1
11SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
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
22SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
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
2
33SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
A A frequency distributionfrequency distribution is a tabular summary ofis a tabular summary ofdata showing the frequency (or number) of itemsdata showing the frequency (or number) of itemsin each of several nonoverlapping classes.in each of several nonoverlapping classes.
The objective is to The objective is to provide insightsprovide insights about the dataabout the datathat cannot be quickly obtained by looking only atthat cannot be quickly obtained by looking only atthe original data.the original data.
Frequency DistributionFrequency Distribution
44SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
Example: Marada InnExample: Marada Inn
Guests staying at Marada Inn wereGuests staying at Marada Inn were
asked 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, or
poorpoor. The ratings provided by a sample of 20 guests are:. The 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
AverageAverage
3
55SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
Frequency DistributionFrequency Distribution
PoorPoor
Below AverageBelow Average
AverageAverage
Above AverageAbove Average
ExcellentExcellent
22
33
55
99
11
TotalTotal 2020
RatingRating FrequencyFrequency
66SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
The The relative frequencyrelative frequency of a class is the fraction orof a class is the fraction orproportion of the total number of data itemsproportion of the total number of data itemsbelonging to the class.belonging to the class.
A A relative frequency distributionrelative frequency distribution is a tabularis a tabularsummary of a set of data showing the relativesummary of a set of data showing the relativefrequency for each class.frequency for each class.
Relative Frequency DistributionRelative Frequency Distribution
4
77SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
Percent Frequency DistributionPercent Frequency Distribution
The The percent frequencypercent frequency of a class is the relativeof a class is the relativefrequency multiplied by 100.frequency multiplied by 100.
AA percent frequency distributionpercent frequency distribution is a tabularis a tabularsummary of a set of data showing the percentsummary of a set of data showing the percentfrequency for each class.frequency for each class.
88SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
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 1.001.00
1010
1515
2525
4545
55
100100
RelativeRelative
FrequencyFrequency
PercentPercent
FrequencyFrequencyRatingRating
.10(100) = 10.10(100) = 10
1/20 = .051/20 = .05
5
99SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
Bar GraphBar Graph
�� A A bar graphbar graph is a graphical device for depictingis a graphical device for depictingqualitative data.qualitative data.
�� On one axis (usually the horizontal axis), we specifyOn one axis (usually the horizontal axis), we specifythe 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 frequencyscale can be used for the other axis (usually thescale can be used for the other axis (usually thevertical axis).vertical axis).
�� Using a Using a bar of fixed widthbar of fixed width drawn above each classdrawn above each classlabel, we extend the height appropriately.label, we extend the height appropriately.
�� The The bars are separatedbars are separated to emphasize the fact that eachto emphasize the fact that eachclass is a separate category.class is a separate category.
1010SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
PoorPoor BelowAverageBelow
AverageAverageAverage Above
AverageAbove
AverageExcellentExcellent
Fre
qu
ency
Fre
qu
ency
RatingRating
Bar GraphBar Graph
11
22
33
44
55
66
77
88
99
1010Marada Inn Quality RatingsMarada Inn Quality Ratings
6
1111SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
Pie ChartPie Chart
�� The The pie chartpie chart is a commonly used graphical deviceis a commonly used graphical devicefor presenting relative frequency distributions forfor presenting relative frequency distributions forqualitative data.qualitative data.
■■ First draw a First draw a circlecircle; then use the relative; then use the relative
frequencies to subdivide the circlefrequencies to subdivide the circle
into sectors that correspond to theinto sectors that correspond to the
relative frequency for each class.relative frequency for each class.
■■ Since there are 360 degrees in a circle, Since there are 360 degrees in a circle,
a class with a relative frequency of .25 woulda class with a relative frequency of .25 would
consume .25(360) = 90 degrees of the circle.consume .25(360) = 90 degrees of the circle.
1212SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
BelowAverage
15%
BelowAverage
15%
Average25%
Average25%
AboveAverage
45%
AboveAverage
45%
Poor10%Poor10%
Excellent5%
Excellent5%
Marada InnMarada Inn Quality RatingsQuality Ratings
Pie ChartPie Chart
7
1313SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
■■ Insights Gained from the Preceding Pie ChartInsights Gained from the Preceding Pie Chart
Example: Marada InnExample: Marada Inn
•• OneOne--half of the customers surveyed gave Maradahalf of the customers surveyed gave Maradaa 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 mightplease 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 shoulddisplease the manager.displease the manager.
