1、Organizing and Visualizing VariablesChapter 2ObjectivesIn this chapter you learn:nMethods to organize variables.nMethods to visualize variables.nMethods to organize or visualize more than one variable at the same time.nPrinciples of proper visualizations.Categorical Data Are Organized By Utilizing T
2、ablesDCOVACategorical DataTallying Data Summary Table One Categorical Variable Two Categorical VariablesContingency TableOrganizing Categorical Data:Summary TableA summary table tallies the frequencies or percentages of items in a set of categories so that you can see differences between categories.
3、Reason For Shopping Online?PercentBetter Prices37%Avoiding holiday crowds or hassles29%Convenience18%Better selection13%Ships directly 3%DCOVAMain Reason Young Adults Shop OnlineSource:Data extracted and adapted from“Main Reason Young Adults Shop Online?”USA Today,December 5,2012,p.1A.A Contingency
4、Table Helps Organize Two or More Categorical VariablesnUsed to study patterns that may exist between the responses of two or more categorical variablesnCross tabulates or tallies jointly the responses of the categorical variablesnFor two variables the tallies for one variable are located in the rows
5、 and the tallies for the second variable are located in the columnsDCOVAContingency Table-ExamplenA random sample of 400 invoices is drawn.nEach invoice is categorized as a small,medium,or large amount.nEach invoice is also examined to identify if there are any errors.nThis data are then organized i
6、n the contingency table to the right.DCOVANoErrorsErrorsTotalSmallAmount17020190MediumAmount10040140LargeAmount65570Total33565400Contingency Table ShowingFrequency of Invoices CategorizedBy Size and The Presence Of ErrorsContingency Table Based On Percentage Of Overall TotalNoErrorsErrorsTotalSmallA
7、mount17020190MediumAmount10040140LargeAmount65570Total33565400DCOVANoErrorsErrorsTotalSmallAmount42.50%5.00%47.50%MediumAmount25.00%10.00%35.00%LargeAmount16.25%1.25%17.50%Total83.75%16.25%100.0%42.50%=170/40025.00%=100/40016.25%=65/40083.75%of sampled invoices have no errors and 47.50%of sampled in
8、voices are for small amounts.Contingency Table Based On Percentage of Row TotalsNoErrorsErrorsTotalSmallAmount17020190MediumAmount10040140LargeAmount65570Total33565400DCOVANoErrorsErrorsTotalSmallAmount89.47%10.53%100.0%MediumAmount71.43%28.57%100.0%LargeAmount92.86%7.14%100.0%Total83.75%16.25%100.0
9、%89.47%=170/19071.43%=100/14092.86%=65/70Medium invoices have a larger chance(28.57%)of having errors than small(10.53%)or large(7.14%)invoices.Contingency Table Based On Percentage Of Column TotalsNoErrorsErrorsTotalSmallAmount17020190MediumAmount10040140LargeAmount65570Total33565400DCOVANoErrorsEr
10、rorsTotalSmallAmount50.75%30.77%47.50%MediumAmount29.85%61.54%35.00%LargeAmount19.40%7.69%17.50%Total100.0%100.0%100.0%50.75%=170/33530.77%=20/65There is a 61.54%chance that invoices with errors are of medium size.Tables Used For Organizing Numerical DataDCOVANumerical DataOrdered ArrayCumulativeDis
11、tributionsFrequencyDistributionsOrganizing Numerical Data:Ordered ArrayAn ordered array is a sequence of data,in rank order,from the smallest value to the largest value.Shows range(minimum value to maximum value)May help identify outliers(unusual observations)Age of Surveyed College StudentsDay Stud
12、ents161717181818191920202122222527323842Night Students181819192021232832334145DCOVAOrganizing Numerical Data:Frequency DistributionThe frequency distribution is a summary table in which the data are arranged into numerically ordered classes.You must give attention to selecting the appropriate number
13、 of class groupings for the table,determining a suitable width of a class grouping,and establishing the boundaries of each class grouping to avoid overlapping.The number of classes depends on the number of values in the data.With a larger number of values,typically there are more classes.In general,
14、a frequency distribution should have at least 5 but no more than 15 classes.To determine the width of a class interval,you divide the range(Highest valueLowest value)of the data by the number of class groupings desired.DCOVAOrganizing Numerical Data:Frequency Distribution ExampleExample:A manufactur
15、er of insulation randomly selects 20 winter days and records the daily high temperature24,35,17,21,24,37,26,46,58,30,32,13,12,38,41,43,44,27,53,27DCOVAOrganizing Numerical Data:Frequency Distribution ExampleSort raw data in ascending order:12,13,17,21,24,24,26,27,27,30,32,35,37,38,41,43,44,46,53,58F
16、ind range:58-12=46Select number of classes:5(usually between 5 and 15)Compute class interval(width):10(46/5 then round up)Determine class boundaries(limits):Class 1:10 but less than 20Class 2:20 but less than 30Class 3:30 but less than 40Class 4:40 but less than 50Class 5:50 but less than 60Compute
17、class midpoints:15,25,35,45,55Count observations&assign to classesDCOVAOrganizing Numerical Data:Frequency Distribution Example Class Midpoints Frequency10 but less than 20 15320 but less than 30 25630 but less than 40 355 40 but less than 50 45450 but less than 60 552 Total 20Data in ordered array:
