Descriptive

1
Part 2

• Descriptive
• Statistics

2
Part 2 Descriptive Statistics

• Chapters 2 through 4 cover Descriptive Statistics
• After we collect the raw data (from a sample
  survey or a designed experiment), we can
• Describe the data using visual methods (charts,
  etc.)
• Describe the data using numeric methods
  (averages, etc.)
• Different methods are appropriate for different
  types of data

3
Chapter 2

• Organizing and
• Summarizing Data

4
Overview

• This is the first of three chapters on
  Descriptive Statistics
• Our first task in Descriptive Statistics is to
  organize and summarize the data
• Organizing qualitative data into tables
• Organizing quantitative data using charts
• We must be careful to not misrepresent the data
• We should recognize misrepresented data

5
Chapter 2 Sections

• Sections in Chapter 2
• Organizing Qualitative Data
• Organizing Quantitative Data The Popular
  Displays
• Additional Displays of Quantitative Data
• Graphical Misrepresentations of Data

6
Chapter 2Section 1

• Organizing
• Qualitative Data

7
Chapter 2 Section 1

• Learning objectives
• Organize qualitative data in tables
• Construct bar graphs
• Construct pie charts

8
Chapter 2 Section 1

• Learning objectives
• Organize qualitative data in tables
• Construct bar graphs
• Construct pie charts

9
Chapter 2 Section 1

• Raw qualitative data comes as a list of values
  each value is one out of a set of categories
• These values can be organized as either a long
  list or in a table

• Raw qualitative data comes as a list of values
  each value is one out of a set of categories
• These values can be organized as either a long
  list or in a table
• Interpreting the list of data can be difficult,
  particularly if there is a lot of data
• Methods are needed to aid interpretation

10
Chapter 2 Section 1

• Qualitative data values can be organized by a
  frequency distribution
• A frequency distribution lists
• Each of the categories
• The frequency for each category

11
Chapter 2 Section 1

• A simple data set is
• blue, blue, green, red, red, blue, red, blue
• A frequency table for this qualitative data is
• The most commonly occurring color is blue

12
Chapter 2 Section 1

• The frequencies are the counts of the observations

• The frequencies are the counts of the
  observations
• The relative frequencies are the proportions (or
  percents) of the observations out of the total

• The frequencies are the counts of the
  observations
• The relative frequencies are the proportions (or
  percents) of the observations out of the total
• A relative frequency distribution lists
• Each of the categories
• The relative frequency for each category

13
Chapter 2 Section 1

• Use the same simple set of data
• blue, blue, green, red, red, green, blue, blue
• A relative frequency table is computed as follows
• Sum of all frequencies 8 (there are 8
  observations)
• Blue has a relative frequency of 4 / 8 .500
• Green has a relative frequency of 1 / 8 .125
• Red has a relative frequency of 3 / 8 .375

14
Chapter 2 Section 1

• A relative frequency table for this qualitative
  data is
• A relative frequency table can also be
  constructed with percents (50, 12.5, and 37.5
  for the above table)

15
Chapter 2 Section 1

• Tables are useful because they provide an exact
  count for the data
• However, if the data set is medium to large in
  size, it may be difficult to understand the data
  when presented in a table
• Additional techniques are needed to give a better
  idea of the big picture

16
Chapter 2 Section 1

• Learning objectives
• Organize qualitative data in tables
• Construct bar graphs
• Construct pie charts

17
Chapter 2 Section 1

• In general, pictures of data send a more powerful
  message than tables
• Visual methods, such as bar graphs, present a
  better summary than just a table

• In general, pictures of data send a more powerful
  message than tables
• Visual methods, such as bar graphs, present a
  better summary than just a table
• A bar graph
• Lists the categories on the horizontal axis
• Draws rectangles above each category where the
  heights are equal to the categorys frequency or
  relative frequency

18
Chapter 2 Section 1

• Bar graphs for our simple data (using Excel)
• Frequency bar graph
• Relative frequency bar graph

19
Chapter 2 Section 1

• Good practices in constructing bar graphs
• The horizontal scale
• The categories should be spaced equally apart
• The rectangles should have the same widths

