Slides Prepared by JOHN S. LOUCKS St. Edward’s University

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© 2003 South-Western /Thomson Learning© 2003 South-Western /Thomson LearningTMTM

Slides Prepared bySlides Prepared byJOHN S. LOUCKSJOHN S. LOUCKS

St. Edward’s UniversitySt. Edward’s University

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Chapter 2Chapter 2Descriptive Statistics:Descriptive Statistics:

Tabular and Graphical MethodsTabular and Graphical Methods Summarizing Qualitative DataSummarizing Qualitative Data Summarizing Quantitative DataSummarizing Quantitative Data Exploratory Data AnalysisExploratory Data Analysis CrosstabulationsCrosstabulations

and Scatter Diagramsand Scatter Diagrams

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Summarizing Qualitative DataSummarizing Qualitative Data

Frequency DistributionFrequency Distribution Relative Frequency Relative Frequency Percent Frequency DistributionPercent Frequency Distribution Bar Graphs and Pie ChartsBar Graphs and Pie Charts

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Frequency DistributionFrequency Distribution

A A frequency distributionfrequency distribution is a tabular summary is a tabular summary of data showing the frequency (or number) of of data showing the frequency (or number) of items in each of several nonoverlapping items in each of several nonoverlapping classes.classes.

The objective is to The objective is to provide insightsprovide insights about the about the data that cannot be quickly obtained by data that cannot be quickly obtained by looking only at the original data.looking only at the original data.

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Guests staying at Marada Inn were asked to rate the Guests staying at Marada Inn were asked to rate the quality of their accommodations as being quality of their accommodations as being excellentexcellent, , above averageabove average, , averageaverage, , below averagebelow average, or , or poorpoor. .

The Theratings provided by a sample of 20 quests are shownratings provided by a sample of 20 quests are shownbelow.below.

Below Average Average Below Average Average Above AverageAbove AverageAbove Average Above Average Above Average Above Average Above Average Above Average Above Average Below Average Above Average Below Average Below Average Below Average Average Poor Average Poor PoorPoorAbove Average ExcellentAbove Average Excellent Above Average Above Average Average Above AverageAverage Above Average AverageAverageAbove Average Average Above Average Average

Example: Marada InnExample: Marada Inn

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RatingRating FrequencyFrequency

PoorPoor 2 2

Below AverageBelow Average 3 3

AverageAverage 5 5

Above AverageAbove Average 9 9

ExcellentExcellent 1 1

TotalTotal 20 20


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Using Excel’s COUNTIF FunctionUsing Excel’s COUNTIF Functionto Construct a Frequency Distributionto Construct a Frequency Distribution

Formula WorksheetFormula Worksheet

Note: Rows 9-21 are not shown.Note: Rows 9-21 are not shown.

A B C D1 Quality Rating Quality Rating Frequency2 Above Average Poor =COUNTIF($A$2:$A$21,C2)3 Below Average Below Average =COUNTIF($A$2:$A$21,C3)4 Above Average Average =COUNTIF($A$2:$A$21,C4)5 Average Above Average =COUNTIF($A$2:$A$21,C5)6 Average Excellent =COUNTIF($A$2:$A$21,C6)7 Above Average Total =SUM(D2:D6)8 Above Average

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Value WorksheetValue Worksheet

A B C D1 Quality Rating Quality Rating Frequency2 Above Average Poor 23 Below Average Below Average 34 Above Average Average 55 Average Above Average 96 Average Excellent 17 Above Average Total 208 Above Average

Using Excel’s COUNTIF FunctionUsing Excel’s COUNTIF Functionto Construct a Frequency Distributionto Construct a Frequency Distribution


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Relative Frequency DistributionRelative Frequency Distribution

The The relative frequencyrelative frequency of a class is the fraction of a class is the fraction or proportion of the total number of data items or proportion of the total number of data items belonging to the class.belonging to the class.

A A relative frequency distributionrelative frequency distribution is a tabular is a tabular summary of a set of data showing the relative summary of a set of data showing the relative frequency for each class.frequency for each class.

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Percent Frequency DistributionPercent Frequency Distribution

The The percent frequencypercent frequency of a class is the relative of a class is the relative frequency multiplied by 100.frequency multiplied by 100.

AA percent frequency distributionpercent frequency distribution is a tabular is a tabular summary of a set of data showing the percent summary of a set of data showing the percent frequency for each class.frequency for each class.

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Relative Frequency and Percent Frequency Relative Frequency and Percent Frequency DistributionsDistributions

RelativeRelative PercentPercent

RatingRating FrequencyFrequency FrequencyFrequency

PoorPoor .10 .10 1010

Below AverageBelow Average .15 .15 1515

AverageAverage .25 .25 2525

Above AverageAbove Average .45 .45 4545

ExcellentExcellent .05 .05 5 5

TotalTotal 1.00 1.00 100 100

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Using Excel to Construct Relative Using Excel to Construct Relative Frequency and Percent Frequency Frequency and Percent Frequency

DistributionsDistributions Formula WorksheetFormula Worksheet

Note: Columns A-B and rows 9-21 and are not shown.Note: Columns A-B and rows 9-21 and are not shown.

