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    Assignment of MB0050

    (Research Methodology)

    (Set-1)

    Indexing:-

    Question Page No.

    1.1 Distinguish between Double sampling andmultiphase sampling. Page No. 2-31.2 What is replicated or interpenetrating

    sampling?Page No. 4

    2. What are the differences between observationand interviewing as methods of data collection?Give two specific examples of situations whereeither observation or interviewing would be more.

    Page No. 5-9

    3. How case study method is useful to BusinessResearch?

    Page No. 10-11

    4. Would case studies be considered as scientificresearch? Why or why not?

    Page No. 12

    5. What are the contents of research reports? Page No. 13-156.1 Write short notes on the following: Median Page No. 16-186.2 Write short notes on the following:StandardDeviation

    Page No. 19

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    1.1 Distinguish between double sampling and multi-phasesampling?

    Ans. Double sampling and multi-phase sampling:-

    A standard form of sample design for industrial inspection purposes. In accordance with thecharacteristics of a particular plan, two samples are drawn, n1 and n2, and the first sampleinspected. The batch can then be accepted or rejected upon the results of this inspection orthe second sample be inspected and the decision made upon the combined result.

    The term has also been used somewhat loosely for what is called multi-phase sampling andthe two-stage version of multi-stage sampling. There is a further usage whereby a first sampleprovides a preliminary estimate of design parameters which govern the size of the secondsample to achieve a desired overall result.

    MULTI-PHASE SAMPLING

    It is sometimes convenient and economical to collect certain items of information from thehole of the units of a sample and other items of usually more detailed information from a su

    sample of the units constituting the original sample. This may be termed two-phase samplinge.g. if the collection of information concerning variate, y, is relatively expensive, and thereexists some other variate, x, correlated with it, which is relatively cheap to investigate, it maybe profitable to carry out sampling in two phases.

    At the first phase, x is investigated, and the information thus obtained is used either (a) tostratify the population at the second phase, when y is investigated, or (b) as supplementaryinformation at the second phase, a ratio or regression estimate being used.

    wo-phase sampling is sometimes called "double sampling".

    Further phases may be added if desired. It may be noted, however, that multiphasesampling does not necessarily imply the use of any relationships between variates x and yThe expression is not to be confused with multi-stage sampling.

    Sample size calculations for a continuous outcome require specification of the anticipatedariance; inaccurate specification can result in an underpowered or overpowered study. Forhis reason, adaptive methods whereby sample size is recalculated using the variance of a

    subsample have become increasingly popular. The first proposal of this type (Stein, 1945,Annals of Mathematical Statistics 16, 243-258) used all of the data to estimate the mean

    difference but only the first stage data to estimate the variance. Stein's procedure is notcommonly used because many people perceive it as ignoring relevant data. This isespecially problematic when the first stage sample size is small, as would be the case if theanticipated total sample size were small. A more naive approach uses in the denominatorof the final test statistic the variance estimate based on all of the data. Applying the Helmertransformation, we show why this naive approach underestimates the true variance and

    how to construct an unbiased estimate that uses all of the data. We prove that the type I

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    error rate of our procedure cannot exceed alpha.

    Double and multiple sampling plans were invented to give a questionable lot anotherchance. For example, if in double sampling the results of the first sample are not conclusive

    ith regard to accepting or rejecting, a second sample is taken. Application of doublesampling requires that a first sample of size n1 is taken at random from the (large) lot. Thenumber of defectives is then counted and compared to the first sample's acceptancenumber a1 and rejection number r1. Denote the number of defectives in sample 1 by d1and in sample 2 by d2, then:

    If d1 a1, the lot is accepted.

    If d1 r1, the lot is rejected.

    If a1 < d1 < r1, a second sample is taken.

    If a second sample of size n2 is taken, the number of defectives, d2, is counted. The totalnumber of defectives is D2 = d1 + d2. Now this is compared to the acceptance number a2and the rejection number r2 of sample 2. In double sampling, r2 = a2 + 1 to ensure adecision on the sample.

    If D2 a2, the lot is accepted.If D2 r2, the lot is rejected.

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    1.2 What is replicated or interpenetrating sampling?

