Preface Statistics is used in two senses, singular and plural. In the singular, it concerns with the whole subject of statistics, as a branch of knowledge. In the plural sense, it relates to the numerical facts, data gathered systematically with some definite object in view. Thus, Statistics is the science, which deals with the collection, analysis and interpretation of data. An understanding of the logic and theory of statistics is essential for the students of agriculture who are expected to know the technique of analyzing numerical data and drawing useful conclusions. It is the intention of the author to keep the practical manual at a readability level at appropriate for students who do not have a mathematical background. This manual has been prepared for the students and teachers as well to acquaint the basic concepts of statistical principles and procedures of calculations as per the syllabi of 5th Dean’s committee of ICAR for undergraduate courses in agriculture and allied sciences. I wish the practical manual would be very much useful for students and teachers.

Frequency Distribution: A tabular presentation of the data in which the fre- quencies of values of a variable are given along with them is called a frequen- cy distribution. Two types of frequency distribution are available Discrete Frequency Distribution or Ungrouped Frequency Distribution Continuous Frequency Distribution or Grouped Frequency Distribution Objective (a): Prepare a discrete frequency distribution from the following sentence in English: “Today there is hardly a phase of Endeavour which does not find statistical devices at least occasionally useful.” Theory: This problem relates to discrete frequency distribution i. e. a frequency distribution which is formed by distinct values of a discrete or continuous variable.

The important convincing appealing and easily understood method of presenting the statistical data is the use of diagrams and graphs. Diagram makes the comparison of the data but graphs makes the relationship among the data. Objective: Aggregated figures for merchandise export in India for eight years are as follows

Pie charts are useful to compare different parts of a whole amount. They are often used to present financial information. E.g. A farmer’s expenditure on farm can be shown to be the sum of its parts including different expense cate- gories such as cultivation, irrigation, sowing, harvesting, and general running costs (i.e. rent, electricity, fuel etc). A pie chart is a circular chart in which the circle is divided into sectors. Each sector visually represents an item in a data set to match the amount of the item as a percentage or fraction of the total data set.

Measures of Central Tendency The term ‘Mean” or ‘average’ is something, we have been familiar with from a very early age when we start analyzing our marks on report card. We add together all of our test results and then divide it by the sum of the total number of marks there are. We often call it the average. However, statistically it’s the Mean! The Median is the ‘middle value’. When the totals of the list are odd, the me- dian is the middle entry in the list after sorting the list into increasing order. When the totals of the list are even, the median is equal to the sum of the two middle (after sorting the list into increasing order) numbers divided by two. Thus, remember to line up values, the middle number is the median! The Mode in a list of numbers refers to the list of numbers that occur most frequently. A trick to remember this one is to remember that mode starts with the same first two letters that most does. Most frequently - Mode.

Objective: The distribution of age of males at the time of marriage was as follows

Paratition Value Median is calculated for dividing the series into two equal parts. If the series is divided into four, eight, ten or hundred parts then the measures are called Quartile, Octile, Decile and Percentile respectively denoted by Q(1,2,3), O(1,2..7), D(1,2…9) and P(1,2…99).

A relative measure of dispersion based on the quartile deviation is called the coefficient of quartile deviation. It is defined as

MEAN DEVIATION The mean deviation is the first measure of dispersion that we will use that ac- tually uses each data value in its computation. It is the mean of the distances between each value and the mean. It gives us an idea of how spread out from the center the set of values is. The mean deviation is based on all the observations, a property which is not possessed by the range and the quartile deviation. The formula of the mean de- viation gives a mathematical impression that is a better way of measuring the variation in the data. Any suitable average among the mean, median or mode can be used in its calculation but the value of the mean deviation is minimum if the deviations are taken from the median.

COEFFICIENT OF THE MEAN DEVIATION A relative measure of dispersion based on the mean deviation is called the coefficient of the mean deviation or the coefficient of dispersion. It is defined as the ratio of the mean deviation to the average used in the calculation of the mean deviation.

STANDARD DEVIATION AND THEIR COEFFICIENT The standard deviation is a measure of the spread of scores within a set of data. Usually, we are interested in the standard deviation of a population. However, as we are often presented with data from a sample only, we can estimate the population standard deviation from a sample standard deviation. These two standard deviations - sample and population standard deviations - are calculat- ed differently. In statistics, we are usually presented with having to calculate sample standard deviations.

FOR GROUPED DATA Objective: The following data shows the marks of students and their frequencies find the standard deviation and its coefficient.

