eChapters

Samples having size, n - 30 are known as large sample. For large values of n (i.e., for n - 30), almost all the distributions are very closely approximated by normal distribution. Thus in this case we apply the normal test, which is based upon the area property of the normal probability curve. Large Sample Test is also known as Z-test or SND (Standard Normal Deviate) test.

A sample is said to be small sample if the sample size(n) is less than 30. Various small sample tests are: (i) t-test (ii) F-test

t-test performs the following tests for small sample(s).

1. To test the equality between a single sample mean and the assumed population mean.
2. To test the equality of two independent sample means.
3. To test the equality of two paired sample means.
4. To test the equality between sample correlation coefficient and the assumed population correlation coefficient

The aggregate of data forming a subject of investigation is known as population. Example, All the crop plants grown in a plot; all the fruit plants in an orchard, etc. The population may be:

1. Finite: No. of observations are definite. e.g., number of farms in a village, no. of house in a locality, no. of trees in an orchard are all finite population.
2. Infinite: Number of observation are indefinite e.g., (a) Number of insects in a region, number of grasses in a field etc. are all infinite population.

The Population may be:

1. hypothetical or theoretical

   Hypothetical population consists of unit that has no physical existence. This population exists only in imagination. For example, Result of tossing coin, throwing die,

2. Existent or real population consists of units that are physically present. This population actually exists. For example, Number of employees in a factory, Number of farms in a village.

Concept of correlation was given by Karl Pearson in 1890.

The study of strength and nature of relationship between two variables is called correlation. Two variables are said to be correlated, if the change in one variable affects the other variable. Correlation is also known as covariation.

Bivariate Distribution: Distribution involving two variables is known as bivariate distribution. For example, Height and weight of persons, yield and production of a crop in.

On basis of type of association between variables, correlation is of two types:

Random variable: A random variable is a variable that assumes numerical values associated with the events of a random experiment whose outcomes cannot be predicted with certainty. For example, Height of rice plants in a field; seed weight; No. of tillers per rice plant.

A random variable is of two types:-

1. Discrete random variable: The random variable which takes only countable values i.e., integral values. For Example, Number of tillers per rice plant. Here there can be finite no. of values between any two values.
2. Continuous random variables: The random variable which takes values within intervals. These values cannot be counted but can only be measured. Here there can be infinite no. of values between any two values for example, Height of rice plants in field; seed weight.

In some experiments the result can be predicted with certainty. For example, the bulb will stop glowing when switched off, the evolution of hydrogen gas when zinc reacts with dilute hydrochloric acid. Such phenomenon where the result can be predicted with certainty is known as deterministic phenomenon. But in some experiments the result cannot be predicted with certainty. For example, Sales of a firm, profit made by a company, yield of a crop, etc. Such phenomenon where the result cannot be predicted with certainty are known as probabilistic or random phenomenon. Such phenomena are frequently observed in economics, business, agriculture, social science, biology, etc., and in our day to day life.

The theory of probability is a very important branch of statistics which provides a numerical measure of uncertainty.

Literal meaning of dispersion is scatterdness. We study dispersion to have an idea about the homogeneity and heterogeneity of the distribution. Dispersion gives an idea about how much the values of the observation varies from their averages. The series in which the values of the observation differ much from their averages is said to be heterogeneous or highly dispersed. The series in which the value of the observation does not differ much from their averages is said to be homogenous or less dispersed.

There are two types of measures of dispersion:

Measures of central tendency are also known as averages. Averages are defined as statistical constants which enable us to understand in a single effort, the significance of the whole i.e., they are the representative value of the distribution. They give an idea about the concentration of the values in the central part of the distribution.

Five Types of Averages

Arithmetic Mean

It is the sum of all the observations divided by the number of observation.

For individual data or simple series (i.e., series in which all values have single frequency)

Presentation of data is done in form of diagram and graphs.

Diagrammatic Presentation: The data are presented in form of diagrams

The important diagrammatic presentations are simple bar diagram, multiple bar diagram, component bar diagram, percentage bar diagram & pie diagram.

Selection of diagram depends upon the nature of classification, number of characters involved and the type of comparison required to be made. In all the diagrams, the groups or classes are represented on the X-axis and the volumes or frequencies are represented in the Y-axis. The title of these visual aids should be self-explanatory.

1. Simple Bar Diagram: Simple bar diagrams consist of vertical bars of equal width. The height of these bars is proportional to the magnitude of the attribute. All bars are separated from each other by equal intervals. Simple bar diagram is used if the classification is based on attributes and if the attributes are to be compared with respect to a single character. For example, the area under different crops in a state, the production of mangoes in different years for a state, the yield of a variety at different fertilizer level, etc, can be represented by the simple bar diagram. Also the yield performance of different varieties of a crop, the effects of different treatments etc, can be compared using simple bar diagrams.

Frequency distribution table is a statistical table which shows the sets of all distinct values of the variables arranged in order of their magnitude either individually or in groups with their corresponding frequencies side by side.

Need for frequency distribution table

1. To condense the data
2. To provides quick reference for the entire data
3. To facilitate quick & easy statistical analysis of data
4. To reveal the pattern within the data

The word ‘statistics’ has been derived from the Latin Word ‘Status’ which means ‘Political State’. The word statistics has been coined by Gottfried Aschenwall, a German Political Scientist. Ronald Aylmer Fisher is regarded as father of modern statistics. The word statistics has meaning both in plural and singular sense. In plural sense, it means quantitative/qualitative figures. For example, the area under food grains in different states of India, the production of guava in different states of India, germination percentage of seeds, etc.. In singular sense, it means as a subject i.e., method of dealing with the quantitative/qualitative information.

Statistics is defined as scientific method which deals with collection, organization presentation, analysis and interpretation of data. It is included under both science and arts. Statistics is used as a tool to obtain knowledge on other subjects of science.

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.

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.

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.

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:

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:

1. Range
2. Quartile Deviation
3. Mean Deviation
4. 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.

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.

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:

2.1 Some Useful Formulas Related to Farm Power

1.1 Some Useful Formulas Related to Farm Machinery