Descriptive Statistics:Tabular and Graphical Presentations

� Summarizing Qualitative Data� Summarizing Quantitative Data

Summarizing Qualitative Data

� Frequency Distribution� Relative Frequency Distribution� Cumulative Frequency� Cumulative Relative Frequency � Bar Graph� Pie Chart

Example: Marada Inn

Guests staying at Marada Inn wereasked to rate the quality of their accommodations as being excellent,above average, average, below average, orpoor. The ratings provided by a sample of 20 guests are:

Below AverageAbove AverageAbove AverageAverageAbove Average AverageAbove Average

Average Above AverageBelow AveragePoorExcellent Above AverageAverage

Above AverageAbove AverageBelow AveragePoorAbove Average Average

Frequency Distribution

PoorBelow AverageAverageAbove AverageExcellent

23591

Total 20

Rating FrequencyCumulativeFrequency

25101920 2+3+5 =10

Relative Frequency andPercent Frequency Distributions

PoorBelow AverageAverageAbove AverageExcellent

.10

.15

.25

.45

.05Total 1.00

.10

.25

.50

.951.0

RelativeFrequency

CumulativeRelativeFrequencyRating

.10+.15 =.25

1/20 = .05

PoorPoor BelowAverageBelow

AverageAverageAverage Above

AverageAbove

AverageExcellentExcellent

Freq

uenc

yFr

eque

ncy

RatingRating

Bar Graph

112233445566778899

1010Marada Inn Quality Ratings

BelowAverage

15%

BelowAverage

15%

Average25%

Average25%

AboveAverage

45%

AboveAverage

45%

Poor10%Poor10%

Excellent5%

Excellent5%

Marada Inn Quality Ratings

Pie Chart

Summarizing Quantitative Data

� Frequency Distribution� Relative Frequency Distributions� Histogram� Cumulative Distributions

Example: Hudson Auto Repair

The manager of Hudson Autowould like to have a betterunderstanding of the costof parts used in the enginetune-ups performed in theshop. She examines 50customer invoices for tune-ups. The costs of parts,rounded to the nearest dollar, are listed on the nextslide.

Example: 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

Frequency Distribution

� 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

Frequency Distribution

� Guidelines 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.•Approximate Class Width =

Frequency Distribution

For Hudson Auto Repair, if we choose six classes:

50-5960-69 70-7980-8990-99

100-109

21316775

Total 50

Parts Cost ($) Frequency

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

Relative Frequency Distribution

50-5960-69 70-7980-8990-99

100-109

PartsCost ($)

.04

.26

.32

.14

.14

.10Total 1.00

RelativeFrequency

2/50

Cumulative Distributions

� Hudson Auto Repair

50 - 5960 - 69 70 - 7980 - 8990 - 99

100 - 109

Cost ($)CumulativeFrequency

CumulativeRelative

Frequency

CumulativePercent

Frequency2

1531384550

.04

.30

.62

.76

.901.00

430627690

100

2 + 13 15/50 .30(100)

Frequency2

1316775

Histogram

22

44

66

88

1010

1212

1414

1616

1818

PartsCost ($)

PartsCost ($)

Freq

uenc

yFr

eque

ncy

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

� Symmetric• Left tail is the mirror image of the right tail• Examples: heights and weights of people

Histogram

Rela

tive

Freq

uenc

yRe

lativ

e Fr

eque

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.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

Histogram

� Moderately Skewed Left• A longer tail to the left• Example: exam scores

Rela

tive

Freq

uenc

yRe

lativ

e Fr

eque

ncy

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

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

Histogram

Rela

tive

Freq

uenc

yRe

lativ

e Fr

eque

ncy

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

Histogram

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

Rela

tive

Freq

uenc

yRe

lativ

e Fr

eque

ncy

.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

Scatter Diagram

� A Positive Relationship

x

y

Scatter Diagram

� A Negative Relationship

x

y

Scatter Diagram

� No Apparent Relationship

x

y

Example: Panthers Football Team

� Scatter DiagramThe Panthers football team is interested

in investigating the relationship, if any,between interceptions made and points scored.

13213

1424181730

x = Number ofInterceptions

y = Number ofPoints Scored

Scatter Diagram

y

x

Number of Interceptions

Num

ber o

f Poi

nts S

core

d

51015202530

0

35

1 2 30 4

Tabular and Graphical Procedures

Qualitative Data Quantitative Data

TabularMethods

TabularMethods

GraphicalMethods

GraphicalMethods

•FrequencyDistribution

•Rel. Freq. Dist.•Cumulative Freq.

Distribution•Cumulative Rel. Freq.

Distribution

•Bar Graph•Pie Chart •Frequency

Distribution•Rel. Freq. Dist.•Cum. Freq. Dist.•Cum. Rel. Freq.

Distribution

•Histogram•Scatter

Diagram

Data