    Ans. Replicated or Interpenetrating Sampling:-

    When two or more samples are taken from the same population by the same processof selection the samples are called interpenetrating samples. The samples may ormay not be drawn independently, linked interpenetrating samples being an exampleof the latter. There may be different levels of interpenetration corresponding todifferent stages in a multi-stage sampling scheme.

    Interpenetrating Sampling: interpenetrating sampling (IPS), also known asinterpenetrating sub sampling and replicated sampling. IPS was introduced in thepioneering contribution of P.C.

    Mahalanobis. It was originally proposed in assessing the non-sampling errors as theso-called interviewer errors. IPS provides a quick, simple, and effective way ofestimating the variance of an estimator even in a complex survey. In fact, IPS is thefoundation of modern re-sampling methods like Jackknife, bootstrap, and replicationmethods. In IPS, three basic principles of experimental designs, namely,randomization, replication, and local control, are used. IPS is used extensively notonly in agriculture, but also in social sciences, demography, epidemiology, publichealth, and many other fields.

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    2. What are the differences between observation andinterviewing as methods of data collection? Give two specificexamples of situations where either observation or

    interviewing would be more.

    Ans. Differences between observation and interviewing:-

    The most complete form of the sociological datum, after all, is the form in which the

    participant observer gathers it: An observation of some social event, the events which

    precede and follow it, and explanations of its meaning by participants and spectators,

    before, during, and after its occurrence. Such a datum gives us more information

    about the event under study than data gathered by any other sociological method.

    Participant observation can thus provide us with a yardstick against which to measure

    the completeness of data gathered in other ways, a model which can serve to let usknow what orders of information escape us when we use other methods.

    By participant observation we mean that method in which the observer participates in

    the daily life of the people under study, either openly in the role of researcher or

    covertly in some disguised role, observing things that happen, listening to what is said

    and questioning person, over some length of time. We want, in this paper, to compare

    the results of intensive field work with what might be regarded as the first step in the

    other direction along this continuum: the detailed and conversational interview (often

    referred to as the unstructured or undirected interview).3 In this kind of interview, the

    interviewer explores many facets of his interviewee's concerns, treating subjects as

    they come up in conversation, pursuing interesting leads, allowing his imagination and

    ingenuity full rein as he tries to develop new hypotheses and test them in the course

    of the interview.

    In the course of our current participant observation among medical students? we have

    thought a good deal about the kinds of things we were discovering which might

    ordinarily be missed or misunderstood in such an interview. We have no intention of

    denigrating the interview or even such less precise modes of data gathering as the

    questionnaire, for there can always be good reasons of practicality, economy, or

    research design for their use. We simply wish to make explicit the difference in the

    data gathered by one or the other method and to suggest the differing uses to which

    they can legitimately be put.

    In general, the shortcomings we attribute to the interview exist when it is used as a

    source of information about events that have occurred elsewhere and are described

    to us by informants. Our criticisms are not relevant when analysis is restricted to

    interpretation of the interviewee's conduct during the interview, in which case the

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    researcher has in fact observed the behavior he is talking about.? The differences weconsider between the two methods involve two interacting factors: the kinds of words

    and acts of the people under study that the researcher has access to, and the kind of

    sensitivity to problems and data produced in him.

    Our comparison may prove useful by suggestive areas in which interviewing (the

    more widely used method at present and likely to continue so) can improve its

    accuracy by taking account of suggestions made from the perspective of the

    participant observer. We begin by considering some concrete problems: learning the

    native language, or the problem of the degree to which the interviewer really

    understands what is said to him; matters interviewees are unable or unwilling to talk

    about; and getting information on matters people see through distorting lenses. We

    then consider some more general differences between the two methods.

    Observation as a method of data collection has certain characteristics.

    1. It is both a physical and a mental activity: The observing eye catches many thingsthat are present. But attention is focused on data that are pertinent to the given study.

    2. Observation is selective: A researcher does not observe anything and everything,but selects the range of things to be observed on the basis of the nature, scope andobjectives of his study. For example, suppose a researcher desires to study thecauses of city road accidents and also formulated a tentative hypothesis thataccidents are caused by violation of traffic rules and over speeding. When heobserved the movements of the vehicles, the persons sitting in them, their hair style,

    etc. all such things which are not relevant to his study are ignored and only overspeeding and traffic violation are keenly observed by him.