C D E F

1 Quality Rating FrequencyRelative

FrequencyPercent

Frequency2 Poor =COUNTIF($A$2:$A$21,C2) =D2/$D$7 =E2*1003 Below Average =COUNTIF($A$2:$A$21,C3) =D3/$D$7 =E3*1004 Average =COUNTIF($A$2:$A$21,C4) =D4/$D$7 =E4*1005 Above Average =COUNTIF($A$2:$A$21,C5) =D5/$D$7 =E5*1006 Excellent =COUNTIF($A$2:$A$21,C6) =D6/$D$7 =E6*1007 Total =SUM(D2:D6) =SUM(E2:E6) =SUM(F2:F6)8

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Using Excel to Construct Relative Using Excel to Construct Relative Frequency and Percent Frequency Frequency and Percent Frequency

DistributionsDistributions

C D E F

1 Quality Rating FrequencyRelative

FrequencyPercent

Frequency2 Poor 2 0.10 103 Below Average 3 0.15 154 Average 5 0.25 255 Above Average 9 0.45 456 Excellent 1 0.05 57 Total 20 1.00 1008

Note: Columns A-B and rows 9-21 and are not shown.Note: Columns A-B and rows 9-21 and are not shown.

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Bar GraphBar Graph

A A bar graphbar graph is a graphical device for depicting is a graphical device for depicting qualitative data that have been summarized in a qualitative data that have been summarized in a frequency, relative frequency, or percent frequency, relative frequency, or percent frequency distribution.frequency distribution.

On the horizontal axis we specify the labels that On the horizontal axis we specify the labels that are used for each of the classes.are used for each of the classes.

A A frequencyfrequency, , relative frequencyrelative frequency, or , or percent percent frequencyfrequency scale can be used for the vertical scale can be used for the vertical axis.axis.

Using a bar of fixed width drawn above each Using a bar of fixed width drawn above each class label, we extend the height appropriately.class label, we extend the height appropriately.

The The bars are separatedbars are separated to emphasize the fact to emphasize the fact that each class is a separate category.that each class is a separate category.

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Bar GraphBar Graph

1122

33

44

55

66

77

88

99

PoorPoor BelowAverageBelow

AverageAverageAverageAbove

AverageAbove

AverageExcellentExcellent

Fre

qu

en

cy

Fre

qu

en

cy

RatingRating

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Using Excel’s Chart WizardUsing Excel’s Chart Wizardto Construct Bar Graphsto Construct Bar Graphs

Step 1Step 1 Select cells C1:D6 Select cells C1:D6Step 2Step 2 Select the Select the Chart Wizard Chart Wizard buttonbuttonStep 3Step 3 When the When the Chart TypeChart Type dialog box appears: dialog box appears:

Choose Choose ColumnColumn in the in the Chart typeChart type list list Choose Choose Clustered ColumnClustered Column from the from the

ChartChart sub-typesub-type display display

Select Select Next >Next >Step 4Step 4 When the When the Chart Source DataChart Source Data dialog box dialog box

appearsappears Select Select Next >Next >

… … continued continued

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Step 5Step 5 When the When the Chart OptionsChart Options dialog box appears: dialog box appears:

Select the Select the TitlesTitles tab and then tab and then

Type Type Customers’ Quality RatingsCustomers’ Quality Ratings in the in the

Chart titleChart title box box

Enter Enter QualityQuality RatingRating in the in the Value (X)Value (X) axis boxaxis box

Enter Enter FrequencyFrequency in the in the Value (Y)Value (Y) axis axis boxbox

Select the Select the LegendLegend tab and then tab and then

Remove the check in the Remove the check in the Show LegendShow Legend boxbox

Select Select Next >Next >

… … continued continued


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Step 6Step 6 When the When the Chart LocationChart Location dialog box dialog box appears:appears:

Specify the location for the new Specify the location for the new chartchart

Select Select FinishFinish to display the bar to display the bar graphgraph


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Cutomers' Quality Ratings

0

2

4

6

8

10

Poor BelowAverage

Average AboveAverage

Excellent

Quality Rating

Fre

qu

en

cy

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Pie ChartPie Chart

The The pie chartpie chart is a commonly used graphical is a commonly used graphical device for presenting relative frequency device for presenting relative frequency distributions for qualitative data.distributions for qualitative data.

First draw a First draw a circlecircle; then use the relative ; then use the relative frequencies to subdivide the circle into sectors frequencies to subdivide the circle into sectors that correspond to the relative frequency for that correspond to the relative frequency for each class.each class.

