© 2006 by Thomson Learning, a division of Thomson Asia Pte Ltd.. 1 Slide Slide Slides Prepared by...

© 2006 by Thomson Learning, a division of Thomson Asia Pte Ltd.. 11 11 SlideSlide

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Slides Prepared bySlides Prepared by

Juei-Chao ChenJuei-Chao ChenFu Jen Catholic UniversityFu Jen Catholic University

Slides Prepared bySlides Prepared by

Juei-Chao ChenJuei-Chao ChenFu Jen Catholic UniversityFu Jen Catholic University


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Chapter 2Chapter 2STATISTICS STATISTICS in in PRACTICEPRACTICE

• The Colgate-Palmolive Company uses statistics in its quality assurance program for home laundry detergent products.• One concern is customer satisfaction with the quantity of detergent in a carton. To control the problem of heavy detergent powder, limits are placed on the acceptable range of powder density. • Statistical samples are taken and the density of each powder sample is measured.• Data summaries are then provided for operating personnel so that corrective action can be taken if necessary to keep the density within the desired quality.


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Chapter 2Chapter 2Descriptive Statistics:Descriptive Statistics:

Tabular and Graphical PresentationsTabular and Graphical Presentations

2.1 Summarizing Qualitative Data

2.2 Summarizing Quantitative Data

2.3 Exploratory Data Analysis:

The Stem-and- Leaf Display

2.4 Crosstabulations and Scatter Diagrams


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Chapter 2Chapter 2Descriptive Statistics:Descriptive Statistics:

Tabular and Graphical PresentationsTabular and Graphical PresentationsPart APart A

• Summarizing Qualitative Data

• Summarizing Quantitative Data


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2.1 Summarizing Qualitative Data2.1 Summarizing Qualitative Data

• Frequency Distribution

• Relative Frequency Distributions

• Percent Frequency Distributions

• Bar Graphs

• Pie Charts


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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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Frequency DistributionFrequency Distribution• Example: Data from a sample of 50 Soft Drink

Purchases

• Frequency Distribution Soft Drink FrequencyCoke Classic 19Diet Coke 8Dr. Pepper 5Pepsi-Cola 13Sprite 5 Total 50


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Example: Marada InnExample: Marada Inn

Guests staying at Marada Inn wereasked to rate the quality of their accommodations as being excellent,above average, average, below average, or poor.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


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PoorPoorBelow AverageBelow AverageAverageAverageAbove AverageAbove AverageExcellentExcellent

22 33 55 99 11

TotalTotal 20 20

RatingRating FrequencyFrequency


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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.

Relative FrequencyRelative Frequency

Frequency of the classRelative frequency of a class= n


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Relative Frequency DistributionsRelative Frequency Distributions

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

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

Example: Relative and Percent Frequency Distribution of Soft Drink Purchases

Soft Drink Relative FrequencyCoke Classic .38Diet Coke .16Dr. Pepper .10Pepsi-Cola .26Sprite .10 Total 1.00


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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.

Example: Percent Frequency Distribution of Soft Drink Purchases

Soft Drink Percent FrequencyCoke Classic 38Diet Coke 16Dr. Pepper 10Pepsi-Cola 26Sprite 10 Total 100


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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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Bar GraphBar Graph• A bar graph is a graphical device for depicting qualitative data.

• On one axis (usually the horizontal axis), we specify the labels that are used for each of the classes.

• A frequency, relative frequency, or percent frequency scale can be used for the other axis (usually the vertical axis).

• Using a bar of fixed width drawn above each class label, we extend the height appropriately.


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Bar GraphBar Graph• Example: Bar Graph of Soft Drink Purchases


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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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Pie ChartPie Chart

• The pie chart is a commonly used graphical device for presenting relative frequency distributions for qualitative data.

• First draw a circle; then use the relative frequencies to subdivide the circle into sectors that correspond to the

relative frequency for each class.

• Since there are 360 degrees in a circle,

• a class with a relative frequency of .25 would

consume .25(360) = 90 degrees of the circle.


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Pie ChartPie Chart

• Example: Pie Chart of Soft Drink Purchases


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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 Chart

Example: Marada InnExample: Marada Inn

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

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


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2.2 Summarizing Quantitative Data2.2 Summarizing Quantitative Data

• Frequency Distribution• Relative Frequency and Percent Frequency

Distributions• Dot Plot• Histogram• Cumulative Distributions• Ogive


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Frequency DistributionFrequency Distribution• The three steps necessary to define the classes for a frequency distribution with quantitative data are:

1. Determine the number of nonoverlapping classes. we recommend using between 5 and 20 classes. 2. Determine the width of each class.

3. Determine the class limits.

Largest data value –Smallest data valueApproximate class width= Number of classes


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• Guidelines for Selecting Number of Classes

• Use between 5 and 20 classes.

