SBE10_02a [Compatibility Mode]

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


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


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


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


AverageAverage

Above Average Above Average

AverageAverage


Average Average



PoorPoor

Excellent Excellent


AverageAverage




PoorPoor

Above Average Above Average

AverageAverage

AverageAverage

3



PoorPoor


AverageAverage


ExcellentExcellent

22

33

55

99

11

TotalTotal 2020

RatingRating FrequencyFrequency


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


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.


Relative Frequency andRelative Frequency andPercent Frequency DistributionsPercent Frequency Distributions

PoorPoor


AverageAverage


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


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.


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


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.


BelowAverage

15%

BelowAverage

15%

Average25%

Average25%

AboveAverage

45%

AboveAverage

45%

Poor10%Poor10%

Excellent5%

Excellent5%

Marada InnMarada Inn Quality RatingsQuality Ratings

Pie ChartPie Chart

7


■■ 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.


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


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.


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



■■ 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



■■ 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



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



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


•• 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



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


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

Cost ($)Cost ($)

Dot PlotDot Plot

TuneTune--up Parts Costup Parts Cost

. . . ..... .......... .. . .. . . ... . .. .. . . ..... .......... .. . .. . . ... . .. .

. .

. .. . . . . .. . . .

. .. .. .. .. . . . .. .. .. .. . .


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


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



■■ 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


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


■■ 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


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


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


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)


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


•• 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


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


18


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

Documents

SBE10_02a [Compatibility Mode]