Since there are 360 degrees in a circle, a class Since there are 360 degrees in a circle, a class with a relative frequency of .25 would with a relative frequency of .25 would consume .25(360) = consume .25(360) =

90 degrees of the circle.90 degrees of the circle.

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Pie ChartPie Chart

Average 25%Average 25%

BelowAverage 15%

BelowAverage 15%

Poor 10%Poor 10%

AboveAverage 45%

AboveAverage 45%

Exc. 5%Exc. 5%

Quality Ratings Quality Ratings

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Using Excel’s Chart WizardUsing Excel’s Chart Wizardto Construct Pie Chartsto Construct Pie Charts

Step 1Step 1 Select cells C2:C6 and F2:F6 Select cells C2:C6 and F2:F6

Step 2Step 2 Select the Select the Chart Wizard Chart Wizard buttonbutton

Step 3Step 3 When the When the Chart TypeChart Type dialog box appears: dialog box appears:

Choose Choose PiePie in the in the Chart typeChart type list list

Choose Choose PiePie from the from the Chart sub-typeChart sub-type displaydisplay


Step 4Step 4 When the When the Chart Source DataChart Source Data dialog box dialog box appearsappears


… … continuedcontinued

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Step 5Step 5 When the When the Chart OptionsChart Options dialog box appears: dialog box appears:


Type Type Customers’ Quality RatingsCustomers’ Quality Ratings at at MaradaMarada in the in the Chart titleChart title box box


Remove the check in the Remove the check in the Show Show LegendLegend box box

Select the Select the Data LabelsData Labels tab and then tab and then

Select Select Show Label and percentShow Label and percent

Select Select Show leader linesShow leader lines




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Select Select FinishFinish to display the pie to display the pie chartchart


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Customers' Quality Ratingsat Marada Inn

Excellent

5%

Average

25%

Poor

10%

Above

Average

45%

Below

Average

15%

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Insights Gained from the Preceding Pie ChartInsights Gained from the Preceding Pie Chart

• One-half of the customers surveyed gave One-half of the customers surveyed gave Marada a quality rating of “above average” Marada a quality rating of “above average” or “excellent” (looking at the left side of the or “excellent” (looking at the left side of the pie). This might please the manager.pie). This might please the manager.

• For For eacheach customer who gave an “excellent” customer who gave an “excellent” rating, there were rating, there were twotwo customers who gave customers who gave a “poor” rating (looking at the top of the a “poor” rating (looking at the top of the pie). This should displease the manager.pie). This should displease the manager.


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Summarizing Quantitative DataSummarizing Quantitative Data

Frequency DistributionFrequency Distribution Relative Frequency and Percent Frequency Relative Frequency and Percent Frequency

DistributionsDistributions Dot PlotDot Plot HistogramHistogram Cumulative DistributionsCumulative Distributions OgiveOgive

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91 78 93 57 75 52 99 80 97 6271 69 72 89 66 75 79 75 72 76104 74 62 68 97 105 77 65 80 10985 97 88 68 83 68 71 69 67 7462 82 98 101 79 105 79 69 62 73

91 78 93 57 75 52 99 80 97 6271 69 72 89 66 75 79 75 72 76104 74 62 68 97 105 77 65 80 10985 97 88 68 83 68 71 69 67 7462 82 98 101 79 105 79 69 62 73

Example: Hudson Auto RepairExample: Hudson Auto Repair

The manager of Hudson Auto would like to get The manager of Hudson Auto would like to get aa

better picture of the distribution of costs for better picture of the distribution of costs for engineengine

tune-up parts. A sample of 50 customer invoices tune-up parts. A sample of 50 customer invoices hashas

been taken and the costs of parts, rounded to been taken and the costs of parts, rounded to thethe

nearest dollar, are listed below.nearest dollar, are listed below.

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Guidelines for Selecting Number of ClassesGuidelines for Selecting Number of Classes

• Use between 5 and 20 classes.Use between 5 and 20 classes.

• Data sets with a larger number of elements Data sets with a larger number of elements usually require a larger number of classes.usually require a larger number of classes.

• Smaller data sets usually require fewer Smaller data sets usually require fewer classes.classes.

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Guidelines for Selecting Width of ClassesGuidelines for Selecting Width of Classes

• Use classes of equal width.Use classes of equal width.