    3. Observation is purposive and not casual: it is made for the specific purpose ofnoting things relevant to the study. It captures the natural social context in whichpersons behavior occur. It grasps the significant events and occurrences that affectsocial relations of the participants.

    4. Observation should be exact and be based on standardized tools of research andsuch as observation schedule, social metric scale etc., and precision instruments, ifany.

    Observation has following advantages:

    (1) The main virtue of observation is its directness: It makes it possible to studybehavior as it occurs. The researcher need not ask people about theirbehavior and interactions he can simply watch what they do and say.

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    (2) Data collected by observation may describe the observed phenomena as they

    occur in their natural settings. Other methods introduce elements or artificiality into

    the researchers situation for instance, in interviews; the respondent may not behave

    in a natural way. There is no such artificiality into the researched situation for

    instance, in interview; the respondent may not behave in a natural way. There is notsuch artificiality in observational studies, especially when the observed persons are

    not aware of their being observed

    (3) Observations is more suitable for studying subjects who are unable to articulate

    meaningfully, e.g. studies of children, tribal, animals, birds etc.

    (4) Observations improve the opportunities for analyzing the contextual back ground

    of behavior. Furthermore verbal resorts can be validated and compared with behavior

    through observation. The validity of what men of position and authority say can be

    verified by observing what they actually do.

    Observation is less demanding of the subjects and has less biasing effect on their

    conduct than questioning.

    Interview method

    Interviewing is one of the prominent methods of data collection. It may be defined as

    a two way systematic conversation between an investigator and an informant, initiated

    for obtaining information relevant to a specific study. It involves not only conversation,

    but also learning from the respondents gesture, facial expressions and pauses, and

    his environment. Interviewing requires face to face contact or contact over telephone

    and calls for interviewing skills. It is done by using a structured schedule or an

    unstructured guide.

    Interviewing may be used either as a main method or as a supplementary one in

    studies of persons. Interviewing is the only suitable method for gathering information

    from illiterate or less educated respondents. It is useful for collecting a wide range of

    data from factual demographic data to highly personal and intimate information

    relating to a persons opinions, attitudes, values, beliefs past experience and future

    intentions. When qualitative information is required or probing is necessary to draw

    out fully, and then interviewing is required. Where the area covered for the survey is a

    compact, or when a sufficient number of qualified interviewers are available, personal

    interview is feasible.

    Interview is often superior to other data-gathering methods. People are usually more

    willing to talk than to write. Once report is established, even confidential information

    may be obtained. It permits probing into the context and reasons for answers to

    questions.

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    Interview can add flesh to statistical information. It enables the investigator to grasp

    the behavioral context of the data furnished by the respondents.

    There are several advantages to personal interviewing.

    First the greatest value of this method is the depth and detail of information that can

    be secured. When used with well conceived schedules, an interview can obtain a

    great deal of information. It far exceeds mail survey in amount and quality of data that

    can be secured. Second, the interviewer can do more to improve the percentage of

    responses and the quality of information received than other method. He can note the

    conditions of the interview situation, and adopt appropriate approaches to overcome

    such problems as the respondents unwillingness, incorrect understanding of

    questions, suspicion, etc.

    Third, the interviewer can gather other supplemental information like economic level,

    living conditions etc. through observation of the respondents environment.

    Fourth, the interviewer can use special scoring devices, visual materials and the like

    in order to improve the quality of interviewing.

    Fifth, the accuracy and dependability of the answers given by the respondent can be

    checked by observation and probing.

    Last, interview is flexible and adaptable to individual situations. Even more, control

    can be exercised over the interview situation.

    Demerits of interview method

    Interviewing is not free limitations. Its greatest drawback is that it is costly both inmoney and time. Second, the interview results are often adversely affected byinterviewers mode of asking questions and interactions, and incorrect recording andalso by the respondents faulty perception, faulty memory, inability to articulate etc.

    Third, certain types of personal and financial information may be refused in face-toface interviews. Such information might be supplied more willingly on mailquestionnaires, especially if they are to be unsigned.

    Fourth, interview poses the problem of recording information obtained from therespondents. No full proof system is available. Note taking in invariably distracting toboth the respondent and the interviewer and affects the thread of the conversation.

