Chapter 2, Part A Descriptive Statistics: Tabular and Graphical Presentations n Summarizing Categorical Data n Summarizing Quantitative Data Categorical

Chapter 2, Part AChapter 2, Part ADescriptive Statistics:Descriptive Statistics:

Tabular and Graphical PresentationsTabular and Graphical Presentations

Summarizing Categorical DataSummarizing Categorical Data Summarizing Quantitative DataSummarizing Quantitative Data

Categorical dataCategorical data use labels or names use labels or names to identify categories of like items.to identify categories of like items. Categorical dataCategorical data use labels or names use labels or names to identify categories of like items.to identify categories of like items.

Quantitative dataQuantitative data are numerical values are numerical values that indicate how much or how many.that indicate how much or how many. Quantitative dataQuantitative data are numerical values are numerical values that indicate how much or how many.that indicate how much or how many.

Summarizing Categorical DataSummarizing Categorical Data

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

DistributionDistribution Bar ChartBar Chart Pie ChartPie Chart

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

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

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

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

Frequency DistributionFrequency Distribution

Guests staying at the Tanoa were asked to rate theGuests staying at the Tanoa were asked to rate thequality 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 guests are:ratings provided by a sample of 20 guests are: Below AverageBelow Average Above AverageAbove Average Above AverageAbove Average AverageAverage Above Average Above Average AverageAverage Above AverageAbove Average

Average Average Above AverageAbove Average Below AverageBelow Average PoorPoor Excellent Excellent Above AverageAbove Average AverageAverage

Above AverageAbove Average Above AverageAbove Average Below AverageBelow Average PoorPoor Above Average Above Average AverageAverage


Example: Tanoa HotelExample: Tanoa Hotel


PoorPoorBelow AverageBelow AverageAverageAverageAbove AverageAbove AverageExcellentExcellent

22 33 55 99 11

TotalTotal 20 20

RatingRating FrequencyFrequency


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

The The relative frequencyrelative frequency of a class is the fraction or of a class is the fraction or proportion of the total number of data itemsproportion 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 relativesummary of a set of data showing the relative frequency for each class.frequency for each class.

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

Relative Frequency DistributionRelative Frequency Distribution

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. 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 percentsummary of a set of data showing the percent frequency for each class.frequency for each class.

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

Relative Frequency andRelative Frequency andPercent Frequency DistributionsPercent Frequency Distributions

PoorPoor

Below AverageBelow Average

AverageAverage

Above AverageAbove Average

ExcellentExcellent

.10.10

.15.15

.25.25

.45.45

.05.05

TotalTotal 1.00 1.00

1010

1515

2525

4545

55

100100

RelativeRelativeFrequencyFrequency

PercentPercentFrequencyFrequencyRatingRating

.10(100) = .10(100) = 1010

.10(100) = .10(100) = 1010

1/20 1/20 = .05= .051/20 1/20 = .05= .05


Bar ChartBar Chart

A A bar chartbar chart is a graphical device for depicting is a graphical device for depicting qualitative data.qualitative data.

On one axis (usually the horizontal axis), we specifyOn one axis (usually the horizontal axis), we specify the labels that are used for each of the classes.the labels that are used for each of the classes. A A frequencyfrequency, , relative frequencyrelative frequency, or , or percent frequencypercent frequency scale can be used for the other axis (usually thescale can be used for the other axis (usually the vertical axis).vertical axis). Using a Using a bar of fixed widthbar of fixed width drawn above each class drawn above each class label, we extend the height appropriately.label, we extend the height appropriately.

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

PoorPoor BelowAverageBelow

AverageAverageAverage Above

Average Above

AverageExcellentExcellent

Fre

qu

en

cy

Fre

qu

en

cy

RatingRating

Bar ChartBar Chart

1122

33

44

55

66

77

88

991010 Tanoa Hotel Quality RatingsTanoa Hotel Quality RatingsTanoa Hotel Quality RatingsTanoa Hotel Quality Ratings

Pie ChartPie Chart

The The pie chartpie chart is a commonly used graphical device is a commonly used graphical device for presenting relative frequency and percentfor presenting relative frequency and percent frequency distributions for categorical data.frequency distributions for categorical data.

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

Since there are 360 degrees in a circle, a class with aSince there are 360 degrees in a circle, a class with a

relative frequency of .25 would consume .25(360) = 90relative frequency of .25 would consume .25(360) = 90

degrees of the circle.degrees of the circle.

