1 1 slide © university of minnesota-duluth, summer 2009-econ-2030(dr. tadesse) chapter 2...

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© University of Minnesota-Duluth, Summer 2009-Econ-2030(Dr. Tadesse)

Chapter 2

Descriptive Statistics

Descriptive Statistics

Learning Objectives:1. Learn how to construct (procedures) and interpret summarized

qualitative and quantitative data using : frequency and relative frequency distributions, bar graphs and pie charts, a dot plot, a histogram, and an ogive

2. Learn how the shape of a data distribution (negatively skewed, symmetric, and positively skewed).

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Descriptive Statistics:

Tabular and Graphical Presentations

Part A

Exploratory Data AnalysisCross-tabulations; Scatter

DiagramsPart-B

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Tabular and Graphical Presentations:

Visual Description of Data

Part A

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Tabular and Graphical Procedures

Qualitative DataQualitative Data Quantitative DataQuantitative Data

TabularMethods TabularMethods

TabularMethods TabularMethods

Graphical MethodsGraphical Methods

Graphical MethodsGraphical Methods

• Frequency Distribution• Rel. Freq. Dist.• Percent Freq. Distribution• Crosstabulation

• Bar Graph• Pie Chart

• Frequency Distribution• Rel. Freq. Dist.• Cum. Freq. Dist.• Cum. Rel. Freq. Distribution • Stem-and-Leaf Display• Crosstabulation

• Dot Plot• Histogram• Ogive• Scatter Diagram

Two Types of DataTwo Types of Data

Overview of Tabular and Graphical Methods

Sum

marizin

g D

ata

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Summarizing Qualitative Data

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2.1) Summarizing Qualitative Data

1. Frequency Distribution2. Relative Frequency Distribution 3. Percent Frequency Distribution

4. Bar Graph5. Pie Chart

Graphical Methods

Tabular Methods

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The objective is to provide insights about the data that cannot be quickly obtained by looking at the original data.

The objective is to provide insights about the data that cannot be quickly obtained by looking at the original data.

2.1) Summarizing Qualitative Data

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A frequency distribution is a tabular summary of data showing the frequency (or number) of items presented in several non-overlapping classes.

A frequency distribution is a tabular summary of data showing the frequency (or number) of items presented in several non-overlapping classes.

2.1. Frequency Distribution

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Example: Days Inn

Guests staying at Days Inn were asked 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 Average Above Average Above Average Average Above Average Average Above Average

Average Above Average Below Average Poor Excellent Above Average Average

Above Average Above Average Below Average Poor Above Average Average

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Frequency Distribution

PoorBelow AverageAverageAbove AverageExcellent

2 3 5 9 1

Total 20

Rating Frequency

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The relative frequency is the fraction or proportion of the total number of data items belonging to a given class (category).

The relative frequency is the fraction or proportion of the total number of data items belonging to a given class (category).

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

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

Relative Frequency Distribution

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Percent Frequency Distribution

The percent frequency of a class is the relative frequency multiplied by 100. The percent frequency of a class is the relative frequency multiplied by 100.

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

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

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Relative Frequency andPercent Frequency Distributions


.10 .15 .25 .45 .05

Total 1.00

10 15 25 45 5 100

RelativeFrequency

PercentFrequencyRating

.10(100) = 10

1/20 = .05

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Bar Graph

A bar graph is a graphical device for depicting qualitative data.

On one axis (usually the horizontal axis), we specify the labels that represent each of class (Category). 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.

The bars are separated to emphasize the fact that each class is a separate category.

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Recall The Frequency Distribution for Days Inn


2 3 5 9 1

Total 20

Rating Frequency

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PoorPoor BelowAverageBelow

AverageAverageAverage Above

Average Above

AverageExcellentExcellent

Fre

qu

en

cy

Fre

qu

en

cy

RatingRating

Bar Graph

1122

33

44

55

66

77

88

991010 Days Inn Quality Ratings

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Pie 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.