    Last interview calls for highly interviewers. The availability of such persons is limitedand the training of interviewers is often a long and costly process.

    Situation where observation is appropriate:

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    Observations make it possible to capture the whole event as it occurs. For exampleonly observation can provide an insight into all the aspects of the process ofnegotiation between union and management representatives.

    Situation where interview method is appropriate: to study the Reading habits ofnewspaper/magazines readers.

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    3. How case study method is useful to Business Research?

    Ans. Case Study Method:-

    While case study writing may seem easy at first glance, developing an effective casestudy (also called a success story) is an art. Like other marketing communicationskills, learning how to write a case study takes time. Whats more, writing casestudies without careful planning usually results in sub optimal results?

    Savvy case study writers increase their chances of success by following these tenproven techniques for writing an effective case study:

    Involve the customer throughout the process. Involving the customer throughoutthe case study development process helps ensure customer cooperation andapproval, and results in an improved case study. Obtain customer permission beforewriting the document, solicit input during the development, and secure approval afterdrafting the document.

    Write all customer quotes for their review. Rather than asking the customerto draft their quotes, writing them for their review usually results in morecompelling material.

    Case Study Writing Ideas

    Establish a document template.A template serves as a roadmap for thecase study process, and ensures that the document looks, feels, and reads

    consistently. Visually, the template helps build the brand; procedurally, itsimplifies the actual writing. Before beginning work, define 3-5 specificelements to include in every case study, formalize those elements, and stick tothem.

    Start with a bang. Use action verbs and emphasize benefits in the case studytitle and subtitle. Include a short (less than 20-word) customer quote in largertext. Then, summarize the key points of the case study in 2-3 succinct bulletpoints. The goal should be to tease the reader into wanting to read more.

    Organize according to problem, solution, and benefits. Regardless oflength, the time-tested, most effective organization for a case study follows the

    problem-solution-benefits flow. First, describe the business and/or technicalproblem or issue; next, describe the solution to this problem or resolution ofthis issue; finally, describe how the customer benefited from the particularsolution (more on this below). This natural story-telling sequence resonateswith readers.

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    Use the general-to-specific-to-general approach. In the problem section,begin with a general discussion of the issue that faces the relevant industry.Then, describe the specific problem or issue that the customer faced. In thesolution section, use the opposite sequence. First, describe how the solution

    solved this specific problem; then indicate how it can also help resolve thisissue more broadly within the industry. Beginning more generally draws thereader into the story; offering a specific example demonstrates, in a concreteway, how the solution resolves a commonly faced issue; and concluding moregenerally allows the reader to understand how the solution can also addresstheir problem.

    Quantify benefits when possible. No single element in a case study is morecompelling than the ability to tie quantitative benefits to the solution. Forexample, Using Solution X saved Customer Y over $ZZZ, ZZZ after just 6months of implementation; or, Thanks to Solution X, employees at CustomerY have realized a ZZ% increase in productivity as measured by standard

    performance indicators. Quantifying benefits can be challenging, but notimpossible. The key is to present imaginative ideas to the customer for waysto quantify the benefits, and remain flexible during this discussion. If benefitscannot be quantified, attempt to develop a range ofqualitativebenefits; thelatter can be quite compelling to readers as well.

    Use photos.Ask the customer if they can provide shots of personnel, ideallyusing the solution. The shots need not be professionally done; in fact, home-grown digital photos sometimes lead to surprisingly good results and oftenappear more genuine. Photos further personalize the story and help form aconnection to readers.

    Reward the customer.After receiving final customer approval and finalizingthe case study, provide a pdf, as well as printed copies, to the customer.Another idea is to frame a copy of the completed case study and present it tothe customer in appreciation for their efforts and cooperation.

    Writing a case study is not easy. Even with the best plan, a case study is doomedto failure if the writer lacks the exceptional writing skills, technical savvy, andmarketing experience that these documents require. In many cases, a talentedwriter can mean the difference between an ineffective case study and one thatprovides the greatest benefit. If a qualified internal writer is unavailable, consideroutsourcing the task to professionals who specialize in case study writing.

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    4. Would case studies be considered as scientific research? Whyor why not?

