Bài giảng Statistical Techniques in Business and Economics - Chapter 2 Frequency Distributions Describing Data & Graphic Presentations

Describing Datarequency DistributionsfGraphic Presentations&Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 2Chapter GoalsOrganize raw data into frequency distribution Produce a histogram, a frequency polygon, and a cumulative frequency polygon from  quantitative data Develop and interpret a stem-and-leaf displayWhen you have completed this chapter, you will be able to:and... Present qualitative data using such graphical  techniques such as a clustered bar chart, a stacked bar chart, and a pie chartDetect graphic deceptions and use a graph to present data with clarity, precision, and efficiency Chapter GoalsFrequencyA Frequency Distribution is a grouping of data into non-overlapping classes (mutually exclusive)showing the number of observations in each category or class.The range of categories includes all values in the data set (collectively exhaustive classes).FrequencyFrequencyDefinitionsClass Midpoint or Class Mark:A point that divides a class into two equal parts, i.e. the average of the upper and lower class limits.12.5Class frequency:The number of observations in each class. Class interval:The class interval is obtained by subtracting the lower limit of a class from the lower limit of the next class, e.g.517.522.527.532.5EXAMPLE Dr. Tillman is Dean of the School of Business. He wishes to prepare a report showing the number of hours per week students spend studying. He selects a random sample of 30 students and determines the number of hours each student studied last week. 15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6. Required:Organize the data into a frequency distribution.QuestionDecide how many classes you wish to use.Frequency Distributions by handDetermine the class width.There are five steps that can be used to Construct a Frequency Distribution:1.3.2.4.5.Set up the individual class limits.Tally the items into the classes.Count the number of items in each class.Decide how many classes you wish to use1.Use the 2 to the K rule.Choose k so that 2 raised to the power of k is greater than the number of data points (n) or 30.Rule of Thumb:For most data sets, you would want between 3 and 12 classes!2k = 30 students25 = 32, so use k = about 5 classesIn this caseDetermine the class width2.Generally, the class width should be the same size for all classes.Class width >= Max - Min K(33.8 – 10.3)/ 5 = 4.7Therefore, use class size of 5 hours15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6. 10.3,33.8,MaxMinK=5Minimum Value is 10.3, therefore, classes should start at 10 hours10.0 – 14.915.0 – 19.920.0 – 24.925.0 – 29.930.0 – 34.9Lower class limits will be: 10, 15, 20, etc.Classesor10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35Classes3. Set up the individual class limits15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6. 10.3,33.8,Class Width 5 hours15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6. 4.Tally the items into the classes10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35ClassesTallyand so on with the remaining hours10.3,13.514.213.714.012.912.9FindCount the number of items in each class5.10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35Hours Studying xFrequency f71273130Using different limits 7.5 to under 12.512.5 to under 17.517.5 to under 22.522.5 to under 27.527.5 to under 32.532.5 to under 37.5Hours Studying xFrequency f1121011305will give you a different distribution, e.g.Construct a Frequency DistributionUsing ExcelClick on MegaStatSeeClick on Frequency DistributionsSeeUsingClick on QuantitativeUsingSee INPUT NEEDS$A:$A510SeeSeeUsingRelative Frequency DistributionRelative Frequency Distributionshows the percent of observations in each class!Hours Studying xf712731Relative f3010.