eChapters

Matrix is a rectangular array of numbers arranged in rows and columns. The order of a matrix is written as r x c, where r is the no. of rows and c is the no. of columns. The numbers included in a matrix are called elements of the matrix. The name of the matrix is denoted by an uppercase English alphabet.

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

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

Introduction

Measurement of pH of a solution and preparation of different types/strength of buffers are common practices in a laboratory, particularly in the Biochemistry and Molecular Biology laboratories. Therefore, in this chapter you will be acquainted with the procedures of measuring pH of a solution as well as preparation of some of the commonly used buffers in a laboratory.

Potential or Power of Hydrogen Ion (pH)

Measurement of pH of a solution is measuring the molar concentration of hydrogen ion [H+] in the solution, which is an indication of the acidity or basicity of the solution. The symbol pH stands for “power of hydrogen” and numerical value for pH is just the negative logarithm of the molar concentration of hydrogen ions [H+]. The ionic product of water (Kw ) is the basis for the pH scale. The total hydrogen ion concentration from all the sources and it can be experimentally measured.

The pH scale designates the concentration of H+ (and thus of OH−) in an aqueous solution, which ranges between 1.0 M H+ and 1.0 M OH−. The term pH is defined by the expression

Thus, the ionic product of water makes it possible to calculate the concentration of H+, with given concentration of OH−, and vice versa. The value of 7 for pH of a neutral solution is not an arbitrarily chosen figure; it is derived from the absolute value of the ionic product of water. A solution having pH greater than 7 is basic, and the concentration of OH− is greater than that of H+. A solution having a pH less than 7 is acidic.

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.

The central tendency measure is defined as the number used to represent the center or middle of a set of data values. The three commonly used measures of central tendency are the mean, median, and mode.

Measures of location (averages)

The term average refers to any one of several measures of the central tendency of a data set.

• The mean has the disadvantage that its value is influenced by outliers. An outlier is an observation whose value is highly inconsistent with the main body of the data. An outlier with an excessively large value will tend to increase the mean unduly, whilst a particularly small value will decrease it.

• The mean is an appropriate measure of central tendency if the distribution of the data is symmetrical.

The mean will be ‘pulled’ to the right (increased in value) if the distribution is skewed to the right, and ‘pulled’ to the left (decreased in value) if the distribution is skewed to the left.

One of the most important aspects of describing distribution is the central value around which the observations are distributed. A statistical measure used for representing the centre of central value of a set of observations is known as “Measure of Central Tendency”. This central value is also called as ‘Average’. The commonly used measures of central tendency are as follows;

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.

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.

The measure of scatteredness of observation around their average is necessary to get a better description of data. The extent or degree to which data tend to spread around an average is called Dispersion or Variation. Measure of Dispersion also called the averages of second order; help us to measure the scatteredness of observation around an average. The commonly used measures of dispersion are as follows;

A spectrophotometer is an analytical instrument that measures the absorbance of a test sample (solution). The term “spectroscopy” comes from the word “spectrum”. “Spectroscopy” therefore implies the use of multiple wavelengths of light. Spectrophotometers have the ability to specifically measure absorbance at specific wavelengths. The most commonly used method to allow this involves a “monochromator”, a device (either a prism, or more commonly, a diffraction grating) that splits the incident light into its component wavelengths, and allows only light of the desired wavelength to reach the sample. The ability to measure absorbance at different wavelengths is very useful, because the extinction coefficient of a compound varies with wavelength. In addition, the absorbance spectrum of a compound can vary depending on the chemical composition of the compound, and depending on the nature of the solvent around the compound.

Absorbance is a useful quantity and follows the Beer-Lambert law. The extinction coefficient of a molecule at a given wavelength can be calculated using the Beer-Lambert equation from absorbance measurements for solutions of known concentration, which states that:

Abstract

Since the beginning of the civilization, medicinal plants have been used to prevent and treat various ailments. At present, most of the pharmaceutical/ nutraceutical industries highly depend on wild medicinal plants for the supply of raw materials for extraction of bioactive compounds. As medicinal plants receive increased scientific and commercial attention, there is increasing pressure on the wild medicinal plant populations.

SEX IS VERY COSTLY

Large amounts of energy required to find a mate and do the mating: specialized structures and behavior required

Intimate contact provides route for infection by parasites (AIDS, syphillis, etc.)