1414SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
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
8
1515SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
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
tunetune--ups performed in theups performed in the
shop. She examines 50shop. She examines 50
customer invoices for tunecustomer invoices for tune--ups. The costs of parts,ups. The costs of parts,
rounded to the nearest dollar, are listed on the nextrounded to the nearest dollar, are listed on the next
slide.slide.
1616SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
Example: Hudson Auto RepairExample: Hudson Auto Repair
■■ Sample of Parts Cost for 50 TuneSample of Parts Cost for 50 Tune--upsups
91 78 93 57 75 52 99 80 97 62
71 69 72 89 66 75 79 75 72 76
104 74 62 68 97 105 77 65 80 109
85 97 88 68 83 68 71 69 67 74
62 82 98 101 79 105 79 69 62 73
9
1717SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
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 elementsusually 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
1818SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
Frequency DistributionFrequency Distribution
■■ Guidelines for Selecting Width of ClassesGuidelines for Selecting Width 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 =
10
1919SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
Frequency DistributionFrequency Distribution
For Hudson Auto Repair, if we choose six classes: For Hudson Auto Repair, if we choose six classes:
5050--5959
6060--69 69
7070--7979
8080--8989
9090--9999
100100--109109
22
1313
1616
77
77
55
Total 50Total 50
Parts Cost ($)Parts Cost ($) FrequencyFrequency
Approximate Class Width = (109 Approximate Class Width = (109 -- 52)/6 = 9.5 52)/6 = 9.5 ≅≅ 1010
2020SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
Relative Frequency andRelative Frequency andPercent Frequency DistributionsPercent Frequency Distributions
5050--5959
6060--69 69
7070--7979
8080--8989
9090--9999
100100--109109
PartsParts
Cost ($)Cost ($)
.04.04
.26.26
.32.32
.14.14
.14.14
.10.10
Total 1.00 Total 1.00
RelativeRelative
FrequencyFrequency
44
2626
3232
1414
1414
1010
100 100
PercentPercent
FrequencyFrequency
2/502/50 .04(100).04(100)
11
2121SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
•• Only 4% of the parts costs are in the $50Only 4% of the parts costs are in the $50--59 class.59 class.
•• The greatest percentage (32% or almost oneThe greatest percentage (32% or almost one--third)third)of the parts costs are in the $70of the parts costs are in the $70--79 class.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 $100 or more.
■■ Insights Gained from the Percent Frequency Insights Gained from the Percent Frequency DistributionDistribution
Relative Frequency andRelative Frequency andPercent Frequency DistributionsPercent Frequency Distributions
2222SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
Dot PlotDot Plot
■■ One of the simplest graphical summaries of data is a One of the simplest graphical summaries of data is a dot plotdot plot..
■■ A horizontal axis shows the range of data values.A horizontal axis shows the range of data values.
■■ Then each data value is represented by a dot placed Then each data value is represented by a dot placed above the axis.above the axis.
12
2323SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
5050 6060 7070 8080 9090 100100 1101105050 6060 7070 8080 9090 100100 110110
Cost ($)Cost ($)
Dot PlotDot Plot
TuneTune--up Parts Costup Parts Cost
. . . ..... .......... .. . .. . . ... . .. .. . . ..... .......... .. . .. . . ... . .. .
. .
. .. . . . . .. . . .
. .. .. .. .. . . . .. .. .. .. . .
2424SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
HistogramHistogram
�� Another common graphical presentation ofAnother common graphical presentation ofquantitative data is a quantitative data is a histogramhistogram..
�� The variable of interest is placed on the horizontalThe variable of interest is placed on the horizontalaxis.axis.
�� A rectangle is drawn above each class interval withA rectangle is drawn above each class interval withits height corresponding to the interval’s its height corresponding to the interval’s frequencyfrequency,,relative frequencyrelative frequency, or , or percent frequencypercent frequency..
�� Unlike a bar graph, a histogram has Unlike a bar graph, a histogram has no naturalno naturalseparation between rectanglesseparation between rectangles of adjacent classes.of adjacent classes.
