STA301 Statistics and Probability Handouts PDF

STA301 – Statistics and Probability Virtual University of Pakistan Statistics and Probability STA301 Virtual University of Pakistan i STA301 – Statistics and Probabilit...

STA301 – Statistics and Probability Virtual University of Pakistan Statistics and Probability STA301 Virtual University of Pakistan i STA301 – Statistics and Probability TABLE OF CONTENTS TITLE PAGE NO LECTURE NO. 1 1 Definition of Statistics Observation and Variable Types of Variables Measurement Sales Error of Measurement LECTURE NO. 2 6 Data collection Sampling LECTURE NO. 3 16 Types of Data Tabulation and Presentation of Data Frequency distribution of Discrete variable LECTURE NO. 4 23 Frequency distribution of continuous variable LECTURE NO. 5 32 Types o frequency Curves Cumulative frequency Distribution LECTURE NO. 6 42 Stem and Leaf Introduction to Measures of Central Tendency Mode LECTURE NO. 7 53 Arithmetic Mean Weighted Mean Median in case of ungroup Data LECTURE NO. 8 62 Median in case of group Data Median in case of an open-ended frequency distribution Empirical relation between the mean, median and the mode Quantiles (quartiles, deciles & percentiles) Graphic location of Quantiles LECTURE NO. 9 70 Geometric mean Harmonic mean Relation between the arithmetic, geometric and harmonic means Some other measures of central tendency LECTURE NO. 10 76 Concept of dispersion Absolute and relative measures of dispersion Range Coefficient of dispersion Quartile deviation Coefficient of quartile deviation LECTURE NO. 11 82 Mean Deviation Standard Deviation and Variance Coefficient of variation LECTURE NO. 12 89 Chebychev’s Inequality The Empirical Rule The Five-Number Summary LECTURE NO. 13 95 Virtual University of Pakistan ii STA301 – Statistics and Probability Box and Whisker Plot Pearson’s Coefficient of Skewness LECTURE NO. 14 106 Bowley’s coefficient of Skewness The Concept of Kurtosis Percentile Coefficient of Kurtosis Moments & Moment Ratios Sheppard’s Corrections The Role of Moments in Describing Frequency Distributions LECTURE NO. 15 115 Simple Linear Regression Standard Error of Estimate Correlation LECTURE NO. 16 128 Basic Probability Theory Set Theory Counting Rules: The Rule of Multiplication LECTURE NO. 17 136 Permutations Combinations Random Experiment Sample Space Events Mutually Exclusive Events Exhaustive Events Equally Likely Events LECTURE NO. 18 143 Definitions of Probability Relative Frequency Definition of Probability LECTURE NO. 19 147 Relative Frequency Definition of Probability Axiomatic Definition of Probability Laws of Probability Rule of Complementation Addition Theorem LECTURE NO. 20 152 Application of Addition Theorem Conditional Probability Multiplication Theorem LECTURE NO. 21 156 Independent and Dependent Events Multiplication Theorem of Probability for Independent Events Marginal Probability LECTURE NO. 22 161 Bayes’ Theorem Discrete Random Variable Discrete Probability Distribution Graphical Representation of a Discrete Probability Distribution Mean, Standard Deviation and Coefficient of Variation of a Discrete Probability Distribution Distribution Function of a Discrete Random Variable LECTURE NO. 23 169 Graphical Representation of the Distribution Function of a Discrete Random Variable Mathematical Expectation Mean, Variance and Moments of a Discrete Probability Distribution Properties of Expected Values LECTURE NO. 24 177 Virtual University of Pakistan iii STA301 – Statistics and Probability Chebychev’s Inequality Concept of Continuous Probability Distribution Mathematical Expectation, Variance & Moments of a Continuous Probability Distribution LECTURE NO. 25 185 Mathematical Expectation, Variance & Moments of a Continuous Probability Distribution BIVARIATE Probability Distribution LECTURE NO. 26 192 BIVARIATE Probability Distributions (Discrete and Continuous) Properties of Expected Values in the case of Bivariate Probability Distributions LECTURE NO. 27 199 Properties of Expected Values in the case of Bivariate Probability Distributions Covariance & Correlation Some Well-known Discrete Probability Distributions: Discrete Uniform Distribution An Introduction to the Binomial Distribution LECTURE NO. 28 207 Binomial Distribution Fitting a Binomial Distribution to Real Data An Introduction to the Hyper geometric Distribution LECTURE NO. 29 215 Hyper geometric Distribution Poisson distribution and limiting approximation to Binomial Poisson Process Continuous Uniform Distribution LECTURE NO. 30 221 Normal Distribution The Standard Normal Distribution Normal Approximation to the Binomial Distribution LECTURE NO. 31 232 Sampling Distribution of X Mean and Standard Deviation of the Sampling Distribution of X Central Limit Theorem LECTURE NO. 32 239 Sampling Distribution of p̂ Sampling Distribution of X1  X 2 LECTURE NO. 33 249 Point Estimation Desirable Qualities of a Good Point Estimator LECTURE NO. 34 256 Methods of Point Estimation Interval Estimation LECTURE NO. 35 263 Confidence Interval for  Confidence Interval for 1-2 LECTURE NO. 36 268 Large Sample Confidence Intervals for p and p1-p2 Determination of Sample Size (with reference to Interval Estimation) Hypothesis-Testing (An Introduction) LECTURE NO. 37 274 Hypothesis-Testing (continuation of basic concepts) Hypothesis-Testing regarding  (based on Z-statistic LECTURE NO. 38 280 Hypothesis-Testing regarding 1 - 2 (based on Z-statistic) Hypothesis Testing regarding p (based on Z-statistic) LECTURE NO. 39 285 Virtual University of Pakistan iv STA301 – Statistics and Probability Hypothesis Testing regarding p1-p2 (based on Z-statistic) The Student’s t-distribution Confidence Interval for  based on the t-distribution LECTURE NO. 40 292 Tests and Confidence Intervals based on the t-distribution t-distribution in case of paired observations LECTURE NO. 41 298 Hypothesis-Testing regarding Two Population Means in the Case of Paired Observations (t-distribution) The Chi-square Distribution Hypothesis Testing and Interval Estimation Regarding a Population Variance (based on Chi-square Distribution) LECTURE NO. 42 306 The F-Distribution Hypothesis Testing and Interval Estimation in order to compare the Variances of Two Normal Populations (based on F-Distribution) LECTURE NO. 43 315 Analysis of Variance Experimental Design LECTURE NO. 44 323 Randomized Complete Block Design The Least Significant Difference (LSD) Test Chi-Square Test of Goodness of Fit LECTURE NO. 45 331 Chi-Square Test of Independence The Concept of Degrees of Freedom P-value Relationship between Confidence Interval and Tests of Hypothesis An Overview of the Science of Statistics in Today’s World (including Latest Virtual University of Pakistan v STA301 – Statistics and Probability LECTURE NO. 1 WHAT IS STATISTICS?  That science which enables us to draw conclusions about various phenomena on the basis of real data collected on sample-basis  A tool for data-based research  Also known as Quantitative Analysis  A lot of application in a wide variety of disciplines Agriculture, Anthropology, Astronomy, Biology, Economic, Engineering, Environment, Geology, Genetics, Medicine, Physics, Psychology, Sociology, Zoology …. Virtually every single subject from Anthropology to Zoology …. A to Z!  Any scientific enquiry in which you would like to base your conclusions and decisions on real-life data, you need to employ statistical techniques!  Now a day, in the developed countries of the world, there is an active movement for of Statistical Literacy. THE NATURE OF THIS DISCIPLINE DESCRIPTIVE STATISTICS PROBABILITY INFERENTIAL STATISTICS MEANINGS OF ‘STATISTICS’ The word “Statistics” which comes from the Latin words status, meaning a political state, originally meant information useful to the state, for example, information about the sizes of population sand armed forces. But this word has now acquired different meanings.  In the first place, the word statistics refers to “numerical facts systematically arranged”. In this sense, the word statistics is always used in plural. We have, for instance, statistics of prices, statistics of road accidents, statistics of crimes, statistics of births, statistics of educational institutions, etc. In all these examples, the word statistics denotes a set of numerical data in the respective fields. This is the meaning the man in the street gives to the word Statistics and most people usually use the word data instead.  