Important Terminologies In Statistical Inference I I

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Important Terminologies In Statistical Inference I I

  1. 1. Lessons in Business Statistics “Quantitative Analysis”
  2. 2. Chapter 2: Classifying Data to Convey Meaning
  3. 3. Introduction When managers are bewildered by plethora ofWhen managers are bewildered by plethora of data, which do not make any sense on the surfacedata, which do not make any sense on the surface of it, they are looking for methods to classify dataof it, they are looking for methods to classify data that would convey meaning. The idea here is tothat would convey meaning. The idea here is to help them draw the right conclusion. This chapterhelp them draw the right conclusion. This chapter provides theprovides the nittynitty--gritty of arranging data intogritty of arranging data into informationinformation..
  4. 4. 1) Meaning and Example of Raw Data Meaning of RawMeaning of Raw Data:Data: Raw Data represent numbersRaw Data represent numbers and facts in the originaland facts in the original format in which the data haveformat in which the data have been Collected. You need tobeen Collected. You need to convert the raw data intoconvert the raw data into information for managerialinformation for managerial decision Making.decision Making. Example of Raw Data:Example of Raw Data: Assume that you know the weeklyAssume that you know the weekly sales of a product in a region over thesales of a product in a region over the past year are: (Figures in '000' units)past year are: (Figures in '000' units) 52 61 59 55 63 70 59 77 8152 61 59 55 63 70 59 77 81 83 69 91 73 83 90 81 77 7783 69 91 73 83 90 81 77 77 74 65 56 77 64 49 60 52 5074 65 56 77 64 49 60 52 50 45 42 46 39 29 38 41 43 2345 42 46 39 29 38 41 43 23 26 27 22 29 31 29 31 30 3026 27 22 29 31 29 31 30 30 29 40 44 45 46 52 5329 40 44 45 46 52 53 Suppose you present this set of dataSuppose you present this set of data as it is to the General Manageras it is to the General Manager (Sales). At best it will be boring to(Sales). At best it will be boring to him.him.
  5. 5. Information is Key Large and massive raw data tend to bewilder you so muchLarge and massive raw data tend to bewilder you so much that the overall patterns are obscured. You cannot see thethat the overall patterns are obscured. You cannot see the wood for the trees. This implies that the raw data must bewood for the trees. This implies that the raw data must be processed to give you useful information.processed to give you useful information. Raw Data Information Process
  6. 6. 2) Frequency Distribution In simple terms,In simple terms, frequency distributionfrequency distribution is a summarizedis a summarized table in which raw data are arranged into classes andtable in which raw data are arranged into classes and frequencies. Classes represent categories or groupings, whichfrequencies. Classes represent categories or groupings, which contain a lower limit and an upper limit. Classes are formedcontain a lower limit and an upper limit. Classes are formed conveniently following certain guidelines. Against each class,conveniently following certain guidelines. Against each class, you count and then place the number of observations that fallyou count and then place the number of observations that fall into it. When you do it for all classes in a given data analysiinto it. When you do it for all classes in a given data analysiss problem, it becomes a frequency distribution.problem, it becomes a frequency distribution. Frequency distribution focuses on classifying raw data intoFrequency distribution focuses on classifying raw data into information. It is the most widely used data reductioninformation. It is the most widely used data reduction technique in descriptive statistics. When you are looking fortechnique in descriptive statistics. When you are looking for pattern that would help you understand the characteristic youpattern that would help you understand the characteristic you measure in a problem situation, frequency distribution comesmeasure in a problem situation, frequency distribution comes to your rescue.to your rescue.
  7. 7. Guidelines for Constructing a Frequency Distribution Table 1)1) Identify the Minimum ValueIdentify the Minimum Value (Min) and Maximum Value (Max)(Min) and Maximum Value (Max) in the given Data Set. Calculatein the given Data Set. Calculate RangeRange = Max= Max--MinMin 2)2) Decide on theDecide on the Number of ClassesNumber of Classes you would like to have. Theyou would like to have. The number of classes can benumber of classes can be determined as the square root ofdetermined as the square root of the number of observations in thethe number of observations in the data set.. Also for any problem itdata set.. Also for any problem it is recommended that you have notis recommended that you have not less than 5 classes and not moreless than 5 classes and not more than 15 classes.than 15 classes. 3)3) Determine theDetermine the WidthWidth of theof the Class Interval =Class Interval = Range/ Number of ClassesRange/ Number of Classes 4)4) Formulate the Boundaries ofFormulate the Boundaries of the Classesthe Classes in such a mannerin such a manner that it will include all thethat it will include all the observations in the data set.observations in the data set. Avoid overlapping of classes.Avoid overlapping of classes. Once class boundary for eachOnce class boundary for each class is ready, all you need toclass is ready, all you need to do is to tally the number ofdo is to tally the number of observations in each class.observations in each class.
  8. 8. Histogram (also known as frequency histogram) is a snap shot photograph of the frequency distribution. Histogram is a graphical representation of the frequency distribution in which the X-axis represents the classes and the Y-axis represents the frequencies. Rectangular bars are constructed at the boundaries of each class with heights proportional to the frequency. 3) HISTOGRAM Histogram depicts the pattern of the distribution emerging from the characteristic being measured. If the pattern is symmetrical and bell shaped, then it reflects the normal distribution curve. In the quality control parlance, the system is stable; only chance causes are present and the assignable causes are absent.
  9. 9. Role of Histogram in Practice
