eChapters

4.1 Learning Objectives
By the end of this chapter, you will be able to:
i) Understand the concepts of Central Tendency and Partition Values;
ii) Identify appropriate measures for different data sets;
iii) Distinguish between different types of means;
iv) Apply Partition Values in data analysis.

4.2 Introduction

Nowadays, the ability to proficiently analyze and interpret data sets is indispensable for strategic decision-making. This chapter, ‘Measures of Central Tendency and Partition Values,’ extends beyond the basics of mean, median, and mode to explore the deeper layers of data analysis through quartiles, deciles, and percentiles. These measures offer a more comprehensive understanding of data distribution, crucial for identifying trends and making precise business predictions. We will unravel how these statistical tools can be effectively employed to dissect and interpret complex data sets, providing business professionals with an enhanced toolkit for informed analysis and decision-making in various business scenarios.

5.1 Learning Objectives
By the end of this chapter, you will be able to:
i) Understand and apply Absolute Measures of Dispersion;
ii) Explore Relative Measures of Dispersion;
iii) Understand the importance of these measures in comparing the variability of data sets;
iv) Familiarize with the concept of moments in statistical distributions;
v) Learn about skewness and kurtosis as measures of distribution shape;
vi) Develop the ability to apply these concepts to real-world data.

5.2 Introduction

Nowadays, understanding the dispersion and shape of data distributions is as crucial as knowing their central tendency. This chapter, ‘Measures of Dispersion and Shape of Distribution,’ delves into the essential statistical tools that quantify the spread and shape of data sets. It covers key concepts such as range, variance, standard deviation, skewness, and kurtosis. These measures provide deeper insights into the consistency, reliability, and nature of business data, enabling professionals to make more nuanced and informed decisions. Through this chapter, readers will learn not just to describe data, but to understand the facts it tells about variability and distribution trends in a business context.

6.1 Learning Objectives
After going through this chapter, you will be able to:
i) Understand the meaning, significance, and properties of correlation;
ii) Know the various types of correlation;
iii) Comprehend the computation process of various types of correlation;
iv) Appreciate the limitations of correlation.

6.2 Introduction

Correlation is a fundamental statistical concept that measures the degree of association or relationship between two or more variables. Correlation helps us to understand how changes in one variable correspond to changes in another. Correlation values range from -1 to 1; where -1 indicates a perfect negative correlation, +1 represents a perfect positive correlation and 0 suggests no linear correlation between the variables. The sign and magnitude of the correlation coefficient provide information about the strength and direction of the relationship between the variables. Correlation analysis is a multipurpose tool with various applications, such as identifying the strength of relationships between stock prices and market indices, assessing the impact of education on income levels, or determining the association between weather conditions and crop yields. Understanding correlation is essential for making informed decisions, drawing conclusions from data, and developing predictive models in numerous fields of study and practice.

7.1 Learning Objectives
After comprehending this unit, we will be capable to:
i) Understand the conception of regression;
ii) Appreciate the significance of regression;
iii) Understand the difference between regression and correlation;
iv) Know about the regression lines;
v) Anticipate the difference between simple regression and multiple regression.

7.2 Introduction

Correlation is followed by linear regression. Regression is applied for forecasting the value of an unknown variable with the help of a known variable. The forecasted variable is called the dependent variable. On the contrary, an independent variable is used for forecasting the value of a dependent variable. If you desire to know, for instance, if exam achievement can be anticipated based on review time, whether cigarette consumption can be predicted based on smoking length, and so on, you may use linear regression. You must utilize multiple regressions if we deal with more than one independent variable rather than simply one.

8.1 Learning Objectives
After completing this chapter, you should be able to:
i) Understand the concept and significance of index numbers;
ii) Identify different construction methods;
iii) Understand the importance of selecting appropriate base and current periods;
iv) Recognize the properties and tests of adequacy;
v) Recognize the challenges and limitations of index numbers;
vi) Stay informed of recent advances;
vii) Synthesize practical examples and calculations for creating index numbers.

8.2 Introduction

In this chapter, we embark on a journey through the intricate world of index numbers, a cornerstone in the realm of statistics. We will explore what index numbers are; their various types; and the methodologies employed in their construction. Delving into real-world applications, we’ll uncover how these numerical tools are pivotal in measuring economic trends, informing policy decisions, and shaping our understanding of complex data sets. By the chapter’s end, you’ll gain insights into not only the mechanics but also the profound impact of index numbers in our data-driven society.