In the second place, the word statistics is defined as a discipline that includes procedures and techniques used to collect process and analyze numerical data to make inferences and to research decisions in the face of Virtual University of Pakistan 1 STA301 – Statistics and Probability uncertainty. It should of course be borne in mind that uncertainty does not imply ignorance but it refers to the incompleteness and the instability of data available. In this sense, the word statistics is used in the singular. As it embodies more of less all stages of the general process of learning, sometimes called scientific method, statistics is characterized as a science. Thus the word statistics used in the plural refers to a set of numerical information and in the singular, denotes the science of basing decision on numerical data. It should be noted that statistics as a subject is mathematical in character.  Thirdly, the word statistics are numerical quantities calculated from sample observations; a single quantity that has been so collected is called a statistic. The mean of a sample for instance is a statistic. The word statistics is plural when used in this sense. CHARACTERISTICS OF THE SCIENCE OF STATISTICS Statistics is a discipline in its own right. It would therefore be desirable to know the characteristic features of statistics in order to appreciate and understand its general nature. Some of its important characteristics are given below:  Statistics deals with the behaviour of aggregates or large groups of data. It has nothing to do with what is happening to a particular individual or object of the aggregate.  Statistics deals with aggregates of observations of the same kind rather than isolated figures.  Statistics deals with variability that obscures underlying patterns. No two objects in this universe are exactly alike. If they were, there would have been no statistical problem.  Statistics deals with uncertainties as every process of getting observations whether controlled or uncontrolled, involves deficiencies or chance variation. That is why we have to talk in terms of probability.  Statistics deals with those characteristics or aspects of things which can be described numerically either by counts or by measurements.  Statistics deals with those aggregates which are subject to a number of random causes, e.g. the heights of persons are subject to a number of causes such as race, ancestry, age, diet, habits, climate and so forth.  Statistical laws are valid on the average or in the long run. There is n guarantee that a certain law will hold in all cases. Statistical inference is therefore made in the face of uncertainty.  Statistical results might be misleading the incorrect if sufficient care in collecting, processing and interpreting the data is not exercised or if the statistical data are handled by a person who is not well versed in the subject mater of statistics. THE WAY IN WHICH STATISTICS WORKS As it is such an important area of knowledge, it is definitely useful to have a fairly good idea about the way in which it works, and this is exactly the purpose of this introductory course. The following points indicate some of the main functions of this science:  Statistics assists in summarizing the larger set of data in a form that is easily understandable.  Statistics assists in the efficient design of laboratory and field experiments as well as surveys.  Statistics assists in a sound and effective planning in any field of inquiry.  Statistics assists in drawing general conclusions and in making predictions of how much of a thing will happen under given conditions. IMPORTANCE OF STATISTICS IN VARIOUS FIELDS As stated earlier, Statistics is a discipline that has finds application in the most diverse fields of activity. It is perhaps a subject that should be used by everybody. Statistical techniques being powerful tools for analyzing numerical data are used in almost every branch of learning. In all areas, statistical techniques are being increasingly used, and are developing very rapidly. Virtual University of Pakistan 2 STA301 – Statistics and Probability  A modern administrator whether in public or private sector leans on statistical data to provide a factual basis for decision.  A politician uses statistics advantageously to lend support and credence to his arguments while elucidating the problems he handles.  A businessman, an industrial and a research worker all employ statistical methods in their work. Banks, Insurance companies and Government all have their statistics departments.  A social scientist uses statistical methods in various areas of socio-economic life a nation. It is sometimes said that “a social scientist without an adequate understanding of statistics, is often like the blind man groping in a dark room for a black cat that is not there”. THE MEANING OF DATA The word “data” appears in many contexts and frequently is used in ordinary conversation. Although the word carries something of an aura of scientific mystique, its meaning is quite simple and mundane. It is Latin for “those that are given” (the singular form is “datum”). Data may therefore be thought of as the results of observation. EXAMPLES OF DATA  Data are collected in many aspects of everyday life.  Statements given to a police officer or physician or psychologist during an interview are data.  So are the correct and incorrect answers given by a student on a final examination.  Almost any athletic event produces data.  The time required by a runner to complete a marathon,  The number of errors committed by a baseball team in nine innings of play.  And, of course, data are obtained in the course of scientific inquiry:  the positions of artifacts and fossils in an archaeological site,  The number of interactions between two members of an animal colony during a period of observation,  The spectral composition of light emitted by a star. OBSERVATIONS AND VARIABLES In statistics, an observation often means any sort of numerical recording of information, whether it is a physical measurement such as height or weight; a classification such as heads or tails, or an answer to a question such as yes or no. VARIABLES A characteristic that varies with an individual or an object is called a variable. For example, age is a variable as it varies from person to person. A variable can assume a number of values. The given set of all possible values from which the variable takes on a value is called its Domain. If for a given problem, the domain of a variable contains only one value, then the variable is referred to as a constant. QUANTITATIVE AND QUALITATIVE VARIABLES Variables may be classified into quantitative and qualitative according to the form of the characteristic of interest. A variable is called a quantitative variable when a characteristic can be expressed numerically such as age, weight, income or number of children. On the other hand, if the characteristic is non-numerical such as education, sex, eye- colour, quality, intelligence, poverty, satisfaction, etc. the variable is referred to as a qualitative variable. A qualitative characteristic is also called an attribute. An individual or an object with such a characteristic can be counted or enumerated after having been assigned to one of the several mutually exclusive classes or categories. Virtual University of Pakistan 3 STA301 – Statistics and Probability DISCRETE AND CONTINUOUS VARIABLES A quantitative variable may be classified as discrete or continuous. A discrete variable is one that can take only a discrete set of integers or whole numbers, which is the values, are taken by jumps or breaks. A discrete variable represents count data such as the number of persons in a family, the number of rooms in a house, the number of deaths in an accident, the income of an individual, etc. A variable is called a continuous variable if it can take on any value-fractional or integral––within a given interval, i.e. its domain is an interval with all possible values without gaps. A continuous variable represents measurement data such as the age of a person, the height of a plant, the weight of a commodity, the temperature at a place, etc. A variable whether countable or measurable, is generally denoted by some symbol such as X or Y and X i or Xj represents the ith or jth value of the variable. The subscript i or j is replaced by a number such as 1,2,3, … when referred to a particular value. MEASUREMENT SCALES By measurement, we usually mean the assigning of number to observations or objects and scaling is a process of measuring. The four scales of measurements are briefly mentioned below: NOMINAL SCALE The classification or grouping of the observations into mutually exclusive qualitative categories or classes is said to constitute a nominal scale. For example, students are classified as male and female. Number 1 and 2 may also be used to identify these two categories. Similarly, rainfall may be classified as heavy