18、12,13,17,21,24,24,26,27,27,30,32,35,37,38,41,43,44,46,53,58DCOVAOrganizing Numerical Data:Relative&Percent Frequency Distribution Example Class Frequency10 but less than 20 3 .15 15%20 but less than 30 6 .30 30%30 but less than 40 5 .25 25%40 but less than 50 4 .20 20%50 but less than 60 2 .10 10%To
19、tal 20 1.00 100%RelativeFrequency PercentageDCOVARelative Frequency=Frequency/Total,e.g.0.10=2/20Organizing Numerical Data:Cumulative Frequency Distribution ExampleClass10 but less than 20 3 15%3 15%20 but less than 30 6 30%9 45%30 but less than 40 5 25%14 70%40 but less than 50 4 20%18 90%50 but le
20、ss than 60 2 10%20 100%Total 20 100 20100%PercentageCumulative PercentageCumulative Percentage=Cumulative Frequency/Total*100 e.g.45%=100*9/20FrequencyCumulative FrequencyDCOVAWhy Use a Frequency Distribution?nIt condenses the raw data into a more useful formnIt allows for a quick visual interpretat
21、ion of the datanIt enables the determination of the major characteristics of the data set including where the data are concentrated/clusteredDCOVAFrequency Distributions:Some TipsnDifferent class boundaries may provide different pictures for the same data(especially for smaller data sets)nShifts in
22、data concentration may show up when different class boundaries are chosennAs the size of the data set increases,the impact of alterations in the selection of class boundaries is greatly reducednWhen comparing two or more groups with different sample sizes,you must use either a relative frequency or
23、a percentage distributionDCOVAVisualizing Categorical Data Through Graphical DisplaysDCOVACategorical DataVisualizing Data BarChartSummary Table For One VariableContingency Table For Two VariablesSide By Side Bar Chart Pie Chart ParetoChartVisualizing Categorical Data:The Bar ChartThe bar chart visu
24、alizes a categorical variable as a series of bars.The length of each bar represents either the frequency or percentage of values for each category.Each bar is separated by a space called a gap.DCOVAReason For Shopping Online?PercentBetter Prices37%Avoiding holiday crowds or hassles29%Convenience18%B
25、etter selection13%Ships directly 3%Visualizing Categorical Data:The Pie ChartThe pie chart is a circle broken up into slices that represent categories.The size of each slice of the pie varies according to the percentage in each category.DCOVAReason For Shopping Online?PercentBetter Prices37%Avoiding
26、 holiday crowds or hassles29%Convenience18%Better selection13%Ships directly 3%Visualizing Categorical Data:The Pareto ChartnUsed to portray categorical datanA vertical bar chart,where categories are shown in descending order of frequencynA cumulative polygon is shown in the same graphnUsed to separ
27、ate the“vital few”from the“trivial many”DCOVAVisualizing Categorical Data:The Pareto Chart(cont)DCOVA CumulativeCause Frequency PercentPercentWarped card jammed 365 50.41%50.41%Card unreadable 234 32.32%82.73%ATM malfunctions32 4.42%87.15%ATM out of cash28 3.87%91.02%Invalid amount requested 23 3.18
28、%94.20%Wrong keystroke 23 3.18%97.38%Lack of funds in account 19 2.62%100.00%Total 724 100.00%Source:Data extracted from A.Bhalla,“Dont Misuse the Pareto Principle,”Six Sigma ForumMagazine,May 2009,pp.1518.Ordered Summary Table For CausesOf Incomplete ATM TransactionsVisualizing Categorical Data:The
29、 Pareto Chart(cont)DCOVAThe“VitalFew”Visualizing Categorical Data:Side By Side Bar Chartsn The side by side bar chart represents the data from a contingency table.DCOVA0.0%10.0%20.0%30.0%40.0%50.0%60.0%70.0%No ErrorsErrorsInvoice Size Split Out By Errors&No ErrorsLargeMediumSmallInvoices with errors
30、 are much more likely to be ofmedium size(61.54%vs 30.77%and 7.69%)NoErrorsErrorsTotalSmallAmount50.75%30.77%47.50%MediumAmount29.85%61.54%35.00%LargeAmount19.40%7.69%17.50%Total100.0%100.0%100.0%Visualizing Numerical Data By Using Graphical DisplaysNumerical DataOrdered ArrayStem-and-LeafDisplayHis
31、togramPolygonOgiveFrequency Distributions andCumulative DistributionsDCOVAStem-and-Leaf DisplaynA simple way to see how the data are distributed and where concentrations of data existMETHOD:Separate the sorted data series into leading digits(the stems)and the trailing digits(the leaves)DCOVAOrganizi
32、ng Numerical Data:Stem and Leaf DisplayA stem-and-leaf display organizes data into groups(called stems)so that the values within each group(the leaves)branch out to the right on each row.StemLeaf1677888992001225732842 Age of College Students Day Students Night StudentsStemLeaf1889920138323415Age of
33、Surveyed College StudentsDay Students161717181818191920202122222527323842Night Students181819192021232832334145DCOVAVisualizing Numerical Data:The HistogramA vertical bar chart of the data in a frequency distribution is called a histogram.In a histogram there are no gaps between adjacent bars.The cl