10.1 Learning Objectives
After reading this chapter, you will be able to:
i) Understand the concept and types of hypotheses in hypothesis testing;
ii) Differentiate between one-tailed and two-tailed tests and recognize their applications;
iii) Grasp the implications of Type I and Type II errors in hypothesis testing;
iv) Explore the significance level and its connection to confidence limits;
v) Understand the concept of degrees of freedom in hypothesis testing;
vi) Familiarize yourself with the stepwise approach to hypothesis testing for effective application.

10.2 Introduction

In this chapter, we explore the fundamental concept of hypothesis testing, a cornerstone in statistical analysis. We begin by exploring the nature of hypotheses and their types, distinguishing between one-tailed and two-tailed tests and understanding their respective applications. We then explain the intricacies of Type I and Type II errors, crucial pitfalls to avoid in hypothesis testing. Additionally, we probe into the significance level, its relationship to confidence limits, and its role in determining the strength of our conclusions. Furthermore, we explain the concept of degrees of freedom, an essential factor in hypothesis testing that affects the reliability of our results. Finally, we guide you through a stepwise approach to hypothesis testing, ensuring a comprehensive understanding and effective application of these concepts in practical scenarios. Through this journey, you will not only grasp the mechanics but also appreciate the profound impact of hypothesis testing in drawing meaningful conclusions from data in our data-driven society.

11.1 Learning Objectives
After comprehending this unit, we will be capable to:
i) Understand the conception of probability distribution;
ii) Appreciate the significance and applications of Binomial distribution;
iii) Understand the importance and uses of Poisson distribution;
iv) Know about the normal distribution and its’ applications.

11.2 Introduction

A probability distribution is a mathematical function that describes the likelihood of obtaining the possible outcomes of a random variable. In other words, it provides the probabilities of different events or values that a random variable can take. The sum of all probabilities in a probability distribution is equal to 1.

1.1 Carbohydrates Biochemistry
Carbohydrates are essential for life, both in plants and in animals. They are organic chemical compounds made of C, H, and O. These elements get arranged in different ways so that several simple, and stored complex compounds (starch in plants and glycogen in animals) are available in abundance on the Earth. They are involved in enormous biological and nonbiological functions. Their biological functions, found in all living things, are the chief energy source for men and animals, even for herbivorous fishlike carp and rohu fingerlings. Carbohydrates are also essential structural components like nucleic acids, containing genetic information. One must recognize the protective function in cells, plants, and human health management. Even bacteria form a glycocalyx and biofilm for protection. Knowledge about their involvement in blood groups helped to identify universal donors, universal recipients, and genuine donors for the recipient to save a life. Carbohydrates are found as a storage structure for energy in emergency conditions.
Upon combustion, they yield carbon dioxide and water. Autotrophs (plants) synthesize them from carbon dioxide and water to grow, reproduce, and store, but heterotrophs (humans and other animals) use them, too. In plants, they occur as fibers, supporting structures, and simple sugars. The plant carbohydrates are used for preparing adhesives, fermentation, foods, and papers. Cotton fibers are used to make clothes to cover the body and protect humans and animals against adverse climatic conditions.

Carbohydrates are the most abundant class of organic compounds found in living organisms. Carbohydrates are also called sugars and are found in plants and animals. Plants synthesize them through photosynthesis (endothermic) by fixing carbon, hydrogen, and oxygen in the presence of solar energy and chlorophyll pigments.
nCO2 + nH2O + energy → CnH2nOn + nO2
Carbohydrates are stored as starch in plants as a source of energy. They are also stored in animals as glycogen in muscles and the liver. They are also found in skeletal structures of cell walls in plants (cellulose), bacteria (peptidoglycans), insects, and crustaceans. This stored food is consumed by animals and man, and upon their oxidation in the body, energy is released for various metabolic activities (work, growth, and various productive functions). Compositionwise, plants are more prosperous in carbohydrates (30%) than animals (1%), with water at 60% and ash at 4%. Plants are poor in proteins (5%) and lipids (1%) as compared to proteins (20%) and lipids (15%) in animals.
2.1 Introduction to Carbohydrates
Carbohydrates or saccharides (sakcharon=sugar or sweetness) are a chemical term applied to a large group of substances, which are organic compounds containing elements Carbon, Hydrogen, and Oxygen in a ratio of 1:2:1. Hydrogen and Oxygen atoms are in the ratio of two to one as in water molecules. As the ratio of hydrogen to oxygen is like H2O, they are also known as “hydrates of carbon” and are represented as (CH2O)n. But there is an exception to this general formula. Deoxyribose (C5H1OO4) and rhamnose (C6H12O5) do not have the required ratio of 2:1 for hydrogen to oxygen. On the other side, some compounds possess this required ratio but are not carbohydrates, e.g., formaldehyde (CH2O), acetic acid (C2H4O2), and lactic acid (C3H6O3). Further, some carbohydrates have nitrogen(glucosamine), phosphorus, and sulfur but do not satisfy the above general formula. The simplest sugars are called monosugars or monosaccharides. There are two main functional groups in