BelowAverage 15%

BelowAverage 15%

Average 25%Average 25%

AboveAverage 45%

AboveAverage 45%

Poor10%Poor10%

Excellent 5%Excellent 5%

Tanoa HotelQuality Tanoa HotelQuality RatingsRatings

Tanoa HotelQuality Tanoa HotelQuality RatingsRatings

Pie ChartPie Chart

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


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

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

Summarizing Quantitative DataSummarizing Quantitative Data

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

DistributionsDistributions Dot PlotDot Plot HistogramHistogram Cumulative DistributionsCumulative Distributions OgiveOgive

The manager of Nadi Auto would like to gain aThe manager of Nadi Auto would like to gain a

better understanding of the cost of parts used in better understanding of the cost of parts used in thethe

engine tune-ups performed in the shop. She engine tune-ups performed in the shop. She examinesexamines

50 customer invoices for tune-ups. The costs of 50 customer invoices for tune-ups. The costs of parts,parts,

rounded to the nearest dollar, are listed on the rounded to the nearest dollar, are listed on the nextnext

slide.slide.

Example: Nadi Auto RepairExample: Nadi Auto Repair


Sample of Parts Cost($) for 50 Tune-Sample of Parts Cost($) for 50 Tune-upsups91 78 93 57 75 52 99 80 97 62

71 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




2.2. Determine the width of each class. Determine the width of each class.

3.3. Determine the class limits. Determine the class limits.

1.1. Determine the number of non-overlapping classes. Determine the number of non-overlapping classes.

The three steps necessary to define the classes for aThe three steps necessary to define the classes for afrequency distribution with quantitative data are:frequency distribution with quantitative data are:


Guidelines for Determining the Number of ClassesGuidelines for Determining the Number of Classes

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

• Data sets with a larger number of elementsData 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 classes.Smaller data sets usually require fewer classes.

The goal is to use enough classes to show theThe goal is to use enough classes to show thevariation in the data, but not so many classesvariation in the data, but not so many classes

that some contain only a few data items.that some contain only a few data items.

The goal is to use enough classes to show theThe goal is to use enough classes to show thevariation in the data, but not so many classesvariation in the data, but not so many classes

that some contain only a few data items.that some contain only a few data items.


Guidelines for Determining the Width of Each ClassGuidelines for Determining the Width of Each Class

Largest Data Value Smallest Data ValueNumber of Classes

Largest Data Value Smallest Data ValueNumber of Classes

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

• Approximate Class Width =Approximate Class Width =

Making the classes the sameMaking the classes the samewidth reduces the chance ofwidth reduces the chance of

inappropriate interpretations.inappropriate interpretations.

Making the classes the sameMaking the classes the samewidth reduces the chance ofwidth reduces the chance of

inappropriate interpretations.inappropriate interpretations.

Note on Number of Classes and Class WidthNote on Number of Classes and Class Width

• In practice, the number of classes and theIn practice, the number of classes and the appropriate class width are determined by trialappropriate class width are determined by trial and error.and error.• Once a possible number of classes is chosen, theOnce a possible number of classes is chosen, the appropriate class width is found.appropriate class width is found.

• The process can be repeated for a differentThe process can be repeated for a different number of classes.number of classes.


• Ultimately, the analyst uses judgment toUltimately, the analyst uses judgment to determine the combination of the number ofdetermine the combination of the number of classes and class width that provides the best classes and class width that provides the best frequency distribution for summarizing the data.frequency distribution for summarizing the data.


Guidelines for Determining the Class LimitsGuidelines for Determining the Class Limits

• Class limits must be chosen so that each dataClass limits must be chosen so that each data item belongs to one and only one class.item belongs to one and only one class.

• The lower class limit identifies the smallestThe lower class limit identifies the smallest possible data value assigned to the class.possible data value assigned to the class.

• The upper class limit identifies the largestThe upper class limit identifies the largest possible data value assigned to the class.possible data value assigned to the class.

• The appropriate values for the class limitsThe appropriate values for the class limits depend on the level of accuracy of the data.depend on the level of accuracy of the data.

An An open-endopen-end class requires only a class requires only alower class limit or an upper class limit.lower class limit or an upper class limit.

An An open-endopen-end class requires only a class requires only alower class limit or an upper class limit.lower class limit or an upper class limit.


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

50-5950-59

60-69 60-69 70-7970-79 80-8980-89 90-9990-99

100-109100-109

22 1313 1616 77 77 55

Total 50Total 50

Parts Cost ($)Parts Cost ($) FrequencyFrequency

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



50-5950-59

60-69 60-69

70-7970-79

80-8980-89

90-9990-99

100-109100-109

PartsPartsCost ($)Cost ($)

.04.04

.26.26

.32.32

.14.14

.14.14

.10.10

Total 1.00 Total 1.00

RelativeRelativeFrequencyFrequency

44

2626

3232

1414

1414

1010

100 100

PercentPercent FrequencyFrequency

2/502/502/502/50 .04(10.04(100)0)

.04(10.04(100)0)


PercentPercentfrequency isfrequency isthe relativethe relativefrequencyfrequencymultipliedmultiplied by 100.by 100.