    Ans. Case Studies:-Case studies are a tool for discussing scientific integrity. Although one of the mostfrequently used tools for encouraging discussion, cases are only one of manypossible tools. Many of the principles discussed below for discussing case studiescan be generalized to other approaches to encouraging discussion about researchethics. Cases are designed to confront readers with specific real-life problems that donot lend themselves to easy answers. Case discussion demands critical andanalytical skills and, when implemented in small groups, also fosters collaboration(Pimple, 2002). By providing a focus for discussion, cases help trainees to define orrefine their own standards, to appreciate alternative approaches to identifying andresolving ethical problems, and to develop skills for analyzing and dealing with hard

    problems on their own.

    The effective use of case studies is comprised of many factors, including:

    Appropriate selection of case(s) (topic, relevance, length, complexity)

    Method of case presentation (verbal, printed, before or during discussion)

    Format for case discussion (Email or Internet-based, small group, large group)

    Leadership of case discussion (choice of discussion leader, roles andresponsibilities for discussion leader)

    Outcomes for case discussion (answers to specific questions, answers togeneral questions, written or verbal summaries) Research methods don'tseem so intimidating when you're familiar with the terminology. This is

    important whether you're conducting evaluation or merely reading articlesabout other studies to incorporate in your program. To help withunderstanding, here are some basic definitions used.

    Variable:Characteristics by which people or things can be described. Must have more thanone level; in other words, to be able to change over time for the sameperson/object, or from person to person, or object to object. Some variables,called attributes, cannot be manipulated by the researcher (e.g., socioeconomicstatus, IQ)

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    5. What are the contents of research reports?

    Ans. Contents of Research Reports:-

    The information in the first part of this section, Journal Articles, Books, Chaptersin Books, Peer Reviewed Published Conference Proceedings and Non-peerReviewed Published Conference Proceedings is filtered and extracted to formulatethe Preliminary Publication Count for the Department of Education. This is a crucialstep in the subsidy calculation process and it is of the utmost importance that thisinformation must be as comprehensive accurate as possible. All researchpublications, which appeared in 2004, must be included, even if they are "in press".There will be an opportunity early in February to submit volume and page numbers ofpapers, which have appeared in the interim. Publications should be listed using theHarvard referencing system, without using italics or bold. Journal names must appearin full. Abbreviations will be returned to departments for expansion. (Theses passed

    for higher degrees are subsidy earning but this information is not included in thePreliminary Publication Count.)

    The second part of this section, Published Conference Abstracts, Patents,Theses & Dissertations passed for Higher Degrees, University Publications &Publications of a Popular Nature - Unrefereed Journal Articles, NewspaperArticles, CD's, Online Works, Extension and Development Work, Creative Work,Consultancy and Other Activities based on Expertise Developed in Research(gives departments the opportunity to demonstrate the extent and nature of theirextension and development work), Motivation for the addition of a South AfricanJournal to the list of South African approved journals. The work included here isnot subsidy earning but nevertheless is an important aspect of scholarly activity atUCT.

    JOURNAL ARTICLESThis category covers all articles in journals. Articles fall into 2 categories - those forwhich the university earns subsidy and are published in 'approved' journals; and thosefor which the university does not get a subsidy. Please ensure that journal namesappear in full and points below are clearly stated when submitting.1. Surnames & initials of authors 2. Year 3. Title of article 4. Journal name 5. Volumenumber 6. Issue number 7. Page numbers of article

    BOOKSThis section must include ALL books, scholarly peer-reviewed, textbooks, novels,coffee-table books, etc produced by UCT researchers. It should also include editedConference Proceedings produced in a book format where a UCT staff member orresearcher is the editor or part of the editorial panel. Details of year of publication,editor/s, publisher, place, total number of pages of books must be provided.

    1. Surnames & initials of authors 2. Year 3. Title of Chapter 4. Target group/audienceof book (only for subsidy-earning books) 5. If 2nd/3rd etc. edition, provide page

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    numbers of new sections (only for subsidy-earning books) 6. Evidence of peer review(statement frompublisher on peer review policy) (only for subsidy- earning books) 7.ISBN number (only for subsidy-earning books) 8. Page numbers of book 9. Place ofpublication and publisher.

    CHAPTERS IN BOOKSThis section should include bona fidechapters in books only and not conferencepapers published in Proceedings. Details of year of publication. Editor/s, publisher,place, first to last page numbers of the chapter must be provided.