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35Total 7/30 = 0.2333 12/30 = 0.40 7/30 = 0.2333 3/30 = 0.10 1/30 = 0.0333 30/30 =1Using different limits 7.5 to under 12.512.5 to under 17.517.5 to under 22.522.5 to under 27.527.5 to under 32.532.5 to under 37.5Hours Studying xfRelative f30Total1/30 = 0.0333 12/30 = 0.40 10/30 = 0.3333 1/30 = 0.0333 1/30 = 0.0333 30/30 =111210115 5/30 = 0.1666 Stem-and-leaf Displays Each numerical value is divided into two parts: 1. the leading digits become the stem and2. the trailing digits become the leaf. an advantage of the stem-and-leaf display over a frequency distribution is that we retain the value of each observation!A statistical technique for displaying a set of data. Note:EXAMPLE A student achieved the following scores on the twelve accounting quizzes this semester: 86, 79, 92, 84, 69, 88, 91, 83, 96, 78, 82, 85. Construct a stem-and-leaf chart to illustrate the results. Stem-and-leaf DisplaysQuestionStem-and-leaf DisplaysFirst, find the lowest score86, 79, 92, 84, 69, 88, 91, 83, 96, 78, 82, 85. Now list the next scores with the highest leading digits.You should now have the following STEMS:69, 78, 82, 916789Stem6978 82917896SplitLeaf69788291Now, list the remaining ‘leaf’ scores!93458266All 12 ScoresStem-and-leaf DisplaysEXAMPLE86, 79, 92, 84, 69, 88, 91, 83, 96, 78, 82, 85. QuestionThe grades on a statistics exam for a sample of 40 students are as follows:Stem Leaf 3 6 8 4 1 2 7 8 5 0 1 2 5 5 8 9 6 0 1 1 1 2 5 7 8 8 8 9 7 0 0 2 5 6 6 7 8 4 6 8 8 9 9 0 2 4 6How many students earned an A on this test?5What is the most common letter grade earned?FA+ = 90%-100%A = 80%-89%B+ = 75%-79%B = 70%-74%C+ = 65%-69%C = 60%-64%D = 55%-59%F = 0%-54%Alpha-NumericGradingGraphic Presentation of a Frequency DistributionGraphic Presentation of a Frequency Distribution The three commonly used graphic forms are:Histograms Frequency Polygons or Line ChartsCumulative Frequency Distributions The class frequencies are represented by the heights of the bars and the bars are drawn adjacent to each other.A Histogram is a graph in which the classes are marked on the horizontal axis and the class frequencies on the vertical axisFrequencyClass Graphic Presentation of a Frequency Distribution10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35Hours Studying xf712731Graphic Presentation of a Frequency Distribution0 10 15 20 25 30 35 Hours spent studying1412108642FrequencyHistogram A frequency polygon consists of line segments connecting the points formed by the class midpoint and the class frequency.024681012147.512.517.522.527.505101520253035101520253035A cumulative frequency distribution is used to determine how many or what proportion of the data values are below or above a certain value.Graphic Presentation of a Frequency DistributionMaking a Histogram in ExcelUsingClick on DATA ANALYSISSeeClick on HISTOGRAMThe upper limits of the classes you have determinedUsingComplete INPUTTING of DATA must now be entered from Column B (Excel calls these “bins”)To remove the Legend on the right side Right mouse click and Click on ClearUsingTo remove the spaces between the bars Right mouse click on one of the bars and Click on Format Data SeriesUsingNow, Click on the Options tab;To reduce/remove the spaces between the barsAdjust the Gap width down to 0 and Click on OK.UsingEdit the size of the histogram, titles, etc as appropriate. Note that the upper limit values are included in each class – this explains the difference between this Excel Frequency Distribution and the one we did by hand.0 10 15 20 25 30 35 Hours spent studying1412108642FrequencyFrequency Polygon or Line Chart for Hours Spent Studying0 10 15 20 25 30 35 Hours spent studying1412108642Frequency10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35Hours Studying xf712731Notice that the class midpoints (the plotted points) aren’t as “user friendly” in this distribution choice.10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35Hours Studying xf712731Cumulative Frequency Distribution For Hours Studyingunder 15under 20under 25under 30under 35Hours StudyingCumulative f192629307Graph..Cumulative Frequency Distribution For Hours Studying0 10 15 20 25 30 35 Hours spent studying3530252015105Frequencyunder 15under 20under 25under 30under 35192629307Hours StudyingCumulative fNotice that the limits are the plotted points. Pie Bar LineCharts used primarily for Qualitative Datais useful for displaying a Relative Frequency DistributionPieA circle is divided proportionally to the relative frequency and portions of the circle are allocated for the different groups.EXAMPLEChart Pie200 runners were asked to indicate their favourite type of running shoe. TypeNike 92Adidas 49Reebok 37Asics 13Other 9# of runners selecting:Draw a pie chart based on this information.Chart QuestionNike 92Adidas 49Reebok 37Asics 13Other 9Type#200%46.024.518.56.54.5100Adidas24.5%Nike46.0%Reebok18.5%Asics6.5%Other4.5%Relative Frequency Distribution for the running shoesPieChart Nike 92Adidas 49Reebok 37Asics 13Other 9Type#200%46.024.518.56.54.5100Using Excel, follow the steps in the Chart Wizard to construct a Pie Chart!PieChart Barcan be used to