• Approximate Class Width =Approximate Class Width =

Largest Data Value Smallest Data ValueNumber of Classes

Largest Data Value Smallest Data ValueNumber of Classes

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If we choose six classes: If we choose six classes:

Approximate Class Width = (109 - 52)/6 = 9.5 Approximate Class Width = (109 - 52)/6 = 9.5 1010

Cost ($)Cost ($) FrequencyFrequency

50-5950-59 2 2

60-69 60-69 13 13

70-7970-79 16 16

80-8980-89 7 7

90-9990-99 7 7

100-109100-109 5 5

Total 50Total 50

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Using Excel’s FREQUENCY FunctionUsing Excel’s FREQUENCY Functionto Construct a Frequency Distributionto Construct a Frequency Distribution

Formula Worksheet (showing data entered)Formula Worksheet (showing data entered)

A B C D

1Parts Cost

Parts Cost Frequency

2 91 50-59 3 71 60-69 4 104 70-79 5 85 80-89 6 62 90-99 7 78 100-109 8 69


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The FREQUENCY function is not a “simple” The FREQUENCY function is not a “simple” Excel function.Excel function.

FREQUENCY is capable of providing multiple FREQUENCY is capable of providing multiple values.values.

In Excel, a formula that can return multiple In Excel, a formula that can return multiple values is called an values is called an array formulaarray formula..

An array formula must be entered in a special An array formula must be entered in a special way.way.


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Entering the Necessary Array FormulaEntering the Necessary Array Formula

Step 1Step 1 Select D2:D7 (where the frequencies Select D2:D7 (where the frequencies will will

appear)appear)

Step 2Step 2 Type the following formula: Type the following formula:

=FREQUENCY(A2:A51,=FREQUENCY(A2:A51,{59,69,79,89,99,109}){59,69,79,89,99,109})

Step 3Step 3 Hold down CTRL and SHIFT keys while Hold down CTRL and SHIFT keys while pressing ENTER key pressing ENTER key

(Array formula will be entered in (Array formula will be entered in D2:D7)D2:D7)


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A B C D

1Parts Cost

Parts Cost Frequency

2 91 50-59 23 71 60-69 134 104 70-79 165 85 80-89 76 62 90-99 77 78 100-109 58 69


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Relative Frequency and Percent Frequency Relative Frequency and Percent Frequency DistributionsDistributions

Relative PercentRelative Percent

Cost ($)Cost ($) FrequencyFrequency FrequencyFrequency

50-5950-59 .04.04 4 4

60-6960-69 .26.26 2626

70-7970-79 .32.32 3232

80-8980-89 .14.14 1414

90-9990-99 .14.14 1414

100-109100-109 .10.10 1010

Total 1.00Total 1.00 100 100


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Insights Gained from the Percent Frequency Insights Gained from the Percent Frequency DistributionDistribution

• Only 4% of the parts costs are in the $50-59 Only 4% of the parts costs are in the $50-59 class.class.

• 30% of the parts costs are under $70.30% of the parts costs are under $70.

• The greatest percentage (32% or almost The greatest percentage (32% or almost one-third) of the parts costs are in the $70-one-third) of the parts costs are in the $70-79 class.79 class.

• 10% of the parts costs are $100 or more.10% of the parts costs are $100 or more.

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HistogramHistogram

Another common graphical presentation of Another common graphical presentation of quantitative data is a quantitative data is a histogramhistogram..

The variable of interest is placed on the The variable of interest is placed on the horizontal axis and the frequency, relative horizontal axis and the frequency, relative frequency, or percent frequency is placed on frequency, or percent frequency is placed on the vertical axis.the vertical axis.

A rectangle is drawn above each class interval A rectangle is drawn above each class interval with its height corresponding to the interval’s with its height corresponding to the interval’s frequency, relative frequency, or percent frequency, relative frequency, or percent frequency.frequency.

Unlike a bar graph, a histogram has Unlike a bar graph, a histogram has no natural no natural separation between rectanglesseparation between rectangles of adjacent of adjacent classes.classes.

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HistogramHistogram

PartsCost ($) PartsCost ($)

22

44

66

88

1010

1212

1414

1616

1818

Frequency

Frequ

ency

50 60 70 80 90 100 11050 60 70 80 90 100 110

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Using Excel’s Chart WizardUsing Excel’s Chart Wizardto Construct a Histogramto Construct a Histogram

Step 1Step 1 Select cells C1:D7 Select cells C1:D7

Step 2Step 2 Select the Select the Chart Wizard Chart Wizard buttonbutton

Step 3Step 3 When the When the Chart TypeChart Type dialog box dialog box appears:appears:

Choose Choose ColumnColumn in the in the Chart typeChart type listlist

Choose Choose Clustered ColumnClustered Column from the from the ChartChart

sub-typesub-type display display





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Step 5Step 5 When the When the Chart OptionsChart Options dialog box dialog box appears:appears:


Type Type Histogram for Parts Cost Histogram for Parts Cost DataData in the in the

Chart titleChart title box box

Enter Enter Parts Cost ($)Parts Cost ($) in the in the Value Value (X)(X) axis box axis box

Enter Enter FrequencyFrequency in the in the Value (Y)Value (Y) axis boxaxis box


Remove the check in the Remove the check in the Show Show LegendLegend box box



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Select Select FinishFinish to display the to display the histogramhistogram