moderate and light. We may use number 1, 2 and 3 to denote the three classes of rainfall. The numbers when they are used only to identify the categories of the given scale carry no numerical significance and there is no particular order for the grouping. ORDINAL OR RANKING SCALE It includes the characteristic of a nominal scale and in addition has the property of ordering or ranking of measurements. For example, the performance of students (or players) is rated as excellent, good fair or poor, etc. Number 1, 2, 3, 4 etc. are also used to indicate ranks. The only relation that holds between any pair of categories is that of “greater than” (or more preferred). INTERVAL SCALE A measurement scale possessing a constant interval size (distance) but not a true zero point, is called an interval scale. Temperature measured on either the Celsius or the Fahrenheit scale is an outstanding example of interval scale because the same difference exists between 20o C (68o F) and 30o C (86o F) as between 5o C (41o F) and 15o C (59o F). It cannot be said that a temperature of 40 degrees is twice as hot as a temperature of 20 degree, i.e. the ratio 40/20 has no meaning. The arithmetic operation of addition, subtraction, etc. is meaningful. RATIO SCALE It is a special kind of an interval scale where the sale of measurement has a true zero point as its origin. The ratio scale is used to measure weight, volume, distance, money, etc. The, key to differentiating interval and ratio scale is that the zero point is meaningful for ratio scale. ERRORS OF MEASUREMENT Experience has shown that a continuous variable can never be measured with perfect fineness because of certain habits and practices, methods of measurements, instruments used, etc. the measurements are thus always recorded correct to the nearest units and hence are of limited accuracy. The actual or true values are, however, assumed to exist. For example, if a student’s weight is recorded as 60 kg (correct to the nearest kilogram), his true weight in fact lies between 59.5 kg and 60.5 kg, whereas a weight recorded as 60.00 kg means the true weight is known to lie between 59.995 and 60.005 kg. Thus there is a difference, however small it may be between the measured value and the true value. This sort of departure from the true value is technically known as the error of measurement. In other words, if the observed value and the true value of a variable are denoted by x and x +  respectively, then the difference (x + ) – x, i.e.  is the error. This error involves the unit of measurement of x and is therefore called an absolute error. An absolute error  divided by the true value is called the relative error. Thus the relative error  , which when multiplied by 100, x is percentage error. These errors are independent of the units of measurement of x. It ought to be noted that an error has both magnitude and direction and that the word error in statistics does not mean mistake which is a chance inaccuracy. Virtual University of Pakistan 4 STA301 – Statistics and Probability BIASED AND RANDOM ERRORS An error is said to be biased when the observed value is consistently and constantly higher or lower than the true value. Biased errors arise from the personal limitations of the observer, the imperfection in the instruments used or some other conditions which control the measurements. These errors are not revealed by repeating the measurements. They are cumulative in nature, that is, the greater the number of measurements, the greater would be the magnitude of error. They are thus more troublesome. These errors are also called cumulative or systematic errors. An error, on the other hand, is said to be unbiased when the deviations, i.e. the excesses and defects, from the true value tend to occur equally often. Unbiased errors and revealed when measurements are repeated and they tend to cancel out in the long run. These errors are therefore compensating and are also known as random errors or accidental errors. Virtual University of Pakistan 5 STA301 – Statistics and Probability LECTURE NO. 2 Steps involved in a Statistical Research-Project  Collection of Data:  Primary Data  Secondary Data  Sampling:  Concept of Sampling  Non-Random Versus Random Sampling  Simple Random Sampling  Other Types of Random Sampling STEPS INVOLVED IN ANY STATISTICAL RESEARCH  Topic and significance of the study  Objective of your study  Methodology for data-collection  Source of your data  Sampling methodology  Instrument for collecting data As far as the objectives of your research are concerned, they should be stated in such a way that you are absolutely clear about the goal of your study --- EXACTLY WHAT IT IS THAT YOU ARE TRYING TO FIND OUT? As far as the methodology for DATA-COLLECTION is concerned, you need to consider:  Source of your data (the statistical population)  Sampling Methodology  Instrument for collecting data COLLECTION OF DATA The most important part of statistical work is perhaps the collection of data. Statistical data are collected either by a COMPLETE enumeration of the whole field, called CENSUS, which in many cases would be too costly and too time consuming as it requires large number of enumerators and supervisory staff, or by a PARTIAL enumeration associated with a SAMPLE which saves much time and money. PRIMARY AND SECONDARY DATA Data that have been originally collected (raw data) and have not undergone any sort of statistical treatment, are called PRIMARY data. Data that have undergone any sort of treatment by statistical methods at least ONCE, i.e. the data that have been collected, classified, tabulated or presented in some form for a certain purpose, are called SECONDARY data. COLLECTION OF PRIMARY DATA One or more of the following methods are employed to collect primary data:  Direct Personal Investigation  Indirect Investigation  Collection through Questionnaires  Collection through Enumerators  Collection through Local Sources DIRECT PERSONAL INVESTIGATION In this method, an investigator collects the information personally from the individuals concerned. Since he interviews the informants himself, the information collected is generally considered quite accurate and complete. This method may prove very costly and time-consuming when the area to be covered is vast. However, it is useful for laboratory experiments or localized inquiries. Errors are likely to enter the results due to personal bias of the investigator. INDIRECT INVESTIGATION Sometimes the direct sources do not exist or the informants hesitate to respond for some reason or other. In such a case, third parties or witnesses having information are interviewed. Moreover, due allowance is to be made for the personal bias. This method is useful when the information desired is complex or there is reluctance or indifference on the part of the informants. It can be adopted for extensive inquiries. Virtual University of Pakistan 6 STA301 – Statistics and Probability COLLECTION THROUGH QUESTIONNAIRES A questionnaire is an inquiry form comprising of a number of pertinent questions with space for entering information asked. The questionnaires are usually sent by mail, and the informants are requested to return the questionnaires to the investigator after doing the needful within a certain period. This method is cheap, fairly expeditious and good for extensive inquiries. But the difficulty is that the majority of the respondents (i.e. persons who are required to answer the questions) do not care to fill the questionnaires in, and to return them to the investigators. Sometimes, the questionnaires are returned incomplete and full of errors. Students, in spite of these drawbacks, this method is considered as the STANDARD method for routine business and administrative inquiries. It is important to note that the questions should be few, brief, very simple, and easy for all respondents answer, clearly worded and not offensive to certain respondents. COLLECTION THROUGH ENUMERATORS Under this method, the information is gathered by employing trained enumerators who assist the informants in making the entries in the schedules or questionnaires correctly. This method gives the most reliable information if the enumerator is well-trained, experienced and tactful. Students, it is considered the BEST method when a large-scale governmental inquiry is to be conducted. This method can generally not be adopted by a private individual or institution as its cost would be prohibitive to them. COLLECTION THROUGH LOCAL SOURCES In this method, there is no formal collection of data but the agents or local correspondents are directed to collect and send the required information, using their own judgment as to the best way of obtaining it. This method is cheap and expeditious, but gives only the estimates. COLLECTION OF SECONDARY DATA The secondary data may be obtained from the following sources:  Official, e.g. the publications of the Statistical Division, Ministry of Finance, the Federal and Provincial Bureaus of Statistics, Ministries of Food, Agriculture, Industry, Labour, etc.  