Preface The “Applied Statistical Techniques” is a reference book useful for undergraduates, postgraduates and research scholars of biological, ecological and medical sciences. It has been written keeping in view the requirements of students at different levels. Many institutions and universities are offering this course at undergraduate, graduate and Ph.D. level. But the major challenge is unavailability of basic text books suiting the needs of the countries where English is not their native language. Most of the books on these aspects available in the market are written by authors of western countries and designed to fit into their educational programme.   This book grew out of my teaching and research experiences.  The purpose of writing this book is to provide an accessible reference book on statistical techniques whose proper use may help students in withdrawing accurate results and able to interpret them logically. I feel strongly that this can only be accomplished by illustrating the techniques using a variety of datasets. I have incorporated little theory and tried to keep the book relatively short and to the point. The methods described in this book are, of course, the same as those used in different disciplines, but things are made so user’s friendly that even general readers will find this book useful.   It has been experienced that many authors are publishing articles in leading journals after using statistical techniques for analysing their data, therefore, a good number of such innovative techniques have also been included in the work, thereby making this book suitable for post graduates and Ph.D. scholars.

1.1. INTRODUCTION In this chapter, we will know how to classify raw data. The purpose of classifying raw data is to bring order in them so that they can be statistically analysed easily. Classification of objects or things saves our valuable time and effort. It is done in a logical manner and similar things are arranged into a group or class.

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:

3.1. INTRODUCTION In this chapter, we will study the measures of central tendency which is a numerical method to explain the data in brief. We can see examples of summarizing a large set of data in day to day life like average marks obtained by students of a class in a test, average rainfall in an area, average production in a factory, average income of persons living in a locality or working in a firm etc. One of the most important objectives of statistical analysis is to get one single value that describes the characteristic of the entire mass of data. Such a value is called the central value or an ‘average’ or the expected value of the variable. This single value is the point of location around which individual values cluster and, therefore, called the measure of location. Since this single value has a tendency to be somewhere at the center and within the range of all values it is also known as the measure of central tendency. The purpose of computing an average value for a set of observations is to obtain a single value which is representative of all the items.

4.1. INTRODUCTION Dispersion is the extent to which values in a distribution differ from the average of the distribution. To quantify the extent of the variation, there are certain measures namely: Range Quartile Deviation Mean Deviation Standard Deviation Range and Quartile deviation measure the dispersion by calculating the spread within which the values lie. Mean Deviation and Standard Deviation calculate the extent to which the values differ from the average.

5.1. INTRODUCTION Mean and standard deviation summarizes the salient features of a data set. But they do not describe adequately the full characteristics of the data. For an adequate description of the characteristics of a data set, a convenient method is to plot a frequency distribution curve. Such curve is constructed by first drawing a frequency polygon, which may then be smoothened to a frequency curve, indicating the distribution. The frequency curve or distribution of specific biological variables generally has a specific form and plays a very important role in statistical theory and practice. It is customary to describe a curve in terms of its symmetry or assymmetry and its flatness or peakedness. The following types of distribution are commonly seen in the biological world:

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.

7.1. INTRODUCTION The word ‘significance’ in statistics does not describe the strength of association. Statistical significance is simply related to the probability of the result being commonly or rarely encountered (If the probability of chance occurrence is low the result is likely to be determined by the set of circumstances under study and not by chance). If conclusions are to be drawn that a difference exists between two groups of subjects, then we have to ensure that they are equal in all respects except the one under test. ‘P’ values are used to assess the degree of dissimilarity between two (or more) sets of measurements. The ‘P’ value is the probability of obtaining a result as extreme as or more extreme than the one observed. The ‘P’ value is actually a measure of surprise and the smaller the value, the more surprising is the result. When ‘P’ value is between 0.05 and 0.01, the result is usually called statistically significant; when it is less than 0.01, it is usually called highly significant and if, values are lower than 0.005, then it is called very highly significant. Traditionally, the approach had been to indicate the level of significance by rounded off ‘P’ values e.g., P < 0.01. Now it has become a fashion to calculate the exact probabilities and thus arrive at a particular decision.

8.1. INTRODUCTION The word ‘non-parametric’ means “not for any specific parameter”. For example, in the student’s t test we use the parameter ‘mean’. In the non-.parametric tests we check the randomness and trend of the pattern of distribution of a variable directly, without using a population parameter for comparing. In contrast to the parametric tests, where many assumptions are made regarding the distribution of the parameter in the population (i.e. Gaussian features), here we do not make stringent assumptions about the population from which samples are drawn. These techniques are, therefore, also known as ‘distribution free’ tests. We shall be discussing about 10 such non-parametric tests.

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