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Chapter 2 Descriptive Statistics: Tabular and Graphical Methods

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Title: Chapter 2 Descriptive Statistics: Tabular and Graphical Methods


1
Chapter 2Descriptive StatisticsTabular and
Graphical Methods
  • Summarizing Qualitative Data
  • Summarizing Quantitative Data
  • Exploratory Data Analysis
  • Crosstabulations
  • and Scatter Diagrams

2
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

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

6
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.

7
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

9
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.

10
Example Marada Inn
  • Bar Graph

11
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.

12
Example Marada Inn
  • Pie Chart

13
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.

14
Summarizing Quantitative Data
  • Frequency Distribution
  • Relative Frequency and Percent Frequency
    Distributions
  • Dot Plot
  • Histogram
  • Cumulative Distributions
  • Ogive

15
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.

16
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.

17
Frequency Distribution
  • Guidelines for Selecting Width of Classes
  • Use classes of equal width.
  • Approximate Class Width

18
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

19
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

20
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.

21
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.

22
Example Hudson Auto Repair
  • Dot Plot

23
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.

24
Example Hudson Auto Repair
  • Histogram

18
16
14
12
Frequency
10
8
6
4
2
Parts Cost ()
50 60 70 80 90 100
110
25
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.

26
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

27
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.

28
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.

29
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
30
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.

31
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

32
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.

33
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

34
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

35
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

36
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.

37
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

38
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.

39
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.

40
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

41
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.

42
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.

43
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.

44
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

45
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.

46
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

47
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
48
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.

49
Scatter Diagram
  • A Positive Relationship

50
Scatter Diagram
  • A Negative Relationship

51
Scatter Diagram
  • No Apparent Relationship

52
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

53
End of Chapter 2
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