Semi-Official, e.g., State Bank of Pakistan, Railway Board, Central Cotton Committee, Boards of Economic Inquiry, District Councils, Municipalities, etc.  Publications of Trade Associations, Chambers of Commerce, etc  Technical and Trade Journals and Newspapers  Research Organizations such as universities, and other institutions Let us now consider the POPULATION from which we will be collecting our data. In this context, the first important question is: Why do we have to resort to Sampling? The answer is that: If we have available to us every value of the variable under study, then that would be an ideal and a perfect situation. But, the problem is that this ideal situation is very rarely available --- very rarely do we have access to the entire population. The census is an exercise in which an attempt is made to cover the entire population. But, as you might know, even the most developed countries of the world cannot afford to conduct such a huge exercise on an annual basis! More often than not, we have to conduct our research study on a sample basis. In fact, the goal of the science of Statistics is to draw conclusions about large populations on the basis of information contained in small samples. ‘POPULATION’ A statistical population is the collection of every member of a group possessing the same basic and defined characteristic, but varying in amount or quality from one member to another. EXAMPLES  Finite population:  IQ’s of all children in a school.  Infinite population:  Barometric pressure: (There are an indefinitely large number of points on the surface of the earth).  A flight of migrating ducks in Canada (Many finite pops are so large that they can be treated as effectively infinite). The examples that we have just considered are those of existent populations. A hypothetical population can be defined as the aggregate of all the conceivable ways in which a specified event can happen. Virtual University of Pakistan 7 STA301 – Statistics and Probability For Example:  1)All the possible outcomes from the throw of a die – however long we throw the die and record the results, we could always continue to do so far a still longer period in a theoretical concept – one which has no existence in reality.  2) The No. of ways in which a football team of 11 players can be selected from the 16 possible members named by the Club Manager. We also need to differentiate between the sampled population and the target population. Sampled population is that from which a sample is chosen whereas the population about which information is sought is called the target population thus our population will consist of the total no. of students in all the colleges in the Punjab. Suppose on account of shortage of resources or of time, we are able to conduct such a survey on only 5 colleges scattered throughout the province. In this case, the students of all the colleges will constitute the target pop whereas the students of those 5 colleges from which the sample of students will be selected will constitute the sampled population. The above discussion regarding the population, you must have realized how important it is to have a very well-defined population. The next question is: How will we draw a sample from our population? The answer is that: In order to draw a random sample from a finite population, the first thing that we need is the complete list of all the elements in our population. This list is technically called the FRAME. SAMPLING FRAME A sampling frame is a complete list of all the elements in the population. For example:  The complete list of the BCS students of Virtual University of Pakistan on February 15, 2003  Speaking of the sampling frame, it must be kept in mind that, as far as possible, our frame should be free from various types of defects:  does not contain inaccurate elements  is not incomplete  is free from duplication, and  Is not out of date. Next, let’s talk about the sample that we are going to draw from this population. As you all know, a sample is only a part of a statistical population, and hence it can represent the population to only to some extent. Of course, it is intuitively logical that the larger the sample, the more likely it is to represent the population. Obviously, the limiting case is that: when the sample size tends to the population size, the sample will tend to be identical to the population. But, of course, in general, the sample is much smaller than the population. The point is that, in general, statistical sampling seeks to determine how accurate a description of the population the sample and its properties will provide. We may have to compromise on accuracy, but there are certain such advantages of sampling because of which it has an extremely important place in data-based research studies. ADVANTAGES OF SAMPLING 1. Savings in time and money.  Although cost per unit in a sample is greater than in a complete investigation, the total cost will be less (because the sample will be so much smaller than the statistical population from which it has been drawn).  A sample survey can be completed faster than a full investigation so that variations from sample unit to sample unit over time will largely be eliminated.  Also, the results can be processed and analyzed with increased speed and precision because there are fewer of them. 2. More detailed information may be obtained from each sample unit. 3. Possibility of follow-up: (After detailed checking, queries and omissions can be followed up --- a procedure which might prove impossible in a complete survey). 4. Sampling is the only feasible possibility where tests to destruction are undertaken or where the population is effectively infinite. The next two important concepts that need to be considered are those of sampling and non-sampling errors. SAMPLING & NON-SAMPLING ERRORS 1. SAMPLING ERROR The difference between the estimate derived from the sample (i.e. the statistic) and the true population value (i.e. the parameter) is technically called the sampling error. For example, Virtual University of Pakistan 8 STA301 – Statistics and Probability Sampling error = X  Sampling error arises due to the fact that a sample cannot exactly represent the pop, even if it is drawn in a correct manner 2. NON-SAMPLING ERROR Besides sampling errors, there are certain errors which are not attributable to sampling but arise in the process of data collection, even if a complete count is carried out. Main sources of non sampling errors are:  The defect in the sampling frame.  Faulty reporting of facts due to personal preferences.  Negligence or indifference of the investigators  Non-response to mail questionnaires. These (non-sampling) errors can be avoided through  Following up the non-response,  Proper training of the investigators.  Correct manipulation of the collected information, Let us now consider exactly what is meant by ‘sampling error’: We can say that there are two types of non-response --- partial non-response and total non-response. ‘Partial non-response’ implies that the respondent refuses to answer some of the questions. On the other hand, ‘total non-response’ implies that the respondent refuses to answer any of the questions. Of course, the problem of late returns and non-response of the kind that I have just mentioned occurs in the case of HUMAN populations. Although refusal of sample units to cooperate is encountered in interview surveys, it is far more of a problem in mail surveys. It is not uncommon to find the response rate to mail questionnaires as low as 15 or 20%.The provision of INFORMATION ABOUT THE PURPOSE OF THE SURVEY helps in stimulating interest, thus increasing the chances of greater response. Particularly if it can be shown that the work will be to the ADVANTAGE of the respondent IN THE LONG RUN. Similarly, the respondent will be encouraged to reply if a pre-paid and addressed ENVELOPE is sent out with the questionnaire. But in spite of these ways of reducing non-response, we are bound to have some amount of non-response. Hence, a decision has to be taken about how many RECALLS should be made. The term ‘recall’ implies that we approach the respondent more than once in order to persuade him to respond to our queries. Another point worth considering is: How long the process of data collection should be