COEFFICIENT OF VARIATION: The most important of all the relative measure of dispersion is the coefficient of variation. This word is variation not variance. There is no such thing as coefficient of variance. The coefficient of variation( C.V.) is defined

FOR GROUPED DATA Objective: Data regarding lives of two new models of refrigerators collected in a recent survey are

MOMENTS The term moment in mechanics refers to the measure of a force with reference to its tendency to produce rotation. The strength of this tendency is depen- dent upon the amount of the force and the distance from the origin at which the force is applied. If on both the sides of the origin the force are equal than there will be a balance, at a balanced position the positive product equals the negative product. Moment is used in Statistics in a quite analogous sense. The class frequencies are looked upon as the forces and the deviations of the different values from the mean are taken as the distances. In other words, moment is the mean of the first, second, third, fourth, etc. powers of deviations of the different values from the arithmetic mean. The first moment about the mean is always zero and second moment about the mean is variance. If the moment calculated from arithmetic mean are called central moments and denoted by μ with a subscript which indicate the order of the moment. Moments are the mean values of powers of the deviations in any frequency distribution taken about three points

SKEWNESS Skewness means lack of symmetry. Skewness denotes the tendency of a distri- bution to depart from symmetry. A frequency distribution is said to be skewed if the frequencies decrease with markedly greater rapidity on one side of the central maximum than on the other side. This characteristic of a frequency distribution is known as skewness and the measures of asymmetry are usually called measures of skewness. The main object of measuring skewness is to know the direction of the varia- tion from an average and to compare the frequency distribution and the shape of their curves. There are two types of skewness (i) Positive Skewness (ii) Negative Skewness

KURTOSIS Kurtosis refers to the extent to which unimodal frequency curve is peaked. Kurtosis is a measure that refers to the peakedness of the top of the curve. Kurtosis gives the degree of flatness in the region about the mode of a frequency distribution. Types of Kurtosis There are three categories of kurtosis that can be displayed by a set of data. All measures of kurtosis are compared against a standard normal distribution, or bell curve. The first category of kurtosis is a mesokurtic distribution. This type of kurtosis is the most similar to a standard normal distribution in that it also resembles a bell curve. However, a graph that is mesokurtic has fatter tails than a standard normal distribution and has a slightly lower peak. This type of kurtosis is considered normally distributed but is not a standard normal distribution. The second category is a leptokurtic distribution. Any distribution that is leptokurtic displays greater kurtosis than a mesokurtic distribution. Characteristic of this type of distribution is one with extremely thick tails and a very thin and tall peak. The prefix of “lepto-” means “skinny,” making the shape of a leptokurtic distribution easier to remember. T-distributions are leptokurtic. The final type of distribution is a platykurtic distribution. These types of distributions have slender tails and a peak that’s smaller than a mesokurtic distribution. The prefix of “platy-” means “broad,” and it is meant to describe a short and broad-looking peak. Uniform distributions are platykurtic.

CORRELATION In case of bivariate or multivariate normal distributions, we may be interested in discovering and measuring the magnitude and direction of the relationship between two or more variables. For this purpose we use correlation. The sys- tematic interrelationship between the variables is termed as correlation. When only two variables are involved the correlation is known as simple correlation. If more than two variables are involved the correlation is known as multiple correlation. An increase in one variable may cause an increase in the other variable, or a decrease in one variable may cause a decrease in the other vari- able, then they are said to be positively correlated. If a decrease in one variable causes an increase in the other variable or vise versa, the variables are said to be negatively correlated.

TEST OF SIGNIFICANCE FOR CORRELATION COEFFICIENT Coefficient of correlation is a numeric characteristic of a population and is estimated from a sample of the bi-variate population. If ρ be the coefficient of correlation in the population and r be its sample estimate based on n pairs of observations, then standard error of r is given

REGRESSION Regression means to revert or return back. The term was first introduced by Sir Francis Galton in1877. Regression technique is applicable in all those fields where two or more relative variables have the tendency to go back to the mean. Regression analysis refers to the methods by which estimates are made of the values of a variable from the knowledge of the values of one or more other variables and to the measurement of the errors involved in this estimation pro- cess. In statistical modeling, regression analysis is a statistical process for estimat- ing the relationships among variables. It includes many techniques for mod- eling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. Regres- sion analysis helps one understand how the typical value of the dependent variable (or ‘criterion variable’) changes when any one of the independent variables is varied, while the other independent variables are held fixed. Most commonly, regression analysis estimates the conditional expectation of the dependent variable given the independent variables – that is, the average value of the dependent variable when the independent variables are fixed.

