Bstat2 - Lessons in Business Statistics Prepared By P.K...

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Unformatted text preview: Lessons in Business Statistics Prepared By P.K. Viswanathan Chapter 2: Classifying Data to Convey Meaning Introduction When managers are bewildered by plethora of data, which do not make any sense on the surface of it, they are looking for methods to classify data that would convey meaning. The idea here is to help them draw the right conclusion. This chapter provides the nitty-gritty of arranging data into information. 1) Meaning and Example of Raw Data Meaning of Raw Data: Raw Data represent numbers and facts in the original format in which the data have been Collected. You need to convert the raw data into information for managerial decision Making. Example of Raw Data: Assume that you know the weekly sales of a product in a region over the past year are: (Figures in '000' units) 52 61 59 55 63 70 59 77 81 83 69 91 73 83 90 81 77 77 74 65 56 77 64 49 60 52 50 45 42 46 39 29 38 41 43 23 26 27 22 29 31 29 31 30 30 29 40 44 45 46 52 53 Suppose you present this set of data as it is to the General Manager (Sales). At best it will be boring to him. Information is Key Large and massive raw data tend to bewilder you so much that the overall patterns are obscured. You cannot see the wood for the trees. This implies that the raw data must be processed to give you useful information. Raw Data Process Information 2) Frequency Distribution In simple terms, frequency distribution is a summarized table in which raw data are arranged into classes and frequencies. Classes represent categories or groupings, which contain a lower limit and an upper limit. Classes are formed conveniently following certain guidelines. Against each class, you count and then place the number of observations that fall into it. When you do it for all classes in a given data analysis problem, it becomes a frequency distribution. Frequency distribution focuses on classifying raw data into information. It is the most widely used data reduction technique in descriptive statistics. When you are looking for pattern that would help you understand the characteristic you measure in a problem situation, frequency distribution comes to your rescue. Guidelines for Constructing a Frequency Distribution Table 1) Identify the Minimum Value (Min) and Maximum Value (Max) in the given Data Set. Calculate Range = Max-Min 3) Determine the Width of the Class Interval = Range/ Number of Classes 2) Decide on the Number of Classes you would like to have. The number of classes can be determined as the square root of the number of observations in the data set.. Also for any problem it is recommended that you have not less than 5 classes and not more than 15 classes. 4) Formulate the Boundaries of the Classes in such a manner that it will include all the observations in the data set. Avoid overlapping of classes. Once class boundary for each class is ready, all you need to do is to tally the number of observations in each class. 3) HISTOGRAM 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. 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. Role of Histogram in Practice Histogram- Example The inspection records of a hose assembly operation revealed a high level of rejection. An analysis of the records showed that the "leaks" were a major contributing factor to the problem. It was decided to investigate the hose clamping operation. The hose clamping force (torque) was measured on twenty five assemblies. (Figures in foot-pounds). The data are given below: Draw the frequency histogram and comment. 8 13 15 10 16 11 14 11 14 20 15 16 12 15 13 12 13 16 17 17 14 14 14 18 15 Histogram Example Solution Histogram for the Example Fr e qu e nc y You will notice that the Range is 20 -8 =12. You take the number of classes as 15 12 5(Note that the square root of the number of observations is = 5). The width of 25 10 7 the class is Range/Number of classes = 12/5 =2.4. Round it to 3. You can now 3 5 2 1 form the boundaries of the classes starting with 8 and then incrementing by 0 3 successively the lower limit of each 8-11 11-14 14-17 17-20 20-23 class until all the classes are formed. Classes Tally the number of observations under each class. This would give you the Looking at the histogram, it is easy for you to following table of frequency distribution. see that the pattern does not show a bell shape Class 8-11 11-14 14-17 17-20 20-23 Frequency 2 7 12 3 1 curve. The bars adjacent to the class 14-17 cause some distortion to normality. It is also evident that the average is in the range 14 to 17. Corrective action is needed. However, before taking any action, you must be cautious about the fact that the sample size here is only 25 observations. Take more measurements and draw the histogram again before taking corrective steps. Microsoft Excel and Histogram The Microsoft Excel Chart Wizard allows you to create a variety of charts for numerical as well as categorical data. The histogram pictured in the previous slide is an output from Chart Wizard. Also there is a powerful utility as add-in supplied by Microsoft Excel called "Data Analysis" in the Tools Menu. This has a variety of analysis tools, which include Histogram, Cumulative Distribution, Frequency Distribution, Descriptive Statistics, Pareto-Chart and many others. Please get familiarized with these in Excel at the earliest so t hat you could function as a manager taking information based decisions. The po wer of Excel spread sheet software is amazing. 4) Cumulative Frequency Distribution A type of frequency distribution that shows how many observations are above or below the lower boundaries of the classes. You can formulate the following from the previous example of hose clamping force(torque) Class Frequency Relative Frequency 8-11 11-14 14-17 17-20 20-23 2 7 12 3 1 0.08 0.28 0.48 0.12 0.04 Total 25 1.00 Cumulative Frequency 2 9 21 24 25 Cumulative Relative Frequency 0.08 0.36 0.84 0.96 1.00 Ogive Curve The Ogive curve is a graphical representation of the cumulative frequency distribution using numbers or percentages. In this pictorial representation, less than values are in the X-axis and cumulative frequency in numbers or percentages are in the Y-axis. A line graph in the form of a curve is plotted connecting the cumulative frequencies corresponding to the upper boundaries of the classes. Today, this ogive graph is elegantly and efficiently obtained as output from Chart Wizard or Data Analysis in the Toolbox of Microsoft Excel. The Ogive graph for the present torque example obtained from Microsoft Excel is given in the adjacent box: Cumulative Distribution(Ogive Curve) for the Example 30 Cumulative 20 Frequency 10 25 24 21 9 2 0 11 14 17 20 Torque(less than value) 23 ...
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