Hydrates carbons (saccharides) found in nature (animals and plants) are classified into three groups based on the number of monosaccharides found or released upon acid hydrolysis. Monosaccharides the groups of carbohydrates.
3.1 Formulation of Monosaccharides
The molecular formula of any monosaccharide (mono=one and saccharide=sugar) is Cn(H2O)n or (CH2O)n. Rhamnose and Deoxyribose are exceptions to the general formula of carbohydrates. Acetic acid (C2H4O2) and formaldehyde (CH2O) are not carbohydrates but have the formula Cn(H2O)n or (CH2O)n. Fitting and Baeyer proposed the formula CHO-(CHOH)4-CH2OH, which indicates the presence of aldehyde and five hydroxyl groups.
More than one monosaccharide molecule unites to form disaccharides, oligosaccharides, and polysaccharides. Monosaccharides, being simpler units, cannot be broken down further. Generally, they have two to ten carbon atoms per molecule (dioses to decoses). The monosaccharides may have either aldehyde or ketone groups. The one possible aldodiose (CH2OH.CHO) is glycolaldehyde (2-hydroxyethanal), but ketodiose is not possible since there are only two carbons. The glucose molecular formula C6H12O6 represents the number of C, H, and O atoms but not groups. However, the proposed formula, CHO-(CHOH)4-CH2OH, indicates the presence of an aldehyde and five hydroxyl groups. These simple sugars or monosaccharides have carbon backbones linked by single bonds with hydrogen and hydroxyl groups attached in an unbranched manner.

Oligosaccharides comprise 2 to 10 units of monosaccharide molecules, which are released upon hydrolysis. The oligosaccharides are classified depending upon the number of monosaccharide units, e.g., Disaccharides (two monosaccharide units), Trisaccharides (three monosaccharide units), Tetrasaccharides (four monosaccharide units). Disaccharides are the most common oligosaccharides found in nature.
4.1 Disaccharides
Disaccharides are formed when glycosidic linkages covalently bond two monosaccharides together. A glycosidic bond is formed when the hydroxyl group on one of the sugars reacts with the monomeric carbon on the second sugar. A glycosidic bond can be alpha or beta, depending on the steric configuration at carbon one on the monosaccharide unit. During the union of two monosaccharide units, water molecules are eliminated (dehydration reaction or condensation reaction, or dehydration synthesis), and the formation of a glycosidic bond occurs, through which the units are linked via an oxygen bridge.

Polysaccharides comprise several monosaccharide units (more than ten) linked by glycosidic bonds; therefore, they are high molecular weight carbohydrates. They are also called polyanhydrides of simple sugars. The most common monosaccharide unit of polysaccharides is D-glucose. The polysaccharides may differ in their monosaccharide units, molecular weight, chain nature, glycosidic bond types (alpha, beta), and linkages (1-2, 1-3, 1-4, or 1-6). They are tasteless, colorless, amorphous powders that are a little water-soluble. Some may form colloidal solutions. The polysaccharides provide structural stability (structural polysaccharides) to cells, prevent cells and tissues from drying out due to their water-binding properties, and reserve food for the future. Reserved polymeric carbohydrates are osmotically less active and can be stored in large quantities within the cell.
The right and left end of polysaccharides, respectively, are called the reducing and non-reducing ends. Based on the functional aspects, they are grouped in two ways.
• The first way depends on its digestibility—1. Nutrient polysaccharides are (digestible) starch, dextrin, and glycogen, and 2. Structural polysaccharides are (indigestible), chitin and cellulose.
• The second way, depending upon the constituent monosaccharide units, they are 1. Homoglycans (same units), and 2. Heteroglycans (different units).

6.1 Metabolism
Dietary components like carbohydrates, lipids, and proteins are continuously broken down and synthesized through various chemical reactions (exergonic or endergonic) that are concerned with energy. These processes not only break them down but also interconvert them for their synthesis. Dietary components, when utilized, are known as nutrients the body. The number of reactions occurring in living cells maintains the body cells and fluid’s homeostasis and dynamic status. To keep the dynamic state of a cell, many reactions are going on to synthesize or break down the different types and sizes of molecules. The total of these reactions is known as metabolism.
The metabolism may be anabolic or catabolic. Further, it may be aerobic or anaerobic, depending on oxygen availability. These four terms are explained below:
I. Aerobic and Anaerobic Metabolism
• Aerobic metabolism-- In aerobic metabolism, the carbohydrates (substrate) in the presence of oxygen are metabolized to release energy, carbon dioxide, and water. The released energy is converted to ATP by two processes in the mitochondria (the Krebs cycle and oxidative phosphorylation). In oxidation, the electrons released from a glucose molecule are transferred to reduced NAD+ and FAD. The oxygen (electron acceptor) is reduced to water. Water formation [from H2(g) and O2(g)] is an exothermic reaction. The energy released from the chemical bond of glucose is captured as ATP, and heat is released.