• Good practices in constructing bar graphs
• The horizontal scale
• The categories should be spaced equally apart
• The rectangles should have the same widths
• The vertical scale
• Should begin with 0
• Should be incremented in reasonable steps
• Should go somewhat, but not significantly, beyond
  the largest frequency or relative frequency

20
Chapter 2 Section 1

• A Pareto chart is a particular type of bar graph
• A Pareto differs from a bar chart only in that
  the categories are arranged in order
• The category with the highest frequency is placed
  first (on the extreme left)
• The second highest category is placed second
• Etc.

• A Pareto chart is a particular type of bar graph
• A Pareto differs from a bar chart only in that
  the categories are arranged in order
• The category with the highest frequency is placed
  first (on the extreme left)
• The second highest category is placed second
• Etc.
• Pareto charts are often used when there are many
  categories but only the top few are of interest

21
Chapter 2 Section 1

• A Pareto chart for our simple data (using Excel)

22
Chapter 2 Section 1

• An example with more data values
• A data set from the text
• Even with only 30 data values, this table cannot
  be interpreted easily

23
Chapter 2 Section 1

• Graphs for this set of data
• A frequency bar graph
• A relative frequency bar graph
• These graphs are more effective than the table

24
Chapter 2 Section 1

• Graphs for this data (continued)
• A Pareto chart

25
Chapter 2 Section 1

• Two qualitative variables can be compared by
  comparing their bar graphs
• A side-by-side bar graph draws two rectangles for
  each category, one for each variable
• The frequencies (or relative frequencies) for
  each category can be compared

26
Chapter 2 Section 1

• An example side-by-side bar graph comparing
  educational attainment in 1990 versus 2003

27
Chapter 2 Section 1

• Learning objectives
• Organize qualitative data in tables
• Construct bar graphs
• Construct pie charts

28
Chapter 2 Section 1

• A pie chart is a circle divided into sections,
  one for each category
• The area (angle) of each sector is proportional
  to the frequency of that category
• Pie charts are useful to show the relative
  proportions of each category, compared to the
  whole

29
Chapter 2 Section 1

• Good practices for constructing pie charts
• Different colors should be used to distinguish
  the categories
• Each category should be labeled with the category
  name and relative frequency

• Good practices for constructing pie charts
• Different colors should be used to distinguish
  the categories
• Each category should be labeled with the category
  name and relative frequency
• Pie charts are not as effective if there are too
  many categories or if some relative frequencies
  are too small

30
Chapter 2 Section 1

• An example of a pie chart for the 2003 data from
  the side-to-side bar chart

31
Chapter 2 Section 1

• Side-by-side pie charts are used sometimes, but
  can be difficult to interpret (using Excel, with
  substantial modifications)

32
Summary Chapter 2 Section 1

• Qualitative data can be organized in several ways
• Tables are useful for listing the data, its
  frequencies, and its relative frequencies
• Charts such as bar graphs, Pareto charts, and pie
  charts are useful visual methods for organizing
  data
• Side-by-side bar graphs are useful for comparing
  two sets of qualitative data

33
Chapter 2Section 2

• Organizing Quantitative Data
• The Popular Displays

34
Chapter 2 Section 2

• Learning objectives
• Organize discrete data in tables
• Construct histograms of discrete data
• Organize continuous data in tables
• Construct histograms of continuous data
• Draw stem-and-leaf plots
• Draw dot plots
• Identify the shape of a distribution

35
Chapter 2 Section 2

• Learning objectives
• Organize discrete data in tables
• Construct histograms of discrete data
• Organize continuous data in tables
• Construct histograms of continuous data
• Draw stem-and-leaf plots
• Draw dot plots
• Identify the shape of a distribution

36
Chapter 2 Section 2

• Raw quantitative data comes as a list of values
  each value is a measurement, either discrete or
  continuous
• Comparisons (one value being more than or less
  than another) can be performed on the data values
• Mathematical operations (addition, subtraction,
  ) can be performed on the data values

37
Chapter 2 Section 2