continued? Obviously, no such process can be carried out for an indefinite period of time! In fact, the longer the time period over which the survey is conducted, the greater will be the potential VARIATIONS in attitudes and opinions of the respondents. Hence, a well-defined cut-off date generally needs to be established. Let us now look at the various ways in which we can select a sample from our population. We begin by looking at the difference between non-random and RANDOM sampling. First of all, what do we mean by non- random sampling? NONRANDOM SAMPLING ‘Nonrandom sampling’ implies that kind of sampling in which the population units are drawn into the sample by using one’s personal judgment. This type of sampling is also known as purposive sampling. Within this category, one very important type of sampling is known as Quota Sampling. QUOTA SAMPLING In this type of sampling, the selection of the sampling unit from the population is no longer dictated by chance. A sampling frame is not used at all, and the choice of the actual sample units to be interviewed is left to the discretion of the interviewer. However, the interviewer is restricted by quota controls. For example, one particular interviewer may be told to interview ten married women between thirty and forty years of age living in town X, whose husbands are professional workers, and five unmarried professional women of the same age living in the same town. Quota sampling is often used in commercial surveys such as consumer market-research. Also, it is often used in public opinion polls. ADVANTAGES OF QUOTA SAMPLING  There is no need to construct a frame.  It is a very quick form of investigation.  Cost reduction. Virtual University of Pakistan 9 STA301 – Statistics and Probability DISADVANTAGES  It is a subjective method. One has to choose between objectivity and convenience.  If random sampling is not employed, it is no longer theoretically possible to evaluate the sampling error.  (Since the selection of the elements is not based on probability theory but on the personal judgment of the interviewer, hence the precision and the reliability of the estimates can not be determined objectively i.e. in terms of probability.)  Although the purpose of implementing quota controls is to reduce bias, bias creeps in due to the fact that the interviewer is FREE to select particular individuals within the quotas. (Interviewers usually look for persons who either agree with their points of view or are personally known to them or can easily be contacted.)  Even if the above is not the case, the interviewer may still be making unsuitable selection of sample units.  (Although he may put some qualifying questions to a potential respondent in order to determine whether he or she is of the type prescribed by the quota controls, some features must necessarily be decided arbitrarily by the interviewer, the most difficult of these being social class.) If mistakes are being made, it is almost impossible for the organizers to detect these, because follow-ups are not possible unless a detailed record of the respondents’ names, addresses etc. has been kept. Falsification of returns is therefore more of a danger in quota sampling than in random sampling. In spite of the above limitations, it has been shown by F. Edwards that a well-organized quota survey with well-trained interviewers can produce quite adequate results. Next, let us consider the concept of random sampling. RANDOM SAMPLING The theory of statistical sampling rests on the assumption that the selection of the sample units has been carried out in a random manner. By random sampling we mean sampling that has been done by adopting the lottery method. TYPES OF RANDOM SAMPLING  Simple Random Sampling  Stratified Random Sampling  Systematic Sampling  Cluster Sampling  Multi-stage Sampling, etc. In this course, I will discuss with you the simplest type of random sampling i.e. simple random sampling. SIMPLE RANDOM SAMPLING In this type of sampling, the chance of any one element of the parent pop being included in the sample is the same as for any other element. By extension, it follows that, in simple random sampling, the chance of any one sample appearing is the same as for any other. There exists quite a lot of misconception regarding the concept of random sampling. Many a time, haphazard selection is considered to be equivalent to simple random sampling. For example, a market research interviewer may select women shoppers to find their attitude to brand X of a product by stopping one and then another as they pass along a busy shopping area --- and he may think that he has accomplished simple random sampling! Actually, there is a strong possibility of bias as the interviewer may tend to ask his questions of young attractive women rather than older housewives, or he may stop women who have packets of brand X prominently on show in their shopping bags!. In this example, there is no suggestion of INTENTIONAL bias! From experience, it is known that the human being is a poor random selector --- one who is very subject to bias. Fundamental psychological traits prevent complete objectivity, and no amount of training or conscious effort can eradicate them. As stated earlier, random sampling is that in which population units are selected by the lottery method. As you know, the traditional method of writing people’s names on small pieces of paper, folding these pieces of paper and shuffling them is very cumbersome! A much more convenient alternative is the use of RANDOM NUMBERS TABLES. A random number table is a page full of digits from zero to 9. These digits are printed on the page in a TOTALLY random manner i.e. there is no systematic pattern of printing these digits on the page. Virtual University of Pakistan 10 STA301 – Statistics and Probability ONE THOUSAND RANDOM DIGITS 2 3 1 5 7 5 4 8 5 9 0 1 8 3 7 2 5 9 9 3 7 6 2 4 9 7 0 8 8 6 9 5 2 3 0 3 6 7 4 4 0 5 5 4 5 5 5 0 4 3 1 0 5 3 7 4 3 5 0 8 9 0 6 1 1 8 3 7 4 4 1 0 9 6 2 2 1 3 4 3 1 4 8 7 1 6 0 3 5 0 3 2 4 0 4 3 6 2 2 3 5 0 0 5 1 0 0 3 2 2 1 1 5 4 3 8 0 8 3 4 3 8 9 7 6 7 4 9 5 1 9 4 0 5 1 7 5 8 5 3 7 8 8 0 5 9 0 1 9 4 3 2 4 2 8 7 1 6 9 5 9 7 3 1 2 6 1 7 1 8 9 9 7 5 5 3 0 8 7 0 9 4 2 5 1 2 5 8 4 1 5 4 8 8 2 1 0 5 1 3 1 1 7 4 2 6 9 3 8 1 4 4 3 3 9 3 0 8 7 2 3 2 7 9 7 3 3 1 1 8 2 2 6 4 7 0 6 8 5 0 4 3 3 6 1 2 8 8 5 9 1 1 0 1 6 4 5 6 2 3 9 3 0 0 9 0 0 4 9 9 4 3 6 4 0 7 4 0 3 6 9 3 8 0 6 2 0 4 7 8 3 8 2 6 8 0 4 4 9 1 5 5 7 5 1 1 8 9 3 2 5 8 4 7 5 5 2 5 7 1 4 9 5 4 0 1 3 1 8 1 0 8 4 2 9 8 4 1 8 7 6 9 5 3 8 2 9 6 6 1 7 7 7 3 8 0 9 5 2 7 3 6 7 6 8 7 2 6 3 3 3 7 9 4 8 2 1 5 6 9 4 1 9 5 9 6 8 6 7 0 4 5 2 7 4 8 3 8 8 0 0 7 0 9 2 5 2 3 9 2 2 4 6 2 7 1 2 6 0 7 0 6 5 5 8 4 5 3 4 4 6 7 3 3 8 4 5 3 2 0 4 3 3 1 0 0 1 0 8 1 4 4 8 6 3 8 0 3 0 7 5 2 5 5 5 1 6 1 4 8 8 9 7 4 2 9 4 6 4 7 6 1 5 7 0 0 6 3 6 0 0 6 1 7 3 6 3 7 7 5 6 3 1 4 8 9 5 1 2 3 3 5 0 1 7 4 6 9 9 3 3 1 3 5 2 8 3 7 9 9 1 0 7 7 9 1 8 9 4 1 3 1 5 7 9 7 6 4 4 8 6 2 5 8 4 8 6 9 1 9 5 7 0 4 8 8 6 5 2 6 2 7 7 9 5 9 3 6 8 2 9 0 5 2 9 5 6 5 4 6 3 5 0 6 5 3 2 2 5 4 0 9 2 4 3 4 4 2 0 0 6 8 7 2 1 0 7 1 3 7 3 0 7 2 9 7 5 7 3 6 0 9 2 9 8 2 7 6 5 0 9 7 9 5 5 3 5 0 1 8 4 0 8 9 4 8 8 3 2 9 5 2 2 3 0 8 2 5 2 1 2 2 5 3 2 6 1 5 8 7 9 3 7 3 2 5 9 5 7 0 4 3 7 8 1 9 8 8 8 5 5 6 6 7 1 6 6 8 2 6 9 5 9 9 6 4 4 5 6 9 7 2 6 2 1 1 1 2 2 5 0 0 9 2 2 6 8 2 6 4 3 5 6 6 6 5 9 4 3 4 7 1 6 8 7 5 1 8 6 7 6 1 0 2 0 7 4 4 1 8 4 5 3 7 1 2 0 7 9 4 9 5 9 1 7 3 7 8 6 6 9 9 5 3 6 1 9 3 7 8 9 7 8 3 9 8 5 4 7 4 3 3 0 5 5 9 1 7 1 8 4 5 4 7 3 5 4 1 4 4 2 2 0 3 4 2 3 0 0 0 8 9 1 6 0 9 7 1 9 2 2 2 2 3 2 9 0 6 3 7 3 5 0 5 5 4 5 4 8 9 8 8 4 3 8 1 6 3 6 1 2 5 9 6 6 8 8 2 2 0 6 2 8 7 1 7 9 2 6 5 0 2 8 2 3 5 2 8 6 2 8 4 9 1 9 5 4 8 8 3 8 1 4 4 3 3 1 7 1 9 0 5 0 4 9 5 4 8 0 6 7 4 6 9 0 0 7 5 6 7 6 5 0 1 7 1 6 5 4 5 1 1 3 2 2 5 4 9 3 1 4 2 3 6 2 3 4 3 8 6 0 8 6 2 4 9 7 6 6 7 4 2 2 4 5 2 3 2 4 5 Actually, Random Number Tables are constructed according to certain mathematical principles so that each digit has the same chance of selection. Of course, nowadays randomness may be achieved electronically. Computers have all those programmes by which we can generate random numbers. EXAMPLE The following frequency table of distribution gives the ages of a population of 1000 teen-age college students in a particular country. Select a sample of 10 students using the random numbers table. Find the sample mean age and compare with the population mean age. Student-Population of a College Age No. of Students (X) (f) 13 6 14 61 15 270 16 491 17 153 18 15 19 4 1000 How will we proceed to select our sample of size 10 from this population of size 1000? Virtual University of Pakistan 11 STA301 – Statistics and Probability The first step is to allocate to each student in