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It’s my pleasure to present the book, its being published with an aim to spread the knowledge attained by the undersigned in the vast teaching career as Statistician amongst the interested persons. There are manifold purposes of writing this book on the subject. Basically, it caters to the needs of the candidates aspiring for competitive examinations, and for the beginners to understand the intricacies of the subject. It is observed that the very name of the subject, statistics evokes fear in the minds of the students, through this book an effort has been made to dispel that fear and students should be able to approach the subject confidently. It should be learning with pleasure for the students and they should face the competitions any time and at any place with confidence. The latest trend of education is the teaching through multiple choice questions. The MCQ’s in this book are intended to enable students to prioritize and plan their learning through regular practicing. Statistics has assumed a lot of importance in recent times. It is imperative for every student to study and gain knowledge of statistics, its uses as well as its applications. The format and contents of this book has been carefully drafted, the Chapters have been written with a view to explain the concepts clearly and examples have been given in the text to help the students to understand the application of these concepts in real life situations. MCQs at the end of every unit will help the students to make a self-assessment of the knowledge assimilated by going through the Chapters. Through this book, the author has made an effort to provide rationale for the solutions. The book, therefore, meets the expectations of the students as it answers the demand and the quest in their mind and give rise to real learning which would stand in good stead for the student’s career and his life. The quality of the MCQs presented in this book stands out from the other books available in the market. Speciality of this book is that there are varieties of MCQs which comprehend all the related questions on the subjects as well the contents have been so designed as to include in the true spirit of the syllabus of basics of statistics.

Introduction to Statistics The word statistics has been derived from Latin word “status”, the German word “statistik”, the Italian word “statistica” or the French word “statistique”. During early periods, it was used by emperors and kings only for collecting the information about state and other information which was needed about their people, their number, land, status of people, revenue of the state etc. Gradually, it meaning and uses extended and there onwards its nature changed. Some scientist/ researchers were given definition of statistics as per time and place. • An ancient date, A.L. Bowley gives some definitions according their use and place. He seems to be inadequate because these do not include all aspects of statistics. As per time, place or requirement, he gave three definitions of statistics: • Statistics may be called the science of counting • Statistics may rightly be called the science of average • Statistics as the science of counting and average  

Statistical Series: When observations of any characteristic present in the individual of a group are recorded and arranged one after the other in a systematic order, they form a series, known as statistical series. For example: milk yield of 20 cows, height of the students in a class in a systematic order, etc. These are some important types of series: • Discrete series: In this series, observations can take up only exact values and not any fractional values, is called discrete series. That means, this type of series has whole number of counting. For example: number of the students in a class, members of a family, number of births in a certain year, etc • Continuous series: A series which can take up any numerical value (integral/ fractional) within a certain range is called continuous series. For example: milk yield of 20 cows, height, weight, temperature, etc. • Array series: When we arranged the observations of a series either in ascending order of their magnitude or descending order of their magnitude, then it becomes an array series. • Spatial series: When items of a series are given according to space or place or time, then such type of series is known as spatial series. For example: month wise milk yield of a herd of cows, month wise income or expenditure of an industry, population of a country according to states, etc.

Population: The word population or universe in statistics means a set of observations relating to a phenomenon under investigation. Sample: It is a fraction or a subset of population drawn through a valid statistical procedure so that it can be regarded as representative of the entire population. Sampling: The statistical procedure of drawing a sample from the population is called Sampling. Census enquiry or complete enumeration: When each and every unit of the population is investigated for the characteristic under study, known as census enquiry or complete enquiry. Sample enquiry or sample enumeration: It is a part of the population is investigated for the characteristic under study, known as sample enquiry or sample enumeration. Necessity of sampling: Since, the process of collecting information from all the elements of a large population may be, in general, expensive, time consuming and difficult. We help to sampling and generalise the properties of the sample to the entire population. The statistical procedures which are used for drawing inferences or conclusions about the population from sample data are covered under Inferential Statistics or Statistical inference. Thus, sampling theory is the basis of statistical inference in which we wish to obtain maximum information about the population with minimum effort and maximum precision.

Diagrammatic Presentation of Data The quantitative data when classify are converted into averages and if the number of average is large then it is not possible to draw a conclusion from their large numbers of averages. Moreover quantitative data are dull, confusing, boring for many of us. Therefore, a statistical device to make such confusing data more attractive was developed i.e. if such complex data are presented by geometric f igures, pictures, graphs or curves then it will become more attractive and more impressive, is called as diagrammatic representation of data. These diagrams are nothing but the use of geometrical figures to improve the overall presentation and offer visual assistance for the reader. Types of Diagrams Generally, diagrams are made for a special series or a time series and they are divided into five different categories: 1) One dimensional diagrams 2) Two dimensional diagrams 3) Three dimensional diagrams 4) Pictograms 5) Cartograms