Genetic costs: in sex you pass on only half your genes to your children

Males are an expensive luxury- in most species they contribute little to rearing offspring

But There are Some Advantages

More genetic diversity: more potential for survival of species when environmental conditions change

Shuffling of genes in meiosis (random separation of homologous chromosomes)

Crossing-over in meiosis (see below)

Fertilization (genes from 2 individuals brought together)

ALL MOLECULES MOVE CONTINUOUSLY BY SIMPLE DIFFUSION

Heat energy causes molecules to move randomly (sometimes called Brownian motion)

If the concentration of molecules is different in 2 regions (this produces a concentration gradient), diffusion will cause molecules to move from a region of high concentration to one of low concentration

The higher the concentration gradient the more rapid the net diffusion

Diffusion evens out the concentrations so they are equal everywhere (maximum entropy)

If 2 different substances have the same concentration gradient, usually one will move faster than the other

Differences in speed between molecules with the same concentration gradient are given in terms of diffusion constants

A substance with a high diffusion constant moves faster than one with a low diffusion constant

In general, large molecules move slower than small molecules

Diffusion across a membrane is called permeability, and membrane diffusion constants are called permeability constants

About Mendel

Born in what is now the Czeck Republic.

Grew up in an agricultural environment.

Became a priest-teacher at a local monastery.

Took courses in botany, physics, and mathematics at the University of Vienna.

Gregor Mendel was the first person to comprehend some of the most basic principles of genetics. He reached these understandings in the mid-nineteenth century; working in what is now the Czech Republic and using as his experimental subjects a species of garden pea, Pisum sativum..

Carried out “hybridization” experiments with the common garden pea.

Mendel’s hypotheses became the laws of inheritance in modern genetics.

Mendel’s work re-discovered by Correns, DeVries, and Tschermak in 1900. Period of modern “Mendelian” genetics begin.

A fair number of practical breeding experiments were done in ancient times

People knew there was a connection between sex and reproduction

The idea had developed that “like begets like”

By selecting certain plants and animals and breeding them many domestic varieties of plants and animals had been developed

Example: corn was developed from a wild grass in Mexico and Central America, probably about 5000 years ago

149 of the top 150 plant crops were domesticated by primitive man

Mendel’s 3 Laws (Segregation, Independent Assortment, Dominance).In the 1860s, an Austrian monk named Gregor Mendel introduced a new theory of inheritance based on his experimental work with pea plants.Mendel believed that heredity is the result of discrete units of inheritance, and every single unit (or gene) was independent in its actions in an individual’s genome.According to this Mendelian concept, the inheritance of a trait depended on the passingon of these units.For any given trait, an individual inherits one gene from each parent so that the individual has a pairing of two genes. We now understand the alternate forms of these units as ‘alleles’.If the two alleles that form the pair for a trait are identical, then the individual is said to be homozygous and if the two genes are different, then the individual is heterozygous for the trait.The breeding experiments of the monk in the mid-1800s laid the groundwork for the science of genetics.He studied peas plant for 7 years and published his results in 1866 which was ignored until 1900 when three separate botanists, who also were theorizing about heredity in plants, independently cited the work.In appreciation of his work he was considered as the “Father of Genetics”. A new stream of genetics was established after his name as Mendelian genetics which involves the study of heredity of both qualitative (monogenic) and quantitative (polygenic) traits and the influence of environment on their expressions.Mendelian inheritance while is a type of biological inheritance that follows the laws originally proposed by Gregor Mendel in 1865 and 1866 and re-discovered in 1900.
Mendel carried out breeding experiments in his monastery’s garden to test inheritance patterns. He selectively cross-bred common pea plants (Pisum sativum) with selected traits over several generations. After crossing two plants which differed in a single trait (tall stems vs. short stems, round peas vs. wrinkled peas, purple flowers vs. white flowers, etc), Mendel discovered that the next generation, the “F1” (first filial generation), was comprised entirely of individuals exhibiting only one of the traits. However, when this generation was interbred, its offspring, the “F2” (second filial generation), showed a 3:1 ratio- three individuals had the same trait as one parent and one individual had the other parent’s trait.

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.

Despite the benefits that minimally invasive surgery and osteosynthesis have brought to fracture management and bone healing, there are still many circumstances where bone healing remains challenging. Large bone defects are serious complications that are most commonly caused by extensive trauma, tumour, infection, or congenital musculoskeletal disorders. In nonunion cases repairing of bone defects with composite biomaterials as defect filler can promote bone regeneration. Currently, the gold standard for bone regeneration is use of autogenous bone graft. In order to avoid morbidity at the donor site or if large amount of autogenous bone is needed, bone substitution materials can be used.

Bone substitution materials can be combined with cells such as mesenchymal stem cells (MSCs) to increase bone formation. Bone-marrow derived mesenchymal stem cells (BMSCs) represent an attractive cell population for tissue regeneration. Bone marrow stem cells are pluripotent cells that have been used to facilitate bone repair because of their capability of differentiating into osteoblasts. Several studies on the regeneration of bone have shown that cultured BMSCs, seeded on different bioabsorbable implants are able to induce bone formation in vivo and lead to improved healing of critical-size defects.