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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
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
AverageAverage
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Frequency DistributionFrequency Distribution
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
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
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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Frequency DistributionFrequency Distribution
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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Frequency DistributionFrequency Distribution
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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Frequency DistributionFrequency Distribution
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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Relative Frequency andRelative Frequency andPercent Frequency DistributionsPercent Frequency Distributions
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 $100 or more.
Insights Gained from the Percent Frequency Insights Gained from the Percent Frequency DistributionDistribution
Relative Frequency andRelative Frequency andPercent Frequency DistributionsPercent Frequency Distributions
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Dot PlotDot Plot
One of the simplest graphical summaries of One of the simplest graphical summaries of data is a data is a dot plotdot plot..
A horizontal axis shows the range of data A horizontal axis shows the range of data values.values.
Then each data value is represented by a dot Then each data value is represented by a dot placed above the axis.placed above the axis.
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5050 6060 7070 8080 9090 100100 1101105050 6060 7070 8080 9090 100100 110110
Cost ($)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
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