34、ass boundaries(or class midpoints)are shown on the horizontal axis.The vertical axis is either frequency,relative frequency,or percentage.The height of the bars represent the frequency,relative frequency,or percentage.DCOVAVisualizing Numerical Data:The Histogram Class Frequency10 but less than 20 3
35、 .15 1520 but less than 30 6 .30 3030 but less than 40 5 .25 25 40 but less than 50 4 .20 2050 but less than 60 2 .10 10 Total 20 1.00 100RelativeFrequency Percentage(In a percentage histogram the vertical axis would be defined to show the percentage of observations per class)DCOVAVisualizing Numeri
36、cal Data:The PolygonA percentage polygon is formed by having the midpoint of each class represent the data in that class and then connecting the sequence of midpoints at their respective class percentages.The cumulative percentage polygon,or ogive,displays the variable of interest along the X axis,a
37、nd the cumulative percentages along the Y axis.Useful when there are two or more groups to compare.DCOVAVisualizing Numerical Data:The Percentage PolygonDCOVAUseful When Comparing Two or More GroupsVisualizing Numerical Data:The Percentage PolygonDCOVAVisualizing Two Numerical Variables By Using Gra
38、phical DisplaysTwo Numerical VariablesScatter PlotTime-Series PlotDCOVAVisualizing Two Numerical Variables:The Scatter PlotScatter plots are used for numerical data consisting of paired observations taken from two numerical variablesOne variable is measured on the vertical axis and the other variabl
39、e is measured on the horizontal axisScatter plots are used to examine possible relationships between two numerical variablesDCOVAScatter Plot ExampleVolume per dayCost per day231252614029146331603816742170501885519560200DCOVAnA Time-Series Plot is used to study patterns in the values of a numeric va
40、riable over timenThe Time-Series Plot:nNumeric variable is measured on the vertical axis and the time period is measured on the horizontal axisVisualizing Two Numerical Variables:The Time Series PlotDCOVATime Series Plot ExampleYearNumber of Franchises1996 431997 541998 601999 732000 822001 952002 1
41、072003 992004 95DCOVAnA multidimensional contingency table is constructed by tallying the responses of three or more categorical variables.nIn Excel creating a Pivot Table to yield an interactive display of this type.nWhile Minitab will not create an interactive table,it has many specialized statist
42、ical&graphical procedures(not covered in this book)to analyze&visualize multidimensional data.Organizing Many Categorical Variables:The Multidimensional Contingency TableDCOVAUsing Excel Pivot Tables To Organize&Visualize Many VariablesA pivot table:nSummarizes variables as a multidimensional summar
43、y tablenAllows interactive changing of the level of summarization and formatting of the variablesnAllows you to interactively“slice”your data to summarize subsets of data that meet specified criterianCan be used to discover possible patterns and relationships in multidimensional data that simpler ta
44、bles and charts would fail to make apparent.DCOVAA Multidimensional Contingency Table Tallies Responses Of Three or More Categorical VariablesTwo Dimensional Table Showing The Mean 10 Year Return%Broken Out By Type Of Fund&Risk LevelDCOVAThree Dimensional Table Showing The Mean 10 Year Return%Broken
45、 Out By Type Of Fund,Market Cap,&Risk LevelData Discovery Methods Can Yield Initial Insights Into Data nData discovery are methods enable the performance of preliminary analyses by manipulating interactive summarizationsnAre used to:nTake a closer look at historical or status datanReview data for un
46、usual valuesnUncover new patterns in datanDrill-down is perhaps the simplest form of data discoveryDCOVADrill-Down Reveals The Data Underlying A Higher-Level SummaryDCOVAResults of drilling down to the details about smallmarket cap value funds withlow risk.Some Data Discovery Methods Are Primarily V
47、isualnA treemap is such a methodnA treemap visualizes the comparison of two or more variables using the size and color of rectangles to represent valuesnWhen used with one or more categorical variables it forms a multilevel hierarchy or tree that can uncover patterns among numerical variables.DCOVAA
48、n Example Of A TreemapDCOVAA treemap of the numerical variables assets(size)and 10-yearreturn percentage(color)for growth and value funds that havesmall market capitalizations and low riskThe Challenges in Organizing and Visualizing VariablesnWhen organizing and visualizing data need to be mindful o
49、f:nThe limits of others ability to perceive and comprehendnPresentation issues that can undercut the usefulness of methods from this chapter.nIt is easy to create summaries thatnObscure the data ornCreate false impressionsDCOVAAn Example Of Obscuring Data,Information OverloadDCOVAFalse Impressions C
50、an Be Created In Many WaysnSelective summarizationnPresenting only part of the data collectednImproperly constructed chartsnPotential pie chart issuesnImproperly scaled axesnA Y axis that does not begin at the origin or is a broken axis missing intermediate valuesnChartjunkDCOVAAn Example of Selecti