depict any of the levels of measurement (nominal, ordinal, interval, or ratio).Chart (also known as a ‘column chart’)Examples of3-DBarChart Use bar charts also when the order in which qualitative data are presented is meaningful. How could we chart this data?BarChart BarChart Using Excel we can produce thisOther formatsEmployment Rate Canadian CityVictoria 57.7 Halifax 60.5 Montreal 60.4 Sherbrooke 59.2 Quebec 59.7 Toronto 65.1 Hamilton 63.2 Kitchener 66.0 London 63.3 Thunder Bay61.0 Regina 67.4 Saskatoon 63.7 Edmonton 67.1 Vancouver 61.4 Winnipeg 66.7 BarChart HalifaxMontrealSherbrookeQuebecTorontoHamiltonKitchenerLondonThunder BayReginaSaskatoonWinnipegEdmontonVancouverVictoria% employment52545658606264666870Employment Rate in Canadian CitiesEmployment Rate Canadian CityVictoria 57.7 Halifax 60.5 Montreal 60.4 Sherbrooke 59.2 Quebec 59.7 Toronto 65.1 Hamilton 63.2 Kitchener 66.0 London 63.3 Thunder Bay61.0 Regina 67.4 Saskatoon 63.7 Edmonton 67.1 Vancouver 61.4 Winnipeg 66.7 BarChart Employment Rate in Canadian Cities% employment52545658606264666870HalifaxMontrealSherbrookeQuebecTorontoHamiltonKitchenerLondonThunder BayReginaSaskatoonWinnipegEdmontonVancouverVictoria- by ProvinceBarChart Did any of the previous Bar Charts adequately display all the information that was provided?The following has been modified from that data found by Statistics Canada.Does it do an effective job of displaying the StatCan data?Real estate and rental and leasingProfessional, scientific and technical servicesManagement of companies and enterprisesEducational services (private sector)Health care and social assistance (private sector)Administration and support, waste management and remediation servicesArts, entertainment and recreationAccommodation and food servicesAll private sectorInformation and cultural industriesFinance and insuranceManufacturingWholesale tradeRetail trade020406080 100% of enterprisesClustered BarChart Comparison of Internet Use in 2000 and 2001 % of enterprises thatuse the Internet 2000 % of enterprises thatuse the Internet 2001% of enterprises with aWeb site 2000% of enterprises with aWeb site 2001Data Source: Statistics CanadaFull-Time University Faculty By Gender,Canada and Jurisdictions, 1987-88 and 1997-98Chart Stacked BarCanadian Full Time University Faculty0204060801001201987-881997-98% of Total% males% femalesData Source: Statistics CanadaTotal34,651 33,925 12,829 13,910 12,650 12,095 9,172 7,817Full ProfessorAssociate Professor1987-881997-981987-881997-981987-88 1997-98 Other 1987-881997-98% Male% Female1783257579313871783287232684456Make sure that your charts are not overly clutteredThere are four typical shape characteristicsShapesof HistogramsSkewnessSymmetryBell Curve ModalClassSymmetrya balanced effect!Both ‘balanced’ or ‘have symmetry’- or -Skewness occurs when the observations are graphed as being skewed or tilted more to one side of the centre of the observations than the other.The skewness, if on the right side is said to be ‘positive’.The skewness, if on the left side is said to be ‘negative’.ClassModalA modal class is the one with the largest number of observationsThis is a uniModal HistogrambiModalClassModalbiModalThis is a biModal HistogramPopulation distributions are often bell shaped. Drawing a histogram helps verify the shape of the population in question. Bell CurveLineChart Line charts are particularly useful when the trend over time is to be emphasizedExamples 3-DIn combinationEXAMPLESTime PlotLineChart OSAJJMAMFJDNOSAJJMAMFJDNOSAJJMAMFJ8.57.56.55.5MonthMonthly Steel ProductionMillions of Tons200020012002 M o n t h l y S t e e l P r o d u c t i o nEmployment Rate in Canadian Cities52545658606264666870% employmentHalifaxMontrealSherbrookeQuebecTorontoHamiltonKitchenerLondonThunder BayReginaSaskatoonWinnipegEdmontonVancouverVictoriaLineChart Preparing a Line Chart for this type of data is not overly useful!Employment Rate in Canadian Cities52545658606264666870% employmentHalifaxMontrealSherbrookeQuebecTorontoHamiltonKitchenerLondonThunder BayReginaSaskatoonWinnipegEdmontonVancouverVictoriaLineChart Is this combination any better for displaying the data?frequency Polygon and Ogivefrequency Polygon504030201000.30.20.10.0Relative FrequencySalesOgive504030201001.00.50.0Cumulative Relative FrequencySalesLineChart Test your learning www.mcgrawhill.ca/college/lindClick onOnline Learning Centrefor quizzesextra contentdata setssearchable glossaryaccess to Statistics Canada’s E-Stat dataand much more!This completes Chapter 2