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Histogram for Parts Cost Data

0

5

10

15

20

50-59 60-69 70-79 80-89 90-99 100-109

Parts Cost ($)

Fre

qu

en

cy

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Eliminating Gaps Between RectanglesEliminating Gaps Between Rectangles

Step 1Step 1 Right click on any rectangle in the Right click on any rectangle in the column chartcolumn chart

Step 2Step 2 Select the Select the Format Data SeriesFormat Data Series option option

Step 3Step 3 When the Format Data Series Option When the Format Data Series Option dialog box dialog box appears: appears:

Select the Select the OptionsOptions tab and then tab and then

Enter Enter 00 in the in the Gap widthGap width box box

Click Click OKOK


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Histogram for Parts Cost Data

0

5

10

15

20

50-59 60-69 70-79 80-89 90-99 100-109

Parts Cost ($)

Fre

qu

en

cy

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Cumulative DistributionCumulative Distribution

The The cumulative frequency distributioncumulative frequency distribution shows shows the number of items with values less than or the number of items with values less than or equal to the upper limit of each class.equal to the upper limit of each class.

The The cumulative relative frequency distributioncumulative relative frequency distribution shows the proportion of items with values less shows the proportion of items with values less than or equal to the upper limit of each class.than or equal to the upper limit of each class.

The The cumulative percent frequency distributioncumulative percent frequency distribution shows the percentage of items with values less shows the percentage of items with values less than or equal to the upper limit of each class.than or equal to the upper limit of each class.

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Cumulative DistributionsCumulative Distributions

Cumulative CumulativeCumulative Cumulative Cumulative Relative Cumulative Relative PercentPercentCost ($)Cost ($) FrequencyFrequency FrequencyFrequency FrequencyFrequency

<< 59 59 2 2 .04 .04 4 4

<< 69 69 15 15 .30 .30 30 30

<< 79 79 31 31 .62 .62 62 62

<< 89 89 38 38 .76 .76 76 76

<< 99 99 45 45 .90 .90 90 90

<< 109 109 50 50 1.00 1.00 100 100

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OgiveOgive

An An ogiveogive is a graph of a cumulative distribution. is a graph of a cumulative distribution. The data values are shown on the horizontal The data values are shown on the horizontal

axis.axis. Shown on the vertical axis are the:Shown on the vertical axis are the:

• cumulative frequencies, orcumulative frequencies, or

• cumulative relative frequencies, orcumulative relative frequencies, or

• cumulative percent frequenciescumulative percent frequencies The frequency (one of the above) of each class The frequency (one of the above) of each class

is plotted as a point.is plotted as a point. The plotted points are connected by straight The plotted points are connected by straight

lines.lines.

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OgiveOgive

• Because the class limits for the parts-cost Because the class limits for the parts-cost data are 50-59, 60-69, and so on, there data are 50-59, 60-69, and so on, there appear to be one-unit gaps from 59 to 60, appear to be one-unit gaps from 59 to 60, 69 to 70, and so on.69 to 70, and so on.

• These gaps are eliminated by plotting points These gaps are eliminated by plotting points halfway between the class limits.halfway between the class limits.

• Thus, 59.5 is used for the 50-59 class, 69.5 Thus, 59.5 is used for the 50-59 class, 69.5 is used for the 60-69 class, and so on.is used for the 60-69 class, and so on.

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Ogive with Cumulative Percent Frequencies Ogive with Cumulative Percent Frequencies

PartsPartsCost ($)Cost ($) PartsPartsCost ($)Cost ($)

2020

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50 60 70 80 90 100 11050 60 70 80 90 100 11050 60 70 80 90 100 11050 60 70 80 90 100 110

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Exploratory Data AnalysisExploratory Data Analysis

The techniques of The techniques of exploratory data analysisexploratory data analysis consist of simple arithmetic and easy-to-draw consist of simple arithmetic and easy-to-draw pictures that can be used to summarize data pictures that can be used to summarize data quickly.quickly.

One such technique is the One such technique is the stem-and-leaf stem-and-leaf displaydisplay..

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Stem-and-Leaf DisplayStem-and-Leaf Display

A stem-and-leaf display shows both the A stem-and-leaf display shows both the rank rank orderorder and and shape of the distributionshape of the distribution of the data. of the data.

It is It is similar to a histogramsimilar to a histogram on its side, but it on its side, but it has the advantage of showing the actual data has the advantage of showing the actual data values.values.

The first digits of each data item are arranged The first digits of each data item are arranged to the left of a vertical line.to the left of a vertical line.

To the right of the vertical line we record the To the right of the vertical line we record the last digit for each item in rank order.last digit for each item in rank order.

Each line in the display is referred to as a Each line in the display is referred to as a stemstem..