t TEST FOR SINGLE MEAN A t-test is any statistical hypothesis test in which the test statistic follows a Student’s t distribution if the null hypothesis is supported. It can be used to determine if two sets of data are significantly different from each other, and is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known. When the scaling term is unknown and is replaced by an estimate based on the data, the test statistic (under certain conditions) follows a Student’s t distribution.

t TEST FOR TWO SAMPLE MEAN: Comparison of two sample means  and assumed to have been obtained on the basis of random samples of sizes n1 and n2 from the same population which is assumed to be normal.

t TEST FOR PAIRED OBSERVATION This test is used for testing whether two series of paired observations are gen- erated from the same population on the basis of the difference in their sample means.

F TEST (VARIANCE RATIO TEST) F distribution is applied in several tests of significance relating to the equality of two sampling variances drawn on the basis of independent samples from a normal population.

STANDARD NORMAL DISTRIBUTION If μ = 0 and σ = 1, the distribution is called the standard normal distribution or the unit normal distribution, and a random variable with that distribution is a standard normal deviate. The approximate test is given by

CHI- SQUARE TEST (χ2 TEST) The chi square test is used to determine if the two attributes are independent of each other. Chi square is a measure to evaluate the difference between observed frequencies and expected frequencies to examine whether the difference so obtained is due to a chance factor or due to sampling error. In a contingency table if each attribute is divided into two classes it is known as 2×2 contingency table. When one attribute is divided into two classes and another into r or c resultant contingency table is known as r×2 or 2×c contingency table. Here, r denotes the number of rows and c the number of columns. For such data, the statistical hypothesis under test is that the two attribute are independent of one another. To test this hypothesis we use the test statistic

The chi square is a continuous distribution based on the assumption that the ul- timate class frequency is not less than 5 otherwise the value of χ2 will be over- estimated and may cause too many rejections of null hypothesis. Therefore, to maintain the continuity of χ2 and to draw the correct inference in 2×2 we apply a correction which is known as Yates’s correction for continuity.

ANALYSIS OF VARIANCE Analysis of Variance (ANOVA) is a hypothesis-testing technique used to test the equality of two or more population (or treatment) means by examining the variances of samples that are taken. ANOVA allows one to determine whether the differences between the samples are simply due to random error (sampling errors) or whether there are systematic treatment effects that cause the mean in one group to differ from the mean in another. Most of the time ANOVA is used to compare the equality of three or more means, however when the means from two samples are compared using ANOVA it is equivalent to using a t-test to compare the means of independent samples. ANOVA is based on comparing the variance (or variation) between the data samples to variation within each particular sample. If the between variation is much larger than the within variation, the means of different samples will not be equal. If the between and within variations are approximately the same size, then there will be no significant difference between sample means.

The two-way ANOVA compares the mean differences between groups that have been split on two independent variables (called factors). The primary purpose of a two-way ANOVA is to understand if there is an interaction be- tween the two independent variables on the dependent variable. For example, you could use a two-way ANOVA to understand whether there is an interac- tion between gender and educational level on test anxiety amongst university students, where gender (males/females) and education level (undergraduate/ postgraduate) are your independent variables, and test anxiety is your depen- dent variable. Alternately, you may want to determine whether there is an in- teraction between physical activity level and gender on blood cholesterol con- centration in children, where physical activity (low/moderate/high) and gender (male/female) are your independent variables, and cholesterol concentration is your dependent variable. The interaction term in a two-way ANOVA informs you whether the effect of one of your independent variables on the dependent variable is the same for all values of your other independent variable (and vice versa). For example, is the effect of gender (male/female) on test anxiety influenced by educational level (undergraduate/postgraduate)? Additionally, if a statistically significant interaction is found, you need to determine whether there are any “simple main effects”, and if there are, what these effects are (we discuss this later in our guide).

A simple random sample (SRS) of size n consists of n individuals from the population chosen in such a way that every set of n individuals has an equal chance to be the sample actually selected. A simple random sample is a subset of a statistical population in which each member of the subset has an equal probability of being chosen. An example of a simple random sample would be the names of 25 employees being chosen out of a hat from a company of 250 employees. In this case, the population is all 250 employees, and the sample is random because each employee has an equal chance of being chosen.

Appendix Values of the t-distribution (two-tailed)