  10. 10. Histogram- Example The inspection records of a hose assembly operation revealed a hThe inspection records of a hose assembly operation revealed a high leveligh level of rejection. An analysis of the records showed that the "leaks"of rejection. An analysis of the records showed that the "leaks" were awere a major contributing factor to the problem. It was decided to invemajor contributing factor to the problem. It was decided to investigate thestigate the hose clamping operation. The hose clamping force (torque) was mehose clamping operation. The hose clamping force (torque) was measuredasured on twenty five assemblies. (Figures in footon twenty five assemblies. (Figures in foot--pounds). The data are givenpounds). The data are given below: Draw the frequency histogram and comment.below: Draw the frequency histogram and comment. 88 1313 1515 1010 1616 1111 1414 1111 1414 2020 1515 1616 1212 1515 1313 1212 1313 1616 1717 1717 1414 1414 1414 1818 1515
  11. 11. Histogram Example Solution You will notice that theYou will notice that the RangeRange is 20is 20--88 =12.=12. You take theYou take the number of classesnumber of classes asas 5(Note that the square root of the number5(Note that the square root of the number of observations isof observations is 25 = 5). The25 = 5). The widthwidth ofof the class is Range/Number of classes =the class is Range/Number of classes = 12/5 =2.4. Round it to 3. You can now12/5 =2.4. Round it to 3. You can now form the boundaries of the classesform the boundaries of the classes starting with 8 and then incrementing bystarting with 8 and then incrementing by 3 successively the lower limit of each3 successively the lower limit of each class until all the classes are formed.class until all the classes are formed. Tally the number of observations underTally the number of observations under each class. This would give you theeach class. This would give you the following table of frequency distribution.following table of frequency distribution. ClassClass FrequencyFrequency 88--1111 22 1111--1414 77 1414--1717 1212 1717--2020 33 2020--2323 11 Looking at the histogram, it is easy for you toLooking at the histogram, it is easy for you to see that the pattern does not show a bell shapesee that the pattern does not show a bell shape curve. The bars adjacent to the class 14curve. The bars adjacent to the class 14--1717 cause some distortion to normality. It is alsocause some distortion to normality. It is also evident that the average is in the range 14 to 17.evident that the average is in the range 14 to 17. Corrective action is needed. However, beforeCorrective action is needed. However, before taking any action, you must be cautious abouttaking any action, you must be cautious about the fact that the sample size here is only 25the fact that the sample size here is only 25 observations. Take more measurements andobservations. Take more measurements and draw the histogram again before takingdraw the histogram again before taking corrective steps.corrective steps. Histogram for the Example 2 7 12 3 1 0 5 10 15 8-11 11-14 14-17 17-20 20-23 Classes Frequency
  12. 12. Microsoft Excel and Histogram  The Microsoft Excel Chart Wizard allows you to create a varietyThe Microsoft Excel Chart Wizard allows you to create a variety of chartsof charts for numerical as well as categorical data. The histogram picturefor numerical as well as categorical data. The histogram pictured in thed in the previous slide is an output from Chart Wizard.previous slide is an output from Chart Wizard.  Also there is a powerful utility as addAlso there is a powerful utility as add--in supplied by Microsoft Excelin supplied by Microsoft Excel called "Data Analysis" in the Tools Menu. This has a variety ofcalled "Data Analysis" in the Tools Menu. This has a variety of analysisanalysis tools, which includetools, which include HistogramHistogram,, Cumulative DistributionCumulative Distribution,, FrequencyFrequency Distribution,Distribution, Descriptive Statistics,Descriptive Statistics, ParetoPareto--ChartChart and many others.and many others. Please get familiarized with these in Excel at the earliest so tPlease get familiarized with these in Excel at the earliest so that you couldhat you could function as a manager taking information based decisions. The pofunction as a manager taking information based decisions. The power ofwer of Excel spread sheet software is amazing.Excel spread sheet software is amazing.
  13. 13. 4) Cumulative Frequency Distribution A type of frequency distribution that shows how manyA type of frequency distribution that shows how many observations are above or below the lower boundariesobservations are above or below the lower boundaries of the classes. You can formulate the following fromof the classes. You can formulate the following from the previous example of hose clamping force(torque)the previous example of hose clamping force(torque) 0.080.08 0.360.36 0.840.84 0.960.96 1.001.00 22 99 2121 2424 2525 0.080.08 0.280.28 0.480.48 0.120.12 0.040.04 22 77 1212 33 11 88--1111 1111--1414 1414--1717 1717--2020 2020--2323 1.001.002525TotalTotal CumulativeCumulative RelativeRelative FrequencyFrequency CumulativeCumulative FrequencyFrequency RelativeRelative FrequencyFrequency FrequencyFrequencyClassClass
  14. 14. Ogive Curve TheThe OgiveOgive curve is a graphicalcurve is a graphical representation of the cumulative frequencyrepresentation of the cumulative frequency distribution using numbers or percentages.distribution using numbers or percentages. In this pictorial representation, less thanIn this pictorial representation, less than values are in the Xvalues are in the X--axis and cumulativeaxis and cumulative frequency in numbers or percentages arefrequency in numbers or percentages are in the Yin the Y--axis. A line graph in the form of aaxis. A line graph in the form of a curve is plotted connecting the cumulativecurve is plotted connecting the cumulative frequencies corresponding to the upperfrequencies corresponding to the upper boundaries of the classes. Today, thisboundaries of the classes. Today, this ogiveogive graph is elegantly and efficientlygraph is elegantly and efficiently obtained as output from Chart Wizard orobtained as output from Chart Wizard or Data Analysis in the Toolbox of MicrosoftData Analysis in the Toolbox of Microsoft Excel. TheExcel. The OgiveOgive graph for the presentgraph for the present torque example obtained from Microsofttorque example obtained from Microsoft Excel is given in the adjacent box:Excel is given in the adjacent box: CumulativeDistribution(OgiveCurve)for theExample 2 9 21 24 25 0 10 20 30 11 14 17 20 23 Torque(lessthanvalue) Cumulative Frequency

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