Each digit on a stem is a Each digit on a stem is a leafleaf..

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55 2 7 2 7

66 2 2 2 2 5 6 7 8 8 8 9 9 9 2 2 2 2 5 6 7 8 8 8 9 9 9

77 1 1 2 2 3 4 4 5 5 5 6 7 8 9 1 1 2 2 3 4 4 5 5 5 6 7 8 9 9 9 9 9

88 0 0 2 3 5 8 9 0 0 2 3 5 8 9

99 1 3 7 7 7 8 9 1 3 7 7 7 8 9

1010 1 4 5 5 9 1 4 5 5 9

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Stretched Stem-and-Leaf DisplayStretched Stem-and-Leaf Display

If we believe the original stem-and-leaf display If we believe the original stem-and-leaf display has condensed the data too much, we can has condensed the data too much, we can stretch the displaystretch the display by using two more stems by using two more stems for each leading digit(s).for each leading digit(s).

Whenever a stem value is stated twice, the Whenever a stem value is stated twice, the first value corresponds to leaf values of 0-4, first value corresponds to leaf values of 0-4, and the second values corresponds to values and the second values corresponds to values of 5-9.of 5-9.

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Stretched Stem-and-Leaf DisplayStretched Stem-and-Leaf Display

55 2 2 55 7 7 66 2 2 2 2 2 2 2 2 66 5 6 7 8 8 8 9 9 9 5 6 7 8 8 8 9 9 9 77 1 1 2 2 3 4 4 1 1 2 2 3 4 4 77 5 5 5 6 7 8 9 9 9 5 5 5 6 7 8 9 9 9 88 0 0 2 3 0 0 2 3 88 5 8 9 5 8 9 99 1 3 1 3 99 7 7 7 8 9 7 7 7 8 9

1010 1 4 1 4 1010 5 5 9 5 5 9

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Leaf UnitsLeaf Units

• A single digit is used to define each leaf.A single digit is used to define each leaf.

• In the preceding example, the leaf unit was In the preceding example, the leaf unit was 1.1.

• Leaf units may be 100, 10, 1, 0.1, and so Leaf units may be 100, 10, 1, 0.1, and so on.on.

• Where the leaf unit is not shown, it is Where the leaf unit is not shown, it is assumed to equal 1.assumed to equal 1.

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Example: Leaf Unit = 0.1Example: Leaf Unit = 0.1

If we have data with values such asIf we have data with values such as

8.68.6 11.711.7 9.49.4 9.19.1 10.210.2 11.011.0 8.88.8

a stem-and-leaf display of these data will bea stem-and-leaf display of these data will be

Leaf Unit = 0.1Leaf Unit = 0.1

8 6 88 6 8

9 1 49 1 4

10 210 2

11 0 711 0 7

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Example: Leaf Unit = 10Example: Leaf Unit = 10

If we have data with values such asIf we have data with values such as

18061806 17171717 19741974 17911791 16821682 19101910 18381838

a stem-and-leaf display of these data will bea stem-and-leaf display of these data will be

Leaf Unit = 10Leaf Unit = 10

16 16 8 8

17 1 917 1 9

18 0 318 0 3

19 1 719 1 7

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Crosstabulations and Scatter DiagramsCrosstabulations and Scatter Diagrams

Thus far we have focused on methods that are Thus far we have focused on methods that are used to summarize the data for used to summarize the data for one variable at one variable at a timea time..

Often a manager is interested in tabular and Often a manager is interested in tabular and graphical methods that will help understand graphical methods that will help understand the the relationship between two variablesrelationship between two variables..

CrosstabulationCrosstabulation and a and a scatter diagramscatter diagram are two are two methods for summarizing the data for two (or methods for summarizing the data for two (or more) variables simultaneously.more) variables simultaneously.

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CrosstabulationCrosstabulation

CrosstabulationCrosstabulation is a tabular method for is a tabular method for summarizing the data for two variables summarizing the data for two variables simultaneously.simultaneously.

Crosstabulation can be used when:Crosstabulation can be used when:

• One variable is qualitative and the other is One variable is qualitative and the other is quantitativequantitative

• Both variables are qualitativeBoth variables are qualitative

• Both variables are quantitativeBoth variables are quantitative The left and top margin labels define the The left and top margin labels define the

classes for the two variables.classes for the two variables.

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Example: Finger Lakes HomesExample: Finger Lakes Homes

CrosstabulationCrosstabulation

The number of Finger Lakes homes sold for each The number of Finger Lakes homes sold for each style and price for the past two years is shown below.style and price for the past two years is shown below.

