.

Q.2 The standard parametric techniques are not useful for estimating parameters and testing hypothesis about them.

15. 1 Introduction

Data collected in social sciences and economic study, are usually based on answers provided by the respondents. These answers are aften based on re-collection of past memories and often suffer from memory lapses, intentional erros etc. As such they provide only a crude informations which are never exact. The analysis of such data on the pattern of other biological data will not yield much information because many times they will not fulfil the underlying assumptions. For such data, the non-parametric (N.P.) methods are most appropriate.

15.2 Parametric Tests

The tests which deal with the parameters of the populations, are known as parametric test e.g. Z, t & F tests are the parametric tests.