this population a sampling number. For this purpose, we will begin by constructing a column of cumulative frequencies. AGE No. of Students Cumulative Frequency X f cf 13 6 6 14 61 67 15 270 337 16 491 828 17 153 981 18 15 996 19 4 1000 1000 Now that we have the cumulative frequency of each class, we are in a position to allocate the sampling numbers to all the values in a class. As the frequency as well as the cumulative frequency of the first class is 6, we allocate numbers 000 to 005 to the six students who belong to this class. No. of AGE Sampling Students cf X Numbers f 13 6 6 000 – 005 14 61 67 15 270 337 16 491 828 17 153 981 18 15 996 19 4 1000 1000 As the cumulative frequency of the second class is 67 while that of the first class was 6, therefore we allocate sampling numbers 006 to 066 to the 61 students who belong to this class. No. of AGE Sampling Students cf X Numbers f 13 6 6 000 – 005 14 61 67 006 – 066 15 270 337 16 491 828 17 153 981 18 15 996 19 4 1000 1000 Virtual University of Pakistan 12 STA301 – Statistics and Probability As the cumulative frequency of the third class is 337 while that of the second class was 67, therefore we allocate sampling numbers 067 to 336 to the 270 students who belong to this class. No. of AGE Sampling Students cf X Numbers f 13 6 6 000 – 005 14 61 67 006 – 066 15 270 337 067 – 336 16 491 828 17 153 981 18 15 996 19 4 1000 1000 Proceeding in this manner, we obtain the column of sampling numbers. No. of AGE Sampling Students cf X Numbers f 13 6 6 000 – 005 14 61 67 006 – 066 15 270 337 067 – 336 16 491 828 337 – 827 17 153 981 828 – 980 18 15 996 981 – 995 19 4 1000 996 - 999 1000 The column implies that the first student of the first class has been allocated the sampling number 000, the second student has been allocated the sampling 001, and, proceeding in this fashion, the last student i.e. the 1000th student has been allocated the sampling number 999. The question is: Why did we not allot the number 0001 to the first student and the number 1000 to the 1000th student? The answer is that we could do that but that would have meant that every student would have been allocated a four-digit number, whereas by shifting the number backward by 1, we are able to allocate to every student a three-digit number --- which is obviously simpler. The next step is to SELECT 10 RANDOM NUMBERS from the random number table. This is accomplished by closing one’s eyes and letting one’s finger land anywhere on the random number table. In this example, since all our sampling numbers are three-digit numbers, hence we will read three digits that are adjacent to each other at that position where our finger landed. Suppose that we adopt this procedure and our random numbers come out to be 041, 103, 374, 171, 508, 652, 880, 066, 715, 471 Selected Random Numbers: 041, 103, 374, 171, 508, 652, 880, 066, 715, 471 Thus the corresponding ages are: 14, 15, 16, 15, 16, 16, 17, 15, 16, 16 EXPLANATION Our first selected random number is 041 which mean that we have to pick up the 42nd student. The cumulative frequency of the first class is 6 whereas the cumulative frequency of the second class is 67. This means that definitely the 42nd student does not belong to the first class but does belong to the second class. Virtual University of Pakistan 13 STA301 – Statistics and Probability No. of AGE Students cf X f 13 6 6 14 61 67 15 270 337 16 491 828 17 153 981 18 15 996 19 4 1000 1000 The age of each student in this class is 14 years; hence, obviously, the age of the 42nd student is also 14 years. This is how we are able to ascertain the ages of all the students who have been selected in our sampling. You will recall that in this example, our aim was to draw a sample from the population of college students, and to compare the sample’s mean age with the population mean age. The population mean age comes out to be 15.785 years. AGE No. of Students fX X f 13 6 78 14 61 854 15 270 4050 16 491 7856 17 153 2601 18 15 270 19 4 76 1000 15785 The population mean age is :   fx  15785  f 1000  15.785 years The above formula is a slightly modified form of the basic formula that you have done ever-since school-days i.e. the mean is equal to the sum of all the observations divided by the total number of observations. Next, we compute the sample mean age. Adding the 10 values and dividing by 10, we obtain: Ages of students selected in the sample (in years):  X  X  n  156  10      15.6 years  14, 15, 16, 15, 16, 16, 17, 15, 16, 16 Hence the sample mean age is: 15.6, comparing the sample mean age of 15.6 years with the population mean age of 15.785 years, we note that the difference is really quite slight, and hence the sampling error is equal to Sampling Error X    15.6  15.785  0.185 years Virtual University of Pakistan 14 STA301 – Statistics and Probability And the reason for such a small error is that we have adopted the RANDOM sampling method. The basic advantage of random sampling is that the probability is very high that the sample will be a good representative of the population from which it has been drawn, and any quantity computed from the sample will be a good estimate of the corresponding quantity computed from the population! Actually, a sample is supposed to be a MINIATURE REPLICA of the population. As stated earlier, there are various other types of random sampling. OTHER TYPES OF RANDOM SAMPLING  ·Stratified sampling (if the population is heterogeneous)  Systematic sampling (practically, more convenient than simple random sampling)  Cluster sampling (sometimes the sampling units exist in natural clusters)  Multi-stage sampling All these designs rest upon random or quasi-random sampling. They are various forms of PROBABILITY sampling --- that in which each sampling unit has a known (but not necessarily equal) probability of being selected. Because of this knowledge, there exist methods by which the precision and the reliability of the estimates can be calculated OBJECTIVELY. It should be realized that in practice, several sampling techniques are incorporated into each survey design, and only rarely will simple random sample be used, or a multi-stage design be employed, without stratification. The point to remember is that whatever method be adopted, care should be exercised at every step so as to make the results as reliable as possible. Virtual University of Pakistan 15  STA301 – Statistics and Probability LECTURE NO. 3  Tabulation  Simple bar chart  Component bar chart  Multiple bar chart  Pie chart As indicated in the last lecture, there are two broad categories of data … qualitative data and quantitative data. A variety of methods exist for summarizing and describing these two types of data. The tree-diagram below presents an outline of the various techniques TYPES OF DATA Qualitative Quantitative Univariate Bivariate Discrete Continuous Frequency Frequency Table Table Frequency Frequency Distribution Distribution Percentages Component Multiple Line Histogram Bar Chart Bar Chart Pie Chart Chart Frequency Bar Chart Polygon Frequency Curve Virtual University of Pakistan 16 STA301 – Statistics and Probability In today’s lecture, we will be dealing with various techniques for summarizing and describing qualitative data. Qualitative Univariate Bivariate Frequency Frequency Table Table Percentages Component Multiple Pie Chart Bar Chart Bar Chart Bar Chart We will begin with the univariate situation, and will proceed to the bivariate situation. EXAMPLE Suppose that we are carrying out a survey of the students of first year studying in a co-educational college of Lahore. Suppose that in all there are 1200 students of first year in this large college. We wish to determine what proportion of these students have come from Urdu medium schools and what proportion has come from English medium schools. So we will interview the students and we will inquire from each one of them about their schooling. As a result, we will obtain a set of data as you can now see on the screen. We will have an array of observations as follows: U, U, E, U, E, E, E, U, …… (U : URDU MEDIUM) (E : ENGLISH MEDIUM) Now, the question is what should we do with this data? Obviously, the first thing that comes to mind is to count the number of students who said “Urdu medium” as well as the number of students who said “English medium”. This will result in the following table: Medium of No. of Students Institution (f) Urdu 719 English 481 1200 The technical term for the numbers given in the second column of this table is “frequency”. It means “how frequently something happens?” Out of the 1200 students, 719 stated that they had come from Urdu medium schools. So in this example, the frequency of the first