PricePrice Home StyleHome Style

RangeRange Colonial Ranch Split A-Frame TotalColonial Ranch Split A-Frame Total

<< $99,000 18 $99,000 18 6 19 12 6 19 12 55 55

> $99,000 12 14 16 3> $99,000 12 14 16 3 45 45

TotalTotal 3030 20 35 15 20 35 15 100 100

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Using Excel’s PivotTable ReportUsing Excel’s PivotTable Reportto Construct a Crosstabulationto Construct a Crosstabulation



A B C D E1 Home Price ($) Style 2 1 >99K Colonial 3 2 <=99K Ranch 4 3 >99K Ranch 5 4 <=99K A-Frame 6 5 <=99K Colonial 7 6 <=99K Split-Level 8 7 >99K A-Frame 9 8 >99K Colonial

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Changing the Default Order for the PivotTable Changing the Default Order for the PivotTable ReportReport

Step 1Step 1 Select the Select the ToolsTools pull-down menu pull-down menu

Step 2Step 2 Choose Choose OptionsOptions

Step 3Step 3 When the Options dialog box appears: When the Options dialog box appears:

Select the Select the Custom listsCustom lists tab tab

In the In the List entries:List entries: box, type box, type <= <= 99K99K and and press Enter, and type press Enter, and type > 99K> 99K

Select Select AddAdd

Click Click OKOK


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Using the PivotTable ReportUsing the PivotTable Report

Step 1Step 1 Select the Select the DataData pull-down menu pull-down menu

Step 2Step 2 Choose the Choose the PivotTable and PivotChart PivotTable and PivotChart ReportReport

Step 3Step 3 When the When the PivotTable and PivotChart PivotTable and PivotChart Wizard Wizard Step 1 of 3 Step 1 of 3 dialog box appears: dialog box appears:

Choose Choose Microsoft Excel list or Microsoft Excel list or databasedatabase

Choose Choose PivotTablePivotTable



Enter A1:C101 in the Enter A1:C101 in the RangeRange box box



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Using the PivotTable ReportUsing the PivotTable Report


Select Select New WorksheetNew Worksheet

Click on the Click on the LayoutLayout button button

When the When the PivotTable and PivotChart PivotTable and PivotChart Wizard Wizard – Layout– Layout diagram appears: diagram appears:

Drag the Drag the Price ($)Price ($) field button to the field button to the RROW OW section of the diagramsection of the diagram

Drag the Drag the StyleStyle field button to the field button to the CCOLUMNOLUMN section of the diagram section of the diagram


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Using the PivotTable Report (Step 5 continued)Using the PivotTable Report (Step 5 continued)Drag the Drag the HomeHome field button to the field button to the

DDATA ATA section of the diagram section of the diagram Double click the Double click the Sum of HomeSum of Home field field

button in the data sectionbutton in the data sectionWhen the When the PivotTable FieldPivotTable Field dialog dialog

box box appears:appears: Choose Choose CountCount under under

Summarized bySummarized by:: Click Click OKOKClick Click OKOK

When the When the PivotTable and PivotTable and PivotChart Wizard PivotChart Wizard Step 3 of 3 Step 3 of 3 dialog dialog box reappears:box reappears:

Select Select Finish >Finish >


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D E F G H I J1 Count of Home Style 2 Price ($) Colonial Ranch Split-Level A-Frame Grand Total3 <=99K 18 6 19 12 554 >99K 12 14 16 3 455 Grand Total 30 20 35 15 1006 7 8

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Insights Gained from the Preceding Insights Gained from the Preceding CrosstabulationCrosstabulation

• The greatest number of homes in the The greatest number of homes in the sample (19) are a split-level style and priced sample (19) are a split-level style and priced at less than or equal to $99,000.at less than or equal to $99,000.

• Only three homes in the sample are an A-Only three homes in the sample are an A-Frame style and priced at more than Frame style and priced at more than $99,000. $99,000.

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Crosstabulation: Row or Column Crosstabulation: Row or Column PercentagesPercentages

Converting the entries in the table into row Converting the entries in the table into row percentages or column percentages can percentages or column percentages can provide additional insight about the provide additional insight about the relationship between the two variables.relationship between the two variables.

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Row PercentagesRow Percentages


RangeRange Colonial Ranch Split A-Frame Colonial Ranch Split A-Frame TotalTotal

<< $99,000 32.73 $99,000 32.73 10.91 34.55 10.91 34.55 21.8221.82 100 100

> $99,000 26.67 31.11 35.56 6.67> $99,000 26.67 31.11 35.56 6.67 100 100

Note: Row totals are actually 100.01 due to rounding.Note: Row totals are actually 100.01 due to rounding.

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Column PercentagesColumn Percentages


RangeRange Colonial Ranch Split A-FrameColonial Ranch Split A-Frame

<< $99,000 60.00 $99,000 60.00 30.00 54.29 30.00 54.29 80.0080.00

> $99,000 40.00> $99,000 40.00 70.00 45.71 70.00 45.71 20.0020.00

TotalTotal 100 100 100 100 100 100 100100

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Scatter DiagramScatter Diagram

A scatter diagram is a graphical presentation A scatter diagram is a graphical presentation of the relationship between two of the relationship between two quantitativequantitative variables.variables.