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

Correlation In a statistical analysis, when two variables are studied simultaneously, then it is called “Bivariate study’. Now, in a bivariate study, if it is found that change in the values of one variable then in sympathy with the change in the values of other variable, then these two variables are said to be associated and the association is termed as correlation. In other words, the term correlation indicates the relationship between two such variables in which with changes in the values of one variable, the values of the other variable also change. In short, the tendency of simultaneous variation between two variables is called correlation. Types of Correlation Correlation can be classified by- • Positive and Negative correlation: The correlation may be classified according to the direction of change in the two variables. In this regard, correlation may be either positive or negative. Positive correlation refers to the movement of variables in the same direction. Negative correlation efers to the movement of variables in the opposite direction. • Simple, Partial and Multiple correlations: When only two variables are studied it is a problem of simple correlation. When three or more variables are studied simultaneously, it is a problem of multiple correlations. On the other hand, in partial correlation we recognize more than two variables, but consider only two variables to be influencing each other the effect of other influencing variables being kept constant. • Linear and non-linear correlations: In linear correlation for every unit change in the values of one variable, there is constant change in the value of other variable. The perfect positive and negative correlations are also linear correlations. In non-linear correlation the change in the values of one variable does not have a constant ratio to the change in the other variable.

The theory of probability has its origin in the games of chance related to gambling such as throwing a die, tossing a coin, drawing cards from a pack of cards etc. Jerame Cardon (1501-1576), an Italian mathematician was the first man to write a book on the subject entitled “Book on Games and Chance”, which was published after his death in 1663. The systematic and scientific foundation of the mathematical theory of probability was laid in mid-seventeenth century by two French mathematicians B. Pascal (1623-1662) and Pierre de Fermat (1601 1665). Important terms and concepts To understand probability, let us first get acquainted with some important terms and concepts associated with probability. • Trial: An experiment is any natural or planned activity for which it is not possible to predict the particular result called a trial. • Events: The possible outcomes of experiments of a trial are called events. For example: Tossing of a coin in an experiment and one toss of the coin is called trial. In tossing of a coin there are only two outcomes or events, i.e. head or tail. Types of events Events can be of following different types: • Exhaustive events • Equally likely events • Mutually exclusive events • Dependent events • Independent events • Null or impossible events • Exhaustive events: It is the total number of all the possible outcomes of an experiment.

Theoretical Probability Distribution Many random variables associated with statistical experiments have similar properties, and as such can be described by the same probability functions. The distribution obtained from a theoretical listing of outcomes and probabilities, which can be described by a mathematical formula representing same phenomenon of interest is called a theoretical probability distribution. The distributions of discrete and continuous random variables are accordingly called discrete or continuous probability distributions. Discrete Probability Distribution: It can be defined as a probability distribution of discrete random variables. Such a distribution will represent data that has a f inite countable number of outcomes. If the random variable X is a discrete random variable, the probability function P(X) is called probability mass function and its distribution is called discrete probability distribution. There are two conditions that a discrete probability distribution must satisfy. These are given as follows: Types of Discrete Probability Distributions There are several important discrete probability distributions- • Uniform distribution • Binomial distribution • Negative Binomial distribution • Geometric distribution • Poisson distribution • Hyper-Geometric distribution

Time Series A series of observations recorded over time is known as a time series. In other words, a time series is a set of observations taken at specified times, usually at equal intervals. For example: Annual retail sales, rainfall measurements, heartbeats per minute, etc. Such studies are to be based on the analysis of time series data collected over time. Thus, the analysis of time series plays an important role in an investigation of economic, social, commercial and even biological phenomenon. Components of time series In a long time series, generally, there are following four components: • Secular trend or long term movements • Seasonal variations • Cyclic variations • Random or irregular movements • Secular trend or long term movements: It is that characteristic of a time series which extends consistently throughout the entire period of time under consideration. It shows a long term tendency of an activity to grow or to decline. The term long period of time in a relative phenomenon and cannot be defined exactly. For some cases a period as small as a weak may be fairly long while in other cases, a period as long as two years may not be assumed long.

An index number is a statistical measure designed to show changes in a variable or group of related variables with respect to time, geographical location or other characteristics like: income, profession, etc. A collection of index numbers for different years, months, locations, etc. is called an index series. Since index number are usually used to measure the level of business and economic activities over a period of time, so they are rightly called barometers of economic activities or simple called economic barometers. The various problems faced in the construction of an index number are to known the specific purpose of index number, selection of items, data for index numbers, choice of base periods, choice of an average, selection of weights and choice of a suitable formula. Methods of Constructing Index Numbers A number of formulae have been developed for constructing index numbers, which may be grouped into the following categories: • Unweighted index numbers • Weighted index numbers • Unweighted index numbers: In this case, weights are not expressly assigned. 