category of responses is 719 whereas the frequency of the second category of responses is 481. Virtual University of Pakistan 17 STA301 – Statistics and Probability It is evident that this information is not as useful as if we compute the proportion or percentage of students falling in each category. Dividing the cell frequencies by the total frequency and multiplying by 100 we obtain the following: Medium of f % Institution Urdu 719 59.9 = 60% English 481 40.1 = 40% 1200 What we have just accomplished is an example of a univariate frequency table pertaining to qualitative data. Let us now see how we can represent this information in the form of a diagram. One good way of representing the above information is in the form of a pie chart. A pie chart consists of a circle which is divided into two or more parts in accordance with the number of distinct categories that we have in our data. For the example that we have just considered, the circle is divided into two sectors, the larger sector pertaining to students coming from Urdu medium schools and the smaller sector pertaining to students coming from English medium schools. How do we decide where to cut the circle? The answer is very simple! All we have to do is to divide the cell frequency by the total frequency and multiply by 360. This process will give us the exact value of the angle at which we should cut the circle. PIE CHART Medium of f Angle Institution Urdu 719 215.70 English 481 144.30 1200 Urdu 215.70 English 144.30 Virtual University of Pakistan 18 STA301 – Statistics and Probability SIMPLE BAR CHART: The next diagram to be considered is the simple bar chart. A simple bar chart consists of horizontal or vertical bars of equal width and lengths proportional to values they represent. As the basis of comparison is one-dimensional, the widths of these bars have no mathematical significance but are taken in order to make the chart look attractive. Let us consider an example. Suppose we have available to us information regarding the turnover of a company for 5 years as given in the table below: Years 1965 1966 1967 1968 1969 Turnover (Rupees) 35,000 42,000 43,500 48,000 48,500 In order to represent the above information in the form of a bar chart, all we have to do is to take the year along the x- axis and construct a scale for turnover along the y-axis. 50,000 40,000 30,000 20,000 10,000 0 1965 1966 1967 1968 1969 Next, against each year, we will draw vertical bars of equal width and different heights in accordance with the turn-over figures that we have in our table. As a result we obtain a simple and attractive diagram as shown below. When our values do not relate to time, they should be arranged in ascending or descending order before-charting. BIVARIATE FREQUENCY TABLE 50,000 40,000 30,000 20,000 10,000 0 1965 1966 1967 1968 1969 What we have just considered was the univariate situation. In each of the two examples, we were dealing with one single variable. In the example of the first year students of a college, our lone variable of interest was ‘medium of schooling’. And in the second example, our one single variable of interest was turnover. Now let us expand the discussion a little, and consider the bivariate situation. Virtual University of Pakistan 19 STA301 – Statistics and Probability Going back to the example of the first year students, suppose that alongwith the enquiry about the Medium of Institution, you are also recording the sex of the student. Suppose that our survey results in the following information: Student No. Medium Gender 1 U F 2 U M 3 E M 4 U F 5 E M 6 E F 7 U M 8 E M : : : : : : Now this is a bivariate situation; we have two variables, medium of schooling and sex of the student. In order to summarize the above information, we will construct a table containing a box head and a stub as shown below: M Sex A Female Total Med. L E Urdu English Total The top row of this kind of a table is known as the boxhead and the first column of the table is known as stub. Next, we will count the number of students falling in each of the following four categories: 1. Male student coming from an Urdu medium school. 2. Female student coming from an Urdu medium school. 3. Male student coming from an English medium school. 4. Female student coming from an English medium school. As a result, suppose we obtain the following figures: M Sex A Female Total Med. L E Urdu 202 517 719 English 350 131 481 Total 552 648 1200 What we have just accomplished is an example of a bivariate frequency table pertaining to two qualitative variables. Virtual University of Pakistan 20 STA301 – Statistics and Probability COMPONENT BAR CHAR: Let us now consider how we will depict the above information diagrammatically. This can be accomplished by constructing the component bar chart (also known as the subdivided bar chart) as shown below: Urdu 800 English 700 600 500 400 300 200 100 0 Male Female In the above figure, each bar has been divided into two parts. The first bar represents the total number of male students whereas the second bar represents the total number of female students. As far as the medium of schooling is concerned, the lower part of each bar represents the students coming from English medium schools. Whereas the upper part of each bar represents the students coming from the Urdu medium schools. The advantage of this kind of a diagram is that we are able to ascertain the situation of both the variables at a glance. We can compare the number of male students in the college with the number of female students, and at the same time we can compare the number of English medium students among the males with the number of English medium students among the females. MULTIPLE BAR CHARTS The next diagram to be considered is the multiple bar charts. Let us consider an example. Suppose we have information regarding the imports and exports of Pakistan for the years 1970-71 to 1974-75 as shown in the table below: Imports Exports Years (Crores of Rs.) (Crores of Rs.) 1970-71 370 200 1971-72 350 337 1972-73 840 855 1973-74 1438 1016 1974-75 2092 1029 Source: State Bank of Pakistan A multiple bar chart is a very useful and effective way of presenting this kind of information. This kind of a chart consists of a set of grouped bars, the lengths of which are proportionate to the values of our variables, and each of which is shaded or colored differently in order to aid identification. With reference to the above example, we obtain the multiple bar chart shown below: Virtual University of Pakistan 21 STA301 – Statistics and Probability Multiple Bar Chart Showing Imports & Exports of Pakistan 1970-71 to 1974-75 This is a very good device for the comparison of two different kinds of information. If, in addition to information regarding imports and exports, we also had information regarding production, we could have compared them from year to year by grouping the three bars together. The question is, what is the basic difference between a component bar chart and a multiple bar chart? The component bar chart should be used when we have available to us information regarding totals and their components. For example, the total number of male students out of which some are Urdu medium and some are English medium. The number of Urdu medium male students and the number of English medium male students add up to give us the total number of male students. On the contrary, in the example of exports and imports, the imports and exports do not add up to give us the totality of some one thing! Virtual University of Pakistan 22 STA301 – Statistics and Probability LECTURE NO. 4 In THIS Lecture, we will discuss the frequency distribution of a continuous variable & the graphical ways of representing data pertaining to a continuous variable i.e. histogram, frequency polygon and frequency curve. You will recall that in Lecture No. 1, it was mentioned that a continuous variable takes values over a continuous interval (e.g. a normal Pakistani adult male’s height may lie anywhere between 5.25 feet and 6.5 feet). Hence, in such a situation, the method of constructing a frequency distribution is somewhat different from the one that was discussed in the last lecture. EXAMPLE: Suppose that the Environmental Protection Agency of a developed country performs extensive tests on all new car models in order to determine their mileage rating. Suppose that the following 30 measurements are obtained by conducting such tests on a particular new car model. EPA MILEAGE RATINGS ON 30 CARS (MILES PER GALLON) 36.3 42.1 44.9 30.1 37.5 32.9 40.5 40.0 40.2 36.2 35.6 35.9 38.5 38.8 38.6 36.3 38.4 40.5 41.0 39.0 37.0 37.0 36.7 37.1 37.1 34.8 33.9 39.9 38.1 39.8 EPA: Environmental Protection