Chapter 2 Descriptive Statistics: Tabular and Graphical Methods

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Chapter 2Descriptive StatisticsTabular and
Graphical Methods

• Summarizing Qualitative Data
• Summarizing Quantitative Data
• Exploratory Data Analysis
• Crosstabulations
• and Scatter Diagrams

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Summarizing Qualitative Data

• Frequency Distribution
• Relative Frequency
• Percent Frequency Distribution
• Bar Graph
• Pie Chart

3
Frequency Distribution

• A frequency distribution is a tabular summary of
  data showing the frequency (or number) of items
  in each of several nonoverlapping classes.
• The objective is to provide insights about the
  data that cannot be quickly obtained by looking
  only at the original data.

4
Example Marada Inn

• Guests staying at Marada Inn were asked 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 quests are
  shown
• below.
• Below Average Average Above Average
• Above Average Above Average Above
  Average Above Average Below Average Below
  Average Average Poor Poor
• Above Average Excellent Above Average
• Average Above Average Average
• Above Average Average

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Example Marada Inn

• Frequency Distribution
• Rating Frequency
• Poor 2
• Below Average 3
• Average 5
• Above Average 9
• Excellent 1
• Total 20

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Relative Frequency Distribution

• The relative frequency of a class is the fraction
  or proportion of the total number of data items
  belonging to the class.
• A relative frequency distribution is a tabular
  summary of a set of data showing the relative
  frequency for each class.

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Percent Frequency Distribution

• 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.

8
Example Marada Inn

• Relative Frequency and Percent Frequency
  Distributions
• Relative Percent
• Rating Frequency Frequency
• Poor .10 10
• Below Average .15 15
• Average .25 25
• Above Average .45 45
• Excellent .05 5
• Total 1.00 100

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Bar Graph

• A bar graph is a graphical device for depicting
  qualitative data.
• On 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 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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Example Marada Inn

• Bar Graph

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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.
• 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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Example Marada Inn

• Pie Chart

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Example Marada Inn

• Insights Gained from the Preceding Pie Chart
• 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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Summarizing Quantitative Data

• Frequency Distribution
• Relative Frequency and Percent Frequency
  Distributions
• Dot Plot
• Histogram
• Cumulative Distributions
• Ogive

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Example Hudson Auto Repair

• The manager of Hudson Auto would like to get a
• better picture of the distribution of costs for
  engine
• tune-up parts. A sample of 50 customer invoices
  has
• been taken and the costs of parts, rounded to the
• nearest dollar, are listed below.

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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.

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Frequency Distribution

• Guidelines for Selecting Width of Classes
• Use classes of equal width.
• Approximate Class Width

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Example Hudson Auto Repair

• Frequency Distribution
• If we choose six classes
• Approximate Class Width (109 - 52)/6 9.5
  ??10
• Cost () Frequency
• 50-59 2
• 60-69 13
• 70-79 16
• 80-89 7
• 90-99 7
• 100-109 5
• Total 50

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Example Hudson Auto Repair

• Relative Frequency and Percent Frequency
  Distributions
• Relative Percent
• Cost () Frequency Frequency
• 50-59 .04 4
• 60-69 .26 26
• 70-79 .32 32
• 80-89 .14 14
• 90-99 .14 14
• 100-109 .10 10
• Total 1.00 100

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Example Hudson Auto Repair

• Insights Gained from the Percent Frequency
  Distribution
• Only 4 of the parts costs are in the 50-59
  class.
• 30 of the parts costs are under 70.
• The greatest percentage (32 or almost one-third)
  of the parts costs are in the 70-79 class.
• 10 of the parts costs are 100 or more.

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Dot 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.

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Example Hudson Auto Repair

• Dot Plot

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Histogram

• 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 intervals
  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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Example Hudson Auto Repair

• Histogram

18
16
14
12
Frequency
10
8
6
4
2
Parts Cost ()
50 60 70 80 90 100
110
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Cumulative Distributions

• Cumulative frequency distribution -- shows the
  number of items with values less than or equal to
  the upper limit of each class.
• Cumulative relative frequency distribution --
  shows the proportion of items with values less
  than or equal to the upper limit of each class.
• 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 Hudson Auto Repair

• Cumulative Distributions
• Cumulative Cumulative
• Cumulative Relative
  Percent
• Cost () Frequency Frequency
  Frequency
• lt 59 2 .04 4
• lt 69 15 .30 30
• lt 79 31 .62 62
• lt 89 38 .76 76
• lt 99 45 .90 90
• lt 109 50 1.00 100

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Ogive

• 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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Example Hudson Auto Repair

• Ogive
• 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.
• 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.

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Example Hudson Auto Repair

• Ogive with Cumulative Percent Frequencies

100
80
60
Cumulative Percent Frequency
40
20
Parts Cost ()
50 60 70 80 90 100
110
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Exploratory Data Analysis

• The techniques of exploratory data analysis
  consist of simple arithmetic and easy-to-draw
  pictures that can be used to summarize data
  quickly.
• One such technique is the stem-and-leaf display.

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Stem-and-Leaf Display

• A stem-and-leaf display shows both the rank order
  and shape of the distribution of the data.
• It is similar to a histogram on its side, but it
  has the advantage of showing the actual data
  values.
• The first digits of each data item are arranged
  to the left of a vertical line.
• To the right of the vertical line we record the
  last digit for each item in rank order.
• Each line in the display is referred to as a
  stem.
• Each digit on a stem is a leaf.
• 8 5 7
• 9 3 6 7 8

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Stem-and-Leaf Display

• Leaf Units
• A single digit is used to define each leaf.
• In the preceding example, the leaf unit was 1.
• Leaf units may be 100, 10, 1, 0.1, and so on.
• Where the leaf unit is not shown, it is assumed
  to equal 1.