Example: As 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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Recall The Relative Frequency for Days Inn


.10 .15 .25 .45 .05

Total 1.00

RelativeFrequencyRating

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BelowAverage 15%

BelowAverage 15%

Average 25%Average 25%

AboveAverage 45%

AboveAverage 45%

Poor10%Poor10%

Excellent 5%Excellent 5%

Days Inn Quality RatingsDays Inn Quality Ratings

Pie Chart

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Some Insights about Days Inn—from the Preceding Pie Chart

Example: Days Inn

• One-half of the customers surveyed gave Days Inn 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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Summarizing Quantitative Data

Summarizing Data

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2.2) Summarizing Quantitative Data

1. Frequency Distribution2. Relative Frequency and Percent Frequency

Distributions

3. Dot Plot4. Histogram5. Cumulative Distributions6. Ogive

Tabular Methods

Graphical Methods

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Three Important things to remember when working on Frequency Distribution for Quantitative Data:

1. Determine the number of non-overlapping classes (categories)—Quantitative data do not naturally come in categories

2. Determine the Width of each class (Equal class widths are preferred)

3. Specify the Class Limits: (a given observation should belong to one and only one class)

2.2.1) Frequency Distribution

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In Determining the Number of CLASSES

• Use a thumb rule of 5 to 20 classes.

• Larger data sets usually require a largernumber of classes.

• Smaller data sets usually require fewer classes

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To Determine the WIDTH of Classes:

Largest Data Value Smallest Data ValueNumber of Classes

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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

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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

2 13 16 7 7 5Total 50

Parts Cost ($) Frequency

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

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2.2.2) Relative Frequency andPercent Frequency Distributions

50-59

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

100-109

PartsCost ($)

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

Total 1.00

RelativeFrequency

4 26 32 14 14 10

100

Percent Frequency

2/50 .04(100)

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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.

Some Insights ….on the Parts Cost Data

2.2.2) Relative Frequency andPercent Frequency Distributions

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2.2.3) Dot Plot

Is one of the simplest graphical summaries of data. It has two components:

A horizontal axis that shows the range of data values.

And a dot placed above the axis, that represents each data value.

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

Cost ($)Cost ($)

Tune-up Parts Cost

2.2.3) Dot Plot

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2.2.4) Histogram

Histogram is a common graphical presentation of quantitative data .

In describing data using a Histogram, place the variable of interest on the horizontal axis.

Then we draw a rectangle 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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2.2.4) Histogram

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44

66

88

1010

1212

1414

1616

1818

PartsCost ($) PartsCost ($)

Fre

qu

en

cy

Fre

qu

en

cy

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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Shapes of Histogram

Depending upon the data set, we may see different

shapes when we summarize a data using histogram.

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

Various Shapes of Histogram

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.05.05

.10.10

.15.15

.20.20

.25.25

.30.30

.35.35

00

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2. Moderately Skewed Left• A longer tail to the left• Example: exam scoresR

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tive F

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.05.05

.10.10

.15.15

.20.20

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


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.05.05

.10.10

.15.15

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

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.05.05

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1. Cumulative frequency distribution-shows the number of items with values less than or equal to the upper limit of each class..

1. Cumulative frequency distribution-shows the number of items with values less than or equal to the upper limit of each class..

2. Cumulative relative frequency distribution – shows the proportion of items with values less thanor equal to the upper limit of each class.

2. Cumulative relative frequency distribution – shows the proportion of items with values less thanor equal to the upper limit of each class.

2.2.5) Cumulative Distributions

3. Cumulative percent frequency distribution – shows the percentage of items with values less than or equal to the upper limit of each class.

3. Cumulative percent frequency distribution – shows the percentage of items with values less than or equal to the upper limit of each class.

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Example

Recall the Hudson Auto Repair Data:

50-59

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

100-109

2 13 16 7 7 5Total 50

Parts Cost ($) Frequency

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2.2.5) Cumulative Distributions

Hudson Auto Repair

< 59

< 69 < 79 < 89 < 99

< 109

Cost ($) CumulativeFrequency

CumulativeRelative

Frequency

CumulativePercent

Frequency

2 15 31 38 45

50

.04 .30 .62 .76 .90

1.00

4 30 62 76 90

100

2 + 13

15/50 .30(100)

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2.2.6) Ogive

An ogive is a graph of a cumulative distribution. Can be constructed using one of the following

Shown on the vertical axis are the:• cumulative frequencies, or• cumulative relative frequencies, or• cumulative percent frequencies

The plotted points are connected by straight lines.

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PartsCost ($) PartsCost ($)

2020

4040

6060

8080

100100

Cu

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erc

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eq

uen

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um

ula

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eq

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

(89.5, 76)

Example: Ogive with Cumulative Percent Frequencies

Tune-up Parts CostTune-up Parts Cost

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