Agency There are a few steps in the construction of a frequency distribution for this type of a variable. CONSTRUCTION OF A FREQUENCY DISTRIBUTION Step-1 Identify the smallest and the largest measurements in the data set. In our example: Smallest value (X0) = 30.1, Largest Value (Xm) = 44.9, Step-2 Find the range which is defined as the difference between the largest value and the smallest value In our example: Range = Xm – X0 = 44.9 – 30.1 = 14.8 Let us now look at the graphical picture of what we have just computed. Virtual University of Pakistan 23 STA301 – Statistics and Probability 30.1 44.9 R 30 35 40 45 14.8 (Range) Step-3 Decide on the number of classes into which the data are to be grouped. (By classes, we mean small sub-intervals of the total interval which, in this example, is 14.8 units long.)There are no hard and fast rules for this purpose. The decision will depend on the size of the data. When the data are sufficiently large, the number of classes is usually taken between 10 and 20.In this example, suppose that we decide to form 5 classes (as there are only 30 observations). Step-4 Divide the range by the chosen number of classes in order to obtain the approximate value of the class interval i.e. the width of our classes. Class interval is usually denoted by h. Hence, in this example Class interval = h = 14.8 / 5 = 2.96 Rounding the number 2.96, we obtain 3, and hence we take h = 3. This means that our big interval will be divided into small sub-intervals, each of which will be 3 units long. Step-5 Decide the lower class limit of the lowest class. Where should we start from? The answer is that we should start constructing our classes from a number equal to or slightly less than the smallest value in the data. In this example, smallest value = 30.1 So we may choose the lower class limit of the lowest class to be 30.0. Step-6 Determine the lower class limits of the successive classes by adding h = 3 successively. Hence, we obtain the following table: Class Lower Class Limit Number 1 30.0 2 30.0 + 3 = 33.0 3 33.0 + 3 = 36.0 4 36.0 + 3 = 39.0 5 39.0 + 3 = 42.0 Step-7 Determine the upper class limit of every class. The upper class limit of the highest class should cover the largest value in the data. It should be noted that the upper class limits will also have a difference of h between them. Hence, we obtain the upper class limits that are visible in the third column of the following table. Class Lower Class Upper Class Number Limit Limit 1 30.0 32.9 2 30.0 + 3 = 33.0 32.9 + 3 = 35.9 3 33.0 + 3 = 36.0 35.9 + 3 = 38.9 4 36.0 + 3 = 39.0 38.9 + 3 = 41.9 5 39.0 + 3 = 42.0 41.9 + 3 = 44.9 Virtual University of Pakistan 24 STA301 – Statistics and Probability Hence we obtain the following classes: Classes 30.0 – 32.9 33.0 – 35.9 36.0 – 38.9 39.0 – 41.9 42.0 – 44.9 The question arises: why did we not write 33 instead of 32.9? Why did we not write 36 instead of 35.9? and so on. The reason is that if we wrote 30 to 33 and then 33 to 36, we would have trouble when tallying our data into these classes. Where should I put the value 33? Should I put it in the first class, or should I put it in the second class? By writing 30.0 to 32.9 and 33.0 to 35.9, we avoid this problem. And the point to be noted is that the class interval is still 3, and not 2.9 as it appears to be. This point will be better understood when we discuss the concept of class boundaries … which will come a little later in today’s lecture. Step-8 After forming the classes, distribute the data into the appropriate classes and find the frequency of each class, in this example: Class Tally Frequency 30.0 – 32.9 || 2 33.0 – 35.9 |||| 4 36.0 – 38.9 |||| |||| |||| 14 39.0 – 41.9 |||| ||| 8 42.0 – 44.9 || 2 Total 30 This is a simple example of the frequency distribution of a continuous or, in other words, measurable variable. CLASS BOUNDARIES: The true class limits of a class are known as its class boundaries. In this example: Class Limit Class Boundaries Frequency 30.0 – 32.9 29.95 – 32.95 2 33.0 – 35.9 32.95 – 35.95 4 36.0 – 38.9 35.95 – 38.95 14 39.0 – 41.9 38.95 – 41.95 8 42.0 – 44.9 41.95 – 44.95 2 Total 30 Virtual University of Pakistan 25 STA301 – Statistics and Probability It should be noted that the difference between the upper class boundary and the lower class boundary of any class is equal to the class interval h = 3. 32.95 minus 29.95 is equal to 3, 35.95 minus 32.95 is equal to 3, and so on. A key point in this entire discussion is that the class boundaries should be taken up to one decimal place more than the given data. In this way, the possibility of an observation falling exactly on the boundary is avoided. (The observed value will either be greater than or less than a particular boundary and hence will conveniently fall in its appropriate class).Next, we consider the concept of the relative frequency distribution and the percentage frequency distribution. Next, we consider the concept of the relative frequency distribution and the percentage frequency distribution. This concept has already been discussed when we considered the frequency distribution of a discrete variable. Dividing each frequency of a frequency distribution by the total number of observations, we obtain the relative frequency distribution. Multiplying each relative frequency by 100, we obtain the percentage of frequency distribution. In this way, we obtain the relative frequencies and the percentage frequencies shown below Class Relative %age Frequency Limit Frequency Frequency 30.0 – 32.9 2 2/30 = 0.067 6.7 33.0 – 35.9 4 4/30 = 0.133 13.3 36.0 – 38.9 14 14/30 = 0.467 4.67 39.0 – 41.9 8 8/30 = 0.267 26.7 42.0 – 44.9 2 2/30 = 0.067 6.7 30 The term ‘relative frequencies’ simply means that we are considering the frequencies of the various classes relative to the total number of observations. The advantage of constructing a relative frequency distribution is that comparison is possible between two sets of data having similar classes. For example, suppose that the Environment Protection Agency perform tests on two car models A and B, and obtains the frequency distributions shown below: FREQUENCY MILEAGE Model A Model B 30.0 – 32.9 2 7 33.0 – 35.9 4 10 36.0 – 38.9 14 16 39.0 – 41.9 8 9 42.0 – 44.9 2 8 30 50 In order to be able to compare the performance of the two car models, we construct the relative frequency distributions in the percentage form: Virtual University of Pakistan 26 STA301 – Statistics and Probability MILEAGE Model A Model B 30.0-32.9 2/30 x 100 = 6.7 7/50 x 100 = 14 33.0-35.9 4/30 x 100 = 13.3 10/50 x 100 = 20 36.0-38.9 14/30 x 100 = 46.7 16/50 x 100 = 32 39.0-41.9 8/30 x 100 = 26.7 9/50 x 100 = 18 42.0-44.9 2/30 x 100 = 6.7 8/50 x 100 = 16 From the table it is clear that whereas 6.7% of the cars of model A fall in the mileage group 42.0 to 44.9, as many as 16% of the cars of model B fall in this group. Other comparisons can similarly be made. Let us now turn to the visual representation of a continuous frequency distribution. In this context, we will discuss three different types of graphs i.e. the histogram, the frequency polygon, and the frequency curve. HISTOGRAM: A histogram consists of a set of adjacent rectangles whose bases are marked off by class boundaries along the X-axis, and whose heights are proportional to the frequencies associated with the respective classes. It will be recalled that, in the last lecture, we were considering the mileage ratings of the cars that had been inspected by the Environment Protection Agency. Our frequency table came out as shown below: Class Class Frequency Limit Boundaries 30.0 – 32.9 29.95 – 32.95 2 33.0 – 35.9 32.95 – 35.95 4 36.0 – 38.9 35.95 – 38.95 14 39.0 – 41.9 38.95 – 41.95 8 42.0 – 44.9 41.95 – 44.95 2 Total 30 In accordance with the procedure that I just mentioned, we need to take the class boundaries along the X axis We obtain Y 14 12 Number of Cars 10 8 6 4 2 0 X 29.95 32.95 35.95 38.95 41.95 44.95 Miles per gallon Virtual University of Pakistan 27 STA301 – Statistics and Probability Now, as seen in the frequency table, the frequency of the first class is 2. As such, we will draw a rectangle of height equal to 2 units and obtain the following figure: 14 Y 12 Number of Cars 10 8 6 4 2 0 X 5 5 5 5 5 5.9.9.9.9.9.9 29 32 35 38 41 44 Miles per gallon The frequency of the second class is 4. Hence we draw a rectangle of height equal to 4 units against the second class, and thus obtain the following situation: Y 14 12 Number of Cars 10 8 6 4 2 0 X 5 5 5 5 5 5.9.9.9.9.9.9 29 32 35 38 41 44 Miles per gallon The frequency of the third class is 14. Hence we draw a rectangle of height equal to 14 units against the third class, and thus obtain the following picture: Y