PercentPercentfrequency isfrequency isthe relativethe relativefrequencyfrequencymultipliedmultiplied by 100.by 100.

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

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

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

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

Insights Gained from the % Frequency Insights Gained from the % Frequency Distribution:Distribution:



HistogramHistogram

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

The variable of interest is placed on the horizontalThe variable of interest is placed on the horizontal axis.axis. A rectangle is drawn above each class interval withA rectangle is drawn above each class interval with its height corresponding to the interval’s its height corresponding to the interval’s frequencyfrequency,, relative frequencyrelative frequency, or , or percent frequencypercent frequency..

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

HistogramHistogram

22

44

66

88

1010

1212

1414

1616

1818

PartsCost ($) PartsCost ($)

Fre

qu

en

cy

Fre

qu

en

cy

5059 6069 7079 8089 9099 100-1105059 6069 7079 8089 9099 100-110

Tune-up Parts CostTune-up Parts Cost


SymmetricSymmetric

Histograms Showing SkewnessHistograms Showing Skewness

Rela

tive F

req

uen

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ela

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req

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.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

• Left tail is the mirror image of the right tailLeft tail is the mirror image of the right tail

• Examples: heights and weights of peopleExamples: heights and weights of people


Moderately Skewed LeftModerately Skewed Left

Rela

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.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

• A longer tail to the leftA longer tail to the left

• Example: exam scoresExample: exam scores

Moderately Right SkewedModerately Right Skewed


Rela

tive F

req

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ela

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req

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.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

• A Longer tail to the rightA Longer tail to the right

• Example: housing valuesExample: housing values


Highly Skewed RightHighly Skewed RightR

ela

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.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

• A very long tail to the rightA very long tail to the right

• Example: executive salariesExample: executive salaries

Cumulative frequency distributionCumulative frequency distribution shows the shows the numbernumber of items with values less than or equal to the of items with values less than or equal to the upper limit of each class..upper limit of each class..

Cumulative frequency distributionCumulative frequency distribution shows the shows the numbernumber of items with values less than or equal to the of items with values less than or equal to the upper limit of each class..upper limit of each class..

Cumulative relative frequency distributionCumulative relative frequency distribution – shows – shows the the proportionproportion of items with values less than or of items with values less than or equal to the upper limit of each class.equal to the upper limit of each class.

Cumulative relative frequency distributionCumulative relative frequency distribution – shows – shows the the proportionproportion of items with values less than or of items with values less than or equal to the upper limit of each class.equal to the upper limit of each class.

Cumulative DistributionsCumulative Distributions

Cumulative percent frequency distributionCumulative percent frequency distribution – shows – shows the the percentagepercentage of items with values less than or of items with values less than or equal to the upper limit of each class.equal to the upper limit of each class.

Cumulative percent frequency distributionCumulative percent frequency distribution – shows – shows the the percentagepercentage of items with values less than or of items with values less than or equal to the upper limit of each class.equal to the upper limit of each class.


The last entry in a cumulative frequency distributionThe last entry in a cumulative frequency distribution always equals the total number of observations.always equals the total number of observations.

The last entry in a cumulative relative frequencyThe last entry in a cumulative relative frequency distribution always equals 1.00.distribution always equals 1.00. The last entry in a cumulative percent frequencyThe last entry in a cumulative percent frequency distribution always equals 100.distribution always equals 100.


Nadi Auto RepairNadi Auto Repair

<< 59 59

<< 69 69

<< 79 79

<< 89 89

<< 99 99

<< 109 109

Cost ($)Cost ($) CumulativeCumulativeFrequencyFrequency

CumulativeCumulativeRelativeRelative

FrequencyFrequency

CumulativeCumulativePercentPercent

FrequencyFrequency 22

1515

3131

3838

4545

50 50

.04.04

.30.30

.62.62

.76.76

.90.90

1.00 1.00

44

3030

6262

7676

9090

100 100

2 + 2 + 13132 + 2 + 1313 15/5015/5015/5015/50 .30(10.30(10

0)0).30(10.30(10

0)0)

OgiveOgive

An An ogiveogive is a graph of a cumulative is a graph of a cumulative distribution.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.

• 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.

OgiveOgive

• 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.

Nadi Auto RepairNadi Auto Repair

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

2020

4040

6060

8080

100100

Cu

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eq

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um

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um

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eq

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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

(89.5, (89.5, 76)76)

Ogive with Cumulative Percent Ogive with Cumulative Percent Frequencies Frequencies

Tune-up Parts CostTune-up Parts CostTune-up Parts CostTune-up Parts Cost


End of Chapter 2, Part AEnd of Chapter 2, Part A

Documents

Chapter 2, Part A Descriptive Statistics: Tabular and Graphical Presentations n Summarizing Categorical Data n Summarizing Quantitative Data Categorical