• Data sets with a larger number of elements usually require a larger number of classes.

• Smaller data sets usually require fewer classes


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• Guidelines for Selecting Width of Classes

Use classes of equal width.

Approximate Class Width =

Largest data value –Smallest data value

Number of classes


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Frequency DistributionFrequency Distribution• Example: These data show the time in days

required to complete year-end audits for a sample of 20 clients of Sanderson and Clifford, a small public accounting firm with the data rounded to the nearest day.

YEAR-END AUDITTIMES (IN DAYS)12 14 19 18 15 15 18 1720 27 22 2322 21 33 2814 18 16 13


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Frequency DistributionFrequency Distribution• Example:

1. Number of classes = 5

2. provides an approximate class width of

(33 — 12)/5= 4.2.

3. We therefore decided to round up and use a class

width of five days in the frequency distribution.

4. FREQUENCY

DISTRIBUTION

FOR THE

AUDIT TIMES DATA

Audit Time Frequency (days) 10-14 4 15-19 8 20-24 5 25-29 2 30-34 1 Total 20


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Example: Hudson Auto RepairExample: Hudson Auto Repair

The manager of Hudson Auto

would like to have a better

understanding of the cost

of parts used in the engine

tune-ups performed in the

shop. She examines 50

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

rounded to the nearest dollar, are listed on the next

slide.


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Example: Hudson Auto RepairExample: Hudson Auto Repair

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

50-59

60-69 70-79 80-89 90-99

100-109

22 1313 1616 77 77 55

Total 50Total 50

Parts Cost ($)Parts Cost ($) Frequency

Approximate Class Width = (109 - 52)/6 = 9.5 10


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50-59

60-69 70-79 80-89 90-99

100-109

PartsCost ($)

.04 .26 .32 .14 .14 .10

Total 1.00

RelativeRelativeFrequencyFrequency

44 2626

3232

1414

1414

1010

100 100

Percent Frequency

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


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

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

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

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

• Insights Gained from the Percent Frequency Distribution



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Dot PlotDot Plot

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

• A horizontal axis shows the range of data values.

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

• Example: Dot Plot for The Audit Time Data


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50 60 70 80 90 100 11050 60 70 80 90 100 110

Cost ($)Cost ($)

Dot PlotDot Plot

Tune-up Parts CostTune-up Parts Cost

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

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


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HistogramHistogram

• Another common graphical presentation of quantitative data is a histogram.

• The variable of interest is placed on the horizontal axis.

• A rectangle is drawn above each class interval with its height corresponding to the interval’s frequency, relative frequency, or percent frequency.

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


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HistogramHistogram

• Example: Histogram for The Audit Time Data


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HistogramHistogram

22

44

66

88

1010

1212

1414

1616

1818

PartsCost ($) PartsCost ($)

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

Tune-up Parts Cost


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• Histogram provides information about the shape.

• Symmetric

• Left tail is the mirror image of the right tail

• Examples: 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 Left• A longer tail to the left• Example: exam scores

Rel

ativ

e F

requ

ency

Rel

ativ

e F

requ

ency

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00


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• Moderately Right Skewed• A Longer tail to the right• Example: housing values

HistogramHistogram

Rel

ativ

e F

requ

ency

Rel

ativ

e F

requ

ency

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00


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HistogramHistogram

• Highly Skewed Right• A very long tail to the right• Example: executive salaries

Rel

ativ

e F

requ

ency

Rel

ativ

e F

requ

ency

.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 relative frequency distributionCumulative relative frequency distribution – shows – shows the proportion of items with values less than orthe proportion 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 proportion of items with values less than orthe proportion 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 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 frequency distributionCumulative percent frequency 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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• Example: Cumulative Frequency, Cumulative

Relative Frequency and Cumulative Percent

Frequency Distributions for the Audit Data.


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• Hudson 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

5050

.04.04

.30.30

.62.62

.76.76

.90.90

1.00 1.00

44 3030

6262

7676

9090

100100

2 + 2 + 1313

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


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OgiveOgive

• An ogive is a graph of a cumulative distribution.

• The data values are shown on the horizontal axis.

• Shown on the vertical axis are the:

• cumulative frequencies, or

• cumulative relative frequencies, or

• cumulative percent frequencies

• The frequency (one of the above) of each class is plotted as a point.

• The plotted points are connected by straight lines.


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• Because the class limits for the parts-cost data are 50-59, 60-69, and so on, there appear to be one-unit gaps from 59 to 60, 69 to 70, and so on.

OgiveOgive

• These gaps are eliminated by plotting points halfway between the class limits.

• Thus, 59.5 is used for the 50-59 class, 69.5 is used for the 60-69 class, and so on.

• Hudson Auto Repair


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OgiveOgive

• Example: Ogive for the Audit Time Data.


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

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