Glycolysis involves a series of reactions to extract energy from glucose to
release two molecules with three carbons called pyruvate. The process is not
dependent on oxygen but can occur under aerobic and anaerobic conditions,
releasing different end products. Under aerobic conditions, the pyruvate formed
is then oxidized in the mitochondria to CO2 and H2O. The energy released in
glycolysis is in the form of ATP (High Energy Bond). All the reaction steps
occur in the cytoplasm, as it is the extramitochondrial pathway. In anaerobic
conditions and the absence of mitochondria (RBC), the end product of
glycolysis is lactic acid.
The net reaction of Glycolysis (anaerobic) may be summarized as under
Glucose + 2ADP + 2Pi → 2 Lactate + 2ATP
7.1 Steps in EMP Cycle or Glycolytic Pathway
Table 7.1: Steps and enzymes of glycolysis or EMP cycle.

The Hans Krebs cycle, known by different names like the citric acid or tricarboxylic acid cycle, was named after the scientist Hans Krebs. The citric acid cycle is named after the citrate product formed in the cycle. The name tricarboxylic acid is given as citrate itself, which is a tricarboxylic acid due to the presence of three carboxyl functional groups (-COOH).
The tricarboxylic acid (TCA) cycle occurs in aerobic conditions and within mitochondrial ETC, involving various sequential oxidoreductase reactions. The cycle requires Acetyl-CoA (C2 molecule) and oxaloacetate molecule (C4 molecule) to begin with the formation of a citrate molecule (C6 molecule). Acetyl-CoA is generated from the pyruvate (glycolysis), the fatty acids (lipolysis), or by oxidative deamination of specific amino acids. The complete oxidation of a glucose molecule results in the end products like energy (ATP), CO2 and H2O. A molecule of Acetyl-CoA has two carbons. It is formed from a three-carbon pyruvic acid molecule generated from the glycolysis of a sixcarbon glucose molecule. Acetyl-CoA is the substrate for this cycle and is an energy-rich molecule having a thioester bond between coenzyme A and acetic acid.
8.1 Formation of acetyl-CoA from Pyruvic Acid
Pyruvate is obtained by glycolysis or the oxidation of glucogenic amino acids. The pyruvic acid produced in the cytoplasm crosses the inner membrane of the mitochondrion to enter the mitochondrial matrix, where it undergoes oxidative decarboxylation, forming Acetyl-CoA. Acetyl-CoA carries (carrier molecule) the acetyl group to the citric acid cycle. It also links various metabolic products with the TCA cycle. Such products can be from the EMP cycle, beta-oxidation, and oxidative deamination of amino acids. Oxidative decarboxylation of pyruvic acid needs a complex enzyme system known as pyruvate dehydrogenase

This alternative pathway for glucose oxidation has enzymes in the cytosol through which glucose is degraded to carbon dioxide. It is also called a shunt because carbon atoms from glucose 6-phosphate take a long route before they proceed down the EMP or glycolytic pathway. This pathway is anabolic in nature as it produces many intermediates. In this pathway, two NADPH are generated from one glucose 6-phosphate molecule. No ATP is generated or consumed.
Overall, this pathway needs three glucose-6-phosphate molecules to complete all the reactions. Therefore, one cycle run produces 2 mol of fructose-6- phosphate, 1 mol of glyceraldehyde-3-phosphate, and 3 mol of carbon dioxide from 3 mol of glucose-6-phosphate with the reduction of 6 mol of NADP+ to NADPH.
Overall reactions of the HMP pathway are as follows:
3 Glucose-6-P + 6 NADP+→ 3 ribulose-5-P + 3 CO2 + 6 NADPH
3 Ribulose-5-P → 2 xylulose-5-P + Ribose-5-P
2 Xylulose-5-P + Ribose-5-P → 2 fructose-6-P + Glyceraldehyde-3-P