13
2525SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
HistogramHistogram
22
44
66
88
1010
1212
1414
1616
1818
PartsCost ($)Parts
Cost ($)
Fre
qu
ency
Fre
qu
ency
50−59 60−69 70−79 80−89 90−99 100-11050−59 60−69 70−79 80−89 90−99 100-110
TuneTune--up Parts Costup Parts Cost
2626SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
■■ 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
Rel
ativ
e F
req
uen
cyR
elat
ive
Fre
qu
ency
.05.05
.10.10
.15.15
.20.20
.25.25
.30.30
.35.35
00
14
2727SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
HistogramHistogram
■■ Moderately Skewed LeftModerately Skewed Left
•• A longer tail to the leftA longer tail to the left
•• Example: exam scoresExample: exam scoresR
elat
ive
Fre
qu
ency
Rel
ativ
e F
req
uen
cy
.05.05
.10.10
.15.15
.20.20
.25.25
.30.30
.35.35
00
2828SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
■■ Moderately Right SkewedModerately Right Skewed
•• A Longer tail to the rightA Longer tail to the right
•• Example: housing valuesExample: housing values
HistogramHistogram
Rel
ativ
e F
req
uen
cyR
elat
ive
Fre
qu
ency
.05.05
.10.10
.15.15
.20.20
.25.25
.30.30
.35.35
00
15
2929SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
HistogramHistogram
■■ Highly Skewed RightHighly Skewed Right
•• A very long tail to the rightA very long tail to the right
•• Example: executive salariesExample: executive salaries
Rel
ativ
e F
req
uen
cyR
elat
ive
Fre
qu
ency
.05.05
.10.10
.15.15
.20.20
.25.25
.30.30
.35.35
00
3030SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
Cumulative frequency distributionCumulative frequency distribution −− shows theshows thenumber of items with values less than or equal tonumber of items with values less than or equal tothe upper limit of each class.the upper limit of each class.
Cumulative relative frequency distributionCumulative relative frequency distribution –– showsshowsthe proportion of items with values less than orthe proportion of items with values less than orequal 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 –– showsshowsthe percentage of items with values less than orthe percentage of items with values less than orequal to the upper limit of each class.equal to the upper limit of each class.
16
3131SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
Cumulative DistributionsCumulative Distributions
■■ Hudson Auto RepairHudson Auto Repair
<< 5959
<< 69 69
<< 7979
<< 8989
<< 9999
<< 109109
Cost ($)Cost ($)CumulativeCumulativeFrequencyFrequency
CumulativeCumulativeRelativeRelative
FrequencyFrequency
CumulativeCumulativePercentPercent
FrequencyFrequency
22
1515
3131
3838
4545
50 50
.04.04
.30.30
.62.62
.76.76
.90.90
1.00 1.00
44
3030
6262
7676
9090
100 100
2 + 132 + 13 15/5015/50 .30(100).30(100)
3232SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
OgiveOgive
■■ An An ogiveogive is a graph of a cumulative distribution.is a graph of a cumulative distribution.
■■ The data values are shown on the horizontal axis.The data values are shown on the horizontal axis.
■■ Shown on the vertical axis are the:Shown on the vertical axis are the:
•• cumulative frequencies, orcumulative frequencies, or
•• cumulative relative frequencies, orcumulative relative frequencies, or
•• cumulative percent frequenciescumulative percent frequencies
■■ The frequency (one of the above) of each class is The frequency (one of the above) of each class is plotted as a point.plotted as a point.
■■ The plotted points are connected by straight lines.The plotted points are connected by straight lines.
17
3333SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
•• Because the class limits for the partsBecause the class limits for the parts--cost data are cost data are 5050--59, 6059, 60--69, and so on, there appear to be one69, and so on, there appear to be one--unit unit gaps from 59 to 60, 69 to 70, and so on.gaps from 59 to 60, 69 to 70, and so on.
OgiveOgive
•• These gaps are eliminated by plotting points These gaps are eliminated by plotting points halfway between the class limits.halfway between the class limits.
•• Thus, 59.5 is used for the 50Thus, 59.5 is used for the 50--59 class, 69.5 is used 59 class, 69.5 is used for the 60for the 60--69 class, and so on.69 class, and so on.
■■ Hudson Auto RepairHudson Auto Repair
3434SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
PartsPartsCost ($)Cost ($)PartsParts
Cost ($)Cost ($)
2020
4040
6060
8080
100100
Cu
mu
lati
ve
Per
cen
t F
req
uen
cyC
um
ula
tiv
e P
erce
nt
Fre
qu
ency
Cu
mu
lati
ve
Per
cen
t F
req
uen
cyC
um
ula
tiv
e P
erce
nt
Fre
qu
ency
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, 76)(89.5, 76)
Ogive withOgive with
Cumulative Percent Frequencies Cumulative Percent Frequencies
TuneTune--up Parts Costup Parts Cost
18
3535SlideSlide© 2005 Thomson/South© 2005 Thomson/South--WesternWestern
End of Chapter 2, Part AEnd of Chapter 2, Part A