One variable is shown on the horizontal axis One variable is shown on the horizontal axis and the other variable is shown on the vertical and the other variable is shown on the vertical axis.axis.

The general pattern of the plotted points The general pattern of the plotted points suggests the overall relationship between the suggests the overall relationship between the variables.variables.

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A Positive RelationshipA Positive Relationship

xx

yy

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A Negative RelationshipA Negative Relationship

xx

yy

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No Apparent RelationshipNo Apparent Relationship

xx

yy

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Example: Panthers Football TeamExample: Panthers Football Team


The Panthers football team is interested The Panthers football team is interested in investigating the relationship, if any, in investigating the relationship, if any, between interceptions made and points between interceptions made and points scored.scored.

xx = Number of = Number of yy = Number of = Number of InterceptionsInterceptions Points ScoredPoints Scored

11 14 14 33 24 24 22 18 18 11 17 17 33 27 27

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yy

xx

Number of InterceptionsNumber of Interceptions11 22 33

Nu

mb

er

of

Poin

ts S

core

dN

um

ber

of

Poin

ts S

core

d

00

551010

1515202025253030

00

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The preceding scatter diagram indicates a The preceding scatter diagram indicates a positive relationship between the number of positive relationship between the number of interceptions and the number of points scored.interceptions and the number of points scored.

Higher points scored are associated with a Higher points scored are associated with a higher number of interceptions.higher number of interceptions.

The relationship is not perfect; all plotted The relationship is not perfect; all plotted points in the scatter diagram are not on a points in the scatter diagram are not on a straight line.straight line.

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Using Excel’s Chart WizardUsing Excel’s Chart Wizardto Construct a Scatter Diagramto Construct a Scatter Diagram


A B C

1Number of

InterceptionsNumber of

Points Scored 2 1 14 3 3 24 4 2 18 5 1 17 6 3 27 7

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Producing a Scatter DiagramProducing a Scatter Diagram

Step 1 Step 1 Select cells A1:B6Select cells A1:B6

Step 2Step 2 Select the Select the Chart WizardChart Wizard

Step 3Step 3 When the When the Chart TypeChart Type dialog box appears: dialog box appears:

Choose Choose XY (Scatter)XY (Scatter) in the Chart type in the Chart type listlist

Choose Choose ScatterScatter from the Chart sub-type from the Chart sub-type displaydisplay






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Producing a Scatter DiagramProducing a Scatter DiagramStep 5Step 5 When the When the Chart OptionsChart Options dialog box dialog box appears:appears:

Select the Select the TitlesTitles tab and then tab and then Delete Delete Number of Points ScoredNumber of Points Scored in the in the Chart title box Chart title box Enter Enter Number of InterceptionsNumber of Interceptions in the in the

Value Value (X) (X) axis box axis box Enter Enter Number of Points ScoredNumber of Points Scored in the in the

Value Value (Y) (Y) axis box axis box Select the Select the LegendLegend tab and then tab and then Remove the check in the Remove the check in the Show LegendShow Legend box box Select Select Next >Next > … … continuedcontinued


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Producing a Scatter DiagramProducing a Scatter Diagram



Select Select FinishFinish to display the to display the scatter diagramscatter diagram


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A B C 8 9

10 11 12 13 14 15 16 17 18 19 20

Scatter Diagram for the Panthers

0

5

10

15

20

25

30

0 1 2 3 4Number of Interceptions

Nu

mb

er

of

Po

ints

Sco

red

.

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Tabular and Graphical ProceduresTabular and Graphical Procedures

DataData

Qualitative DataQualitative DataQualitative DataQualitative Data Quantitative DataQuantitative Data

TabularTabularMethodsMethods

TabularTabularMethodsMethods

GraphicalGraphical MethodsMethods

GraphicalGraphical MethodsMethods

•FrequencyFrequency DistributionDistribution•Rel. Freq. Dist.Rel. Freq. Dist.•% Freq. Dist.% Freq. Dist.•CrosstabulationCrosstabulation

•Bar GraphBar Graph•Pie ChartPie Chart

•FrequencyFrequency DistributionDistribution•Rel. Freq. Dist.Rel. Freq. Dist.•Cum. Freq. Dist.Cum. Freq. Dist.•Cum. Rel. Freq.Cum. Rel. Freq. Distribution Distribution •Stem-and-LeafStem-and-Leaf DisplayDisplay•CrosstabulationCrosstabulation

•Dot PlotDot Plot•HistogramHistogram•OgiveOgive•ScatterScatter DiagramDiagram

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End of Chapter 2End of Chapter 2

Slides Prepared by JOHN S. LOUCKS St. Edward’s University

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