The uronic acid pathway is a minor and alternative pathway for glucose oxidation and for the production of uronic acid sugar. Uronic acids of sugars are formed when the terminal carbon’s (C6) hydroxyl group (CH2OH) of sugar gets oxidized to a carboxylic acid. Uronic acid names depend on the name of the parent sugars. Uronic acid of glucose is glucuronic acid, hexose uronic acids are hexuronic acids, and pentose uronic acids are named penturonic acids. The uronic acid pathway is located in the cytosol of the liver and adipose tissue.
The glucuronic acid should not be confused with gluconic acid (aldonic acid) and glucaric acid (aldaric acid or dicarboxylic acid). Gluconic acid (aldonic acid or “onic” acid) is obtained from D-glucose by oxidizing its aldehyde group at C1 to a carboxyl group. Glucaric acid (aldaric or dicarboxylic acid) is formed upon oxidation of both the terminal hydroxyl groups (C1 and C6) and the aldehyde. They have two carboxylic acid groups. This pathway yields pentoses & ascorbic acid or vitamin C (not in primates & guinea pigs) from glucose.

Plants, with the help of photosynthesis, convert simple molecules (like carbon dioxide and water) into more complex molecules (anabolism) through various chemical reactions. Animals synthesize more complex molecules (anabolism) from monomer units through multiple chemical reactions. Anabolism refers to a biosynthesis process involving the sequences of biochemical reactions by which complex molecules are formed in living cells from available monomer units. Anabolic processes include the biosynthesis of complex biomolecules like carbohydrates, proteins, and lipids. The anabolic processes require energy from high-energy bond compounds like ATP.
The anabolic processes related to the biosynthesis of carbohydrates include
(1) Gluconeogenesis and
(2) Glycogenesis
11.1 Gluconeogenesis
Gluconeogenesis is the synthesis of new glucose molecules from noncarbohydrate molecules. The non-carbohydrate molecules that are used for gluconeogenesis are derived from glycolysis, lipolysis, and proteolysis. Examples are pyruvate, lactate, glycerol, and glucogenic amino acids. The liver and kidneys are the preferred organs for the gluconeogenesis process. It also occurs to some extent in the brain. Blood glucose levels are maintained at physiological concentrations through the process of gluconeogenesis in humans and ruminants. The blood glucose drops below normal levels during various conditions like fasting, starvation, pregnancy, lactation, fever, and low carbohydrate diets.

Glycogenesis helps to maintain blood glucose, as it is a primary source of energy for every cell. An inadequate supply of glucose to the body may lead to the failure of the vital organs. Glucose, when in excess, is synthesized into glycogen and stored (glycogenesis), mainly in the liver and muscles (including the uterus). It is also found in other tissues, such as the kidney, brain, heart, and adipose tissues. The stored liver glycogen is broken down (glycogenolysis) into blood glucose when needed. Still, the glycogen stored in muscle is broken down to glucose 1-phosphate and converted to blood glucose later. Either the synthesis or breakdown pathway is primarily active at a time in tissues. Glycogen synthesis and its breakdown are not reversible processes.
The percentage of glycogen stored in skeletal muscles is less ( 1% to 2%) than in the liver (10%). Still, the total quantity of glycogen is greater in muscles due to their greater mass than an animal’s total liver mass. Muscle mass, on average, ranges from 40 to 45 % of body weight for the most mature mammals.
Glycogen synthesis (glycogenesis) is an anabolic process forming a branched homopolysaccharide made of several chains of alpha D-glucose units linked by α-1,4 glycosidic bonds in a linear chain and nearly every ten glucose residues; the chain branches off via α-1,6 glycosidic linkages. The α-glycosidic bonds give rise to a helical polymer structure. The number of glucose units in a single glycogen molecule varies (1,700 to -600,000 units). The glycogen granules are formed on the glycogenin protein in the center, so it acts as a primer, and their number determines the total number of glycogen particles.

Glycogen breakdown is known as glycogenolysis. This breakdown occurs in the cytoplasm of cells of the organs, wherein glycogen is stored. The end product of glycolysis is glucose in the liver, which is glucose-1-phosphate in skeletal muscles.
Glycogen is a polysaccharide synthesized from glucose monomers. Glycogen is stored in the cell instead of glucose because glucose requires more water, whereas glycogen can be stored with much less water. Most of the stored glycogen is in the liver (5% of body weight) and muscle (1 to 2 % of body weight). Muscle and liver glycogen undergo glycolysis for different purposes. Muscle glycogen provides energy for muscular activity, whereas liver glycogen helps to maintain the blood glucose level.
Glycogen contains many glucose residues (up to 120,000) linked by alpha 1-4 glycosidic bonds, forming a straight chain. Glycogen has branches at alpha 1-6 glycosidic bonds at every ten glucose residues. At the reducing end, the carbon one of glucose is free, whereas it is not accessible in the case of non-reducing ends.