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Example Leaf Unit 0.1

• If we have data with values such as
• 8.6 11.7 9.4 9.1 10.2 11.0 8.8
• a stem-and-leaf display of these data will be
• Leaf Unit 0.1
• 8 6 8
• 9 1 4
• 10 2
• 11 0 7

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Example Leaf Unit 10

• If we have data with values such as
• 1806 1717 1974 1791 1682 1910 1838
• a stem-and-leaf display of these data will be
• Leaf Unit 10
• 16 8
• 17 1 9
• 18 0 3
• 19 1 7

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Example Hudson Auto Repair

• Stem-and-Leaf Display
• 5 2 7
• 6 2 2 2 2 5 6 7 8 8 8 9 9 9
• 7 1 1 2 2 3 4 4 5 5 5 6 7 8
  9 9 9
• 8 0 0 2 3 5 8 9
• 9 1 3 7 7 7 8 9
• 10 1 4 5 5 9

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Stretched Stem-and-Leaf Display

• If we believe the original stem-and-leaf display
  has condensed the data too much, we can stretch
  the display by using two more stems for each
  leading digit(s).
• Whenever a stem value is stated twice, the first
  value corresponds to leaf values of 0-4, and the
  second values corresponds to values of 5-9.

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Example Hudson Auto Repair

• Stretched Stem-and-Leaf Display
• 5 2
• 5 7
• 6 2 2 2 2
• 6 5 6 7 8 8 8 9 9 9
• 7 1 1 2 2 3 4 4
• 7 5 5 5 6 7 8 9 9 9
• 8 0 0 2 3
• 8 5 8 9
• 9 1 3
• 9 7 7 7 8 9
• 10 1 4
• 10 5 5 9

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Crosstabulations and Scatter Diagrams

• Thus far we have focused on methods that are used
  to summarize the data for one variable at a time.
• Often a manager is interested in tabular and
  graphical methods that will help understand the
  relationship between two variables.
• Crosstabulation and a scatter diagram are two
  methods for summarizing the data for two (or
  more) variables simultaneously.

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Crosstabulation

• Crosstabulation is a tabular method for
  summarizing the data for two variables
  simultaneously.
• Crosstabulation can be used when
• One variable is qualitative and the other is
  quantitative
• Both variables are qualitative
• Both variables are quantitative
• The left and top margin labels define the classes
  for the two variables.

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Example Finger Lakes Homes

• Crosstabulation
• The number of Finger Lakes homes sold for each
  style and price for the past two years is shown
  below.
• Price Home Style
• Range Colonial Ranch Split
  A-Frame Total
• lt 99,000 18 6
  19 12 55
• gt 99,000 12 14
  16 3 45
• Total 30 20 35
  15 100

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Example Finger Lakes Homes

• Insights Gained from the Preceding
  Crosstabulation
• The greatest number of homes in the sample (19)
  are a split-level style and priced at less than
  or equal to 99,000.
• Only three homes in the sample are an A-Frame
  style and priced at more than 99,000.

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Crosstabulation Row or Column Percentages

• Converting the entries in the table into row
  percentages or column percentages can provide
  additional insight about the relationship between
  the two variables.

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Example Finger Lakes Homes

• Row Percentages
• Price Home Style
• Range Colonial Ranch Split
  A-Frame Total
• lt 99,000 32.73 10.91 34.55
  21.82 100
• gt 99,000 26.67 31.11 35.56
  6.67 100
• Note row totals are actually 100.01 due to
  rounding.

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Example Finger Lakes Homes

• Column Percentages
• Price Home Style
• Range Colonial Ranch Split
  A-Frame
• lt 99,000 60.00 30.00 54.29
  80.00
• gt 99,000 40.00 70.00 45.71
  20.00
• Total 100 100 100
  100

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Scatter Diagram

• A scatter diagram is a graphical presentation of
  the relationship between two quantitative
  variables.
• One variable is shown on the horizontal axis and
  the other variable is shown on the vertical axis.
• The general pattern of the plotted points
  suggests the overall relationship between the
  variables.

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Example Panthers Football Team

• Scatter Diagram
• The Panthers football team is interested in
  investigating the relationship, if any, between
  interceptions made and points scored.
• x Number of y Number of
• Interceptions Points Scored
• 1 14
• 3 24
• 2 18
• 1 17
• 3 27

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Example Panthers Football Team

• Scatter Diagram

y
30
25
20
Number of Points Scored
15
10
5
x
0
1
2
3
0
Number of Interceptions
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Example Panthers Football Team

• The preceding scatter diagram indicates a
  positive relationship between the number of
  interceptions and the number of points scored.
• Higher points scored are associated with a higher
  number of interceptions.
• The relationship is not perfect all plotted
  points in the scatter diagram are not on a
  straight line.

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Scatter Diagram

• A Positive Relationship

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Scatter Diagram

• A Negative Relationship

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Scatter Diagram

• No Apparent Relationship

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Tabular and Graphical Procedures
Data
Qualitative Data
Quantitative Data
Tabular Methods
Tabular Methods
Graphical Methods
Graphical Methods

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

• Bar Graph
• Pie Chart

• Dot Plot
• Histogram
• Ogive
• Scatter
• Diagram

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

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End of Chapter 2