Statistical Inference It is the process of drawing valid inference or conclusion about the population on the basis of a sample drawn from the population. In other words, Statistical inference is the process of using a sample to infer the properties of a population. It requires studies pertaining to three aspects: • Point Estimation • Interval Estimation • Testing of Hypothesis • Point Estimation: It is to find a single value, which is used as an estimator of unknown parameter. For example, sample mean may be used as an estimate of the population mean µ. • Interval Estimation: It is to find confidence limits based on sample values, within which the unknown parameter lies with confidence coefficient described in probability term. In other words, we estimate an unknown parameter using an interval of values that is likely to contain the true value of that parameter and state how confident we are that this interval indeed captures the true value of the parameter. • Testing of Hypothesis: The tests used to ascertain whether the differences between estimator and the parameter or between two estimators are real or due to chance are called tests of hypothesis or tests of significance. In other words, the procedures, which enable us to decide whether to accept or reject hypothesis, are called tests of hypothesis or tests of significance. Two types of problems mainly arise in tests of significance: • Testing the significance of the difference between estimator and the pa rameter.

In 1935, Prof. Ronald A. Fisher laid the foundation for the subject in his monumental work entitled “The Designs of Experiments”. Experimental designs concern the arranging of treatments in such a manner that the inferences and conclusions regarding the effects of these treatments can be easily done and their reliability measured. Experiments are made with a view to find the validity of a particular hypothesis and to have an idea about the extent of the reliability that can be placed on a particular conclusion arrived. The selection of the design will have a very great bearing on the accuracy of the ultimate results. By a random selection of experimental units it is possible to remove the ambiguity about the casual interpretation of the observed associations. Random sampling is the most essential ingredient of all experimental designs. Besides, there are many devices for increasing the precision of the inference and the calculations. To understand design of experiments, let us first get acquainted with some important terms and concepts associated with design of experiments: • Experiment: Experiment is a scientifically planned method. The experiment is conducted draw a valid conclusion about a particular problem. The conclusion is based on statistically observation. • Treatment: The object or the procedures under comparison in an experiment are known as treatment. For Example: Fertilizers, varieties, cultivation practices, irrigation level in case of agricultural experiments and drugs, breeds, farms, etc. in case of animal experiments. • Yield: The response of the treatment is measured by some indicator such as crop production, milk production, body temperature, etc. Such an indicator is called yield. The treatments are applied to some units such as field plots, sample of cows, sample of patients, etc. and the effect on the yield is observed. • Experimental material: The experimental material is nothing but a set of experimental units. In other words, the material which is used in experiment is known as experimental material. For Example: A piece of land i.e. Agricultural field in agricultural experiments and a group of cattle i.e. animals in animal experiments.

A non parametric test (sometimes called a distribution free test) is a statistical analysis method that does not assume the population data belongs to some prescribed distribution which is determined by some parameters. That means, a non-parametric test can be defined as a test that is used in statistical analysis when the data under consideration does not belong to a parameterized family of distributions. When the data does not meet the requirements to perform a parametric test, a non-parametric test is used to analyze it. The non-parametric test compared to parametric test, which makes assumptions about a population’s parameters (for example, the mean or standard deviation); When the word “non parametric” is used in stats, it means that population data does not have a normal distribution. If at all possible, we should us parametric tests, as they tend to be more accurate. Parametric tests have greater statistical power, which means they are likely to find a true significant effect. Nonparametric tests can perform well with non-normal continuous data if the data of the sample are sufficiently large. Non parametric do not assume that the data is normally distributed. The only non parametric test we are likely to come across in elementary stats is the Chi-square test. In nonparametric tests have several advantages as compared to parametric tests. It includes: • More statistical power when assumptions for the parametric tests have been violated. • Fewer assumptions, i.e. the assumption of normality don’t apply. • Small sample sizes are acceptable. • It is used for all data types, including nominal data, rank data or ordinal data • Knowledge of the population is not required to conduct this test. • Nonparametric tests are valid when our sample size is small and our data are potentially non-normal.

Vital Statistics It is defined as that branch of biometry which deals with data and the laws of human mortality, morbidity and demography. According to N.B. Ryder, they “provide cumulative summaries for successive time periods of population movements like birth, death, migration, marriage and marital dissolution as well as demographic and other relevant characteristics of the individuals involved in these events.” Measures of Vital Statistics of Population The total number of vital events such as births and deaths are converted into rates and ratios for comparison of vital statistics from place to place or year to year. In order to determine the population at any time ‘t’ after the census or between two censuses, a number of methods have been devised. Therefore, a suitable method which makes use of births, deaths and migration statistics, if we assume that: (i) The census data gives us the total size of the population of a region or community together with age and sex distribution (ii) The birth, death and migration statistics during different periods are obtained from registers.