    1. Surnames & initials of authors 2. Year 3. Title of Chapter 4. Target group/audienceof book (only for subsidy-earning books) 5. If 2nd/3rd etc. edition, provide pagenumbers of new sections (only for subsidy-earning books) 6. Evidence of peer review(statement from publisher on peer review policy) (only for subsidy- earning books) 7.ISBN number (only for subsidy-earning books) 8. Page numbers of chapter 9. Placeof publication and publisher.

    PEER REVIEWED PUBLISHED CONFERENCE PROCEEDINGSThis should include all papers published in Conference Proceedings, which aresubstantial refereed publications, which will be the final and only publications of thatmaterial. This does not include published abstracts. Peer-reviewed publishedconference proceedings count for subsidy purposes and there must be evidence ofthe peer-review procedure. Please note that unpublished conference proceedings arenot included.

    1. Surnames & initials of authors, Title of conference, Page numbers, Year, Evidenceof peer review, ISBN number, Title of paper, Place of conference

    NON-PEER REVIEWED PUBLISHED CONFERENCE PROCEEDINGSThis section includes published proceedings that have not been submitted to a strictpeer-review process, and as such, are not subsidy-earning but are included in theannual research report.

    PUBLISHED CONFERENCE ABSTRACTSThis section should include abstracts of papers published in the formal scientificliterature only. However, authoritative works such as plenary, honorific and invitedkeynote addresses/papers given at conferences may also be included.

    PATENTSThese must be final registered patents, and not provisional patents. They do count forsubsidy-earning purposes.

    THESES AND DISSERTATIONS PASSED FOR HIGHER DEGREESThis section should include all master's dissertations and doctoral theses passed in2004. Please note the technical distinction between them. A doctoral thesis isrequired to be a defense of original research, whereas a master's dissertation couldbe original research or a review of others' work. Please provide full-page numbers.

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    UNIVERSITY PUBLICATIONS AND WORKS OF A POPULAR NATUREThis section should include all non-peer-reviewed publications, UCT publications,educational CD ROMS, etc that have a scholarly nature and are associated with theresearch enterprise. Newspaper articles are included here.

    NEWSPAPER ARTICLESCD'sONLINE WORKS

    EXTENSION AND DEVELOPMENT WORK (UNPUBLISHED)This section gives researchers the opportunity to report briefly on their unpublishedextension and development work. It should include only completed work, and notreports or drafts intended for publication as formal papers at a later date. Each itemshould include some indication of its recipient or ultimate location. What should beincluded must be documents; CD-ROMs, etc. should be listed.

    CREATIVE WORKS, PERFORMANCES, EXHIBITIONSThis section includes original musical, dramatic and other artistic works, or newproductions of existing works. Each item can include a very brief note about itssignificance. The "creation" of course notes, academic programme plans, technicaldevices etc. does not qualify.

    CONSULTANCY AND OTHER ACTIVITIES BASED ON EXPERTISE DEVELOPEDIN RESEARCHThis section should include short descriptions of the nature of the activity and theoutput. Please keep this brief. Please refer to the 2001, 2002 or 2003 ResearchReports as a guide.

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    6.1 Write short notes on the following: Median

    Ans. Median:

    In statistics and probability theory, median is described as the numerical valueseparating the higher half of a sample, a population, or a probability distribution, fromthe lower half. The medianof a finite list of numbers can be found by arranging all theobservations from lowest value to highest value and picking the middle one. If there isan even number of observations, then there is no single middle value; the median isthen usually defined to be the mean of the two middle values.[1][2]

    A median is only defined on one-dimensional data, and is independent of anydistance metric. A geometric median, on the other hand, is defined in any number ofdimensions.In a sample of data, or a finite population, there may be no member of the samplewhose value is identical to the median (in the case of an even sample size), and, if

    there is such a member, there may be more than one so that the median may notuniquely identify a sample member. Nonetheless, the value of the median is uniquelydetermined with the usual definition. A related concept, in which the outcome is forcedto correspond to a member of the sample, is the medoid.

    At most, half the population has values strictly less than the median, and, at most, halfhave values strictly greater than the median. If each group contains less than half thepopulation, then some of the population is exactly equal to the median. For example,ifa< b< c, then the median of the list {a, b, c} is b. Ifa b cas well, then only ais strictly less than the median, and only cis strictly greater than the median. Sinceeach group is less than half (one-third, in fact), the leftoverbis strictly equal to themedian (a truism).

