1 1 slide © 2006 thomson/south-western chapter 2 descriptive statistics: tabular and graphical...

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© 2006 Thomson/South-Western© 2006 Thomson/South-Western

Chapter 2Chapter 2Descriptive Statistics:Descriptive Statistics:

Tabular and Graphical PresentationsTabular and Graphical PresentationsPart APart A

Summarizing Qualitative DataSummarizing Qualitative Data Summarizing Quantitative DataSummarizing Quantitative Data

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Summarizing Qualitative DataSummarizing Qualitative Data

Frequency DistributionFrequency Distribution Relative Frequency Distribution Relative Frequency Distribution Percent Frequency DistributionPercent Frequency Distribution Bar GraphBar Graph Pie ChartPie Chart

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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 nonoverlapping classes.in each of several nonoverlapping 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 nonoverlapping classes.in each of several nonoverlapping 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

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Example: Marada InnExample: Marada Inn

Guests staying at Marada Inn wereGuests staying at Marada Inn wereasked to rate the quality of their asked to rate the quality of their accommodations as being accommodations as being excellentexcellent,,above averageabove average, , averageaverage, , below averagebelow average, or, orpoorpoor. The ratings provided by a sample of 20 . The ratings provided by a sample of 20

guests are:guests are:

Below AverageBelow Average

Above AverageAbove Average


AverageAverage

Above Average Above Average

AverageAverage


Average Average



PoorPoor

Excellent Excellent


AverageAverage




PoorPoor

Above Average Above Average

AverageAverage

AverageAverage

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PoorPoorBelow AverageBelow AverageAverageAverageAbove AverageAbove AverageExcellentExcellent

22 33 55 99 11

TotalTotal 20 20

RatingRating FrequencyFrequency

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Relative Frequency andRelative Frequency andPercent Frequency DistributionsPercent Frequency Distributions

PoorPoor


AverageAverage


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

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

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PoorPoor BelowAverageBelow

AverageAverageAverage Above

Average Above

AverageExcellentExcellent

Fre

qu

en

cy

Fre

qu

en

cy

RatingRating

Bar GraphBar Graph

1122

33

44

55

66

77

88

991010

Marada Inn Quality RatingsMarada Inn Quality Ratings

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BelowAverage 15%

BelowAverage 15%

Average 25%Average 25%

AboveAverage 45%

AboveAverage 45%

Poor10%Poor10%

Excellent 5%Excellent 5%

Marada InnMarada Inn Quality Quality RatingsRatings

Marada InnMarada Inn Quality Quality RatingsRatings

Pie ChartPie Chart

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

Example: Marada InnExample: Marada Inn

• 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.

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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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Example: Hudson Auto RepairExample: Hudson Auto Repair

The manager of Hudson AutoThe manager of Hudson Auto

would like to have a betterwould like to have a better

understanding of the costunderstanding of the cost

of parts used in the engineof parts used in the engine

tune-ups performed in thetune-ups performed in the

shop. She examines 50shop. She examines 50

customer invoices for tune-ups. The costs of 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.

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Example: Hudson Auto RepairExample: Hudson Auto Repair

Sample of Parts Cost for 50 Tune-upsSample of Parts Cost for 50 Tune-ups

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

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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 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 classesSmaller data sets usually require fewer classes

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Guidelines for Selecting Width of ClassesGuidelines for Selecting Width of Classes

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 =

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For Hudson Auto Repair, if we choose six classes: For Hudson Auto Repair, 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

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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

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

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• 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 Percent Frequency Insights Gained from the Percent Frequency DistributionDistribution


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Dot PlotDot Plot

One of the simplest graphical summaries of One of the simplest graphical summaries of data is a data is a dot plotdot plot..

A horizontal axis shows the range of data A horizontal axis shows the range of data values.values.

Then each data value is represented by a dot Then each data value is represented by a dot placed above the axis.placed above the axis.

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5050 6060 7070 8080 9090 100100 1101105050 6060 7070 8080 9090 100100 110110

Cost ($)Cost ($)

Dot PlotDot Plot

Tune-up Parts CostTune-up Parts Cost

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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

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SymmetricSymmetric

• 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

HistogramHistogram

Rela

tive F

req

uen

cyR

ela

tive F

req

uen

cy

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

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HistogramHistogram

Moderately Skewed LeftModerately Skewed Left

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

• Example: exam scoresExample: exam scoresR

ela

tive F

req

uen

cyR

ela

tive F

req

uen

cy

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

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Moderately Right SkewedModerately Right Skewed

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

• Example: housing valuesExample: housing values

HistogramHistogram

Rela

tive F

req

uen

cyR

ela

tive F

req

uen

cy

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

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HistogramHistogram

Highly Skewed RightHighly Skewed Right

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

• Example: executive salariesExample: executive salaries

Rela

tive F

req

uen

cyR

ela

tive F

req

uen

cy

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

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

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

Cumulative DistributionsCumulative Distributions

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

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Cumulative DistributionsCumulative Distributions

Hudson Auto RepairHudson 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 + 1313

15/5015/50 .30(10.30(100)0)

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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. Use the midpoint of the Use the midpoint of the classclass.. The plotted points are connected by straight The plotted points are connected by straight lines.lines.

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

2020

4040

6060

8080

100100

Cu

mu

lati

ve P

erc

en

t Fr

eq

uen

cyC

um

ula

tive P

erc

en

t Fr

eq

uen

cyC

um

ula

tive P

erc

en

t Fr

eq

uen

cyC

um

ula

tive P

erc

en

t Fr

eq

uen

cy

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 withOgive with

Cumulative Percent Frequencies Cumulative Percent Frequencies

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

1 1 slide © 2006 thomson/south-western chapter 2 descriptive statistics: tabular and graphical...

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average average average

frequency distribution

quantitative data slide

ratingfrequency slide

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original data

tabular summary of data

number of items data