    Likewise, ifa< b< c< d, then the median of the list {a, b, c, d} is the mean ofband c;i.e., it is (b+ c)/2.

    The median can be used as a measure of location when a distribution is skewed,when end-values are not known, or when one requires reduced importance to beattached to outliers, e.g., because they may be measurement errors.

    In terms of notation, some authors represent the median of a variable xeither as or as[1] There is no simple, widely accepted standard notation for the median, so the useof these or other symbols for the median needs to be explicitly defined when they areintroduced.

    For any probability distribution on the real line R with cumulative distribution functionF, regardless of whether it is any kind of continuous probability distribution, inparticular an absolutely continuous distribution (and therefore has a probability), or adiscrete probability distribution, a median is by definition any real numbermthatsatisfies the inequalities

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    or, equivalently, the inequalities

    in which a LebesgueStieltjes integral is used. For an absolutely continuousprobability distribution with probability density function , the median satisfies

    Any probability distribution on R has at least one median, but there may be more than

    one median. Where exactly one median exists, statisticians speak of "the median"correctly; even when the median is not unique, some statisticians speak of "themedian" informally.

    Ifnis odd then Median (M) = value of ((n+ 1)/2)th item term.

    Ifnis even then Median (M) = value of [((n)/2)th item term + ((n)/2 + 1)th item term]/2

    For an odd number of values

    As an example, we will calculate the sample median for the following set ofobservations: 1, 5, 2, 8, 7.

    Start by sorting the values: 1, 2, 5, 7, 8.

    In this case, the median is 5 since it is the middle observation in the ordered list.

    The median is the ((n+ 1)/2)th item, where nis the number of values. For example,for the list {1, 2, 5, 7, 8}, we have n= 5, so the median is the ((5 + 1)/2)th item.

    Median = (6/2)th item

    Median = 3rd item

    Median = 5

    For an even number of valuesAs an example, we will calculate the sample median for the following set ofobservations: 1, 6, 2, 8, 7, and 2.

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    Start by sorting the values: 1, 2, 2, 6, 7, and 8.

    In this case, the arithmetic mean of the two middlemost terms is (2 + 6)/2 = 4.Therefore, the median is 4 since it is the arithmetic mean of the middle observationsin the ordered list.

    We also use this formula MEDIAN = {(n+ 1 )/2}th item . n= number of values

    As above example 1, 2, 2, 6, 7, 8 n= 6 Median = {(6 + 1)/2}th item = 3.5th item. In thiscase, the median is average of the 3rd number and the next one (the fourth number).The median is (2 + 6)/2 which is 4.

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    6.2 Write short notes on the following: Standard Deviation

    Ans. Standard Deviation:

    In statistics and probability theory, standard deviation (represented by the symbolsigma, ) shows how much variation or "dispersion" exists from the average (mean,or expected value). A low standard deviation indicates that the data points tend to bevery close to the mean, whereas high standard deviation indicates that the data pointsare spread out over a large range of values.

    The standard deviation of a random variable, statistical population, data set, orprobability distribution is the square root of its variance. It is algebraically simplerthough practically less robust than the average absolute deviation.[1][2] A usefulproperty of standard deviation is that, unlike variance, it is expressed in the sameunits as the data.

    In addition to expressing the variability of a population, standard deviation iscommonly used to measure confidence in statistical conclusions. For example, themargin of error in polling data is determined by calculating the expected standarddeviation in the results if the same poll were to be conducted multiple times. Thereported margin of error is typically about twice the standard deviation the radius ofa 95 percent confidence interval. In science, researchers commonly report thestandard deviation of experimental data, and only effects that fall far outside the rangeof standard deviation are considered statistically significant normal random error orvariation in the measurements is in this way distinguished from causal variation.Standard deviation is also important in finance, where the standard deviation on therate of return on an investment is a measure of the volatility of the investment.

    An estimator forsometimes used is the standard deviation of the sample,denoted by sNand defined as follows:

    In the case where Xtakes random values from a finite data set x1, x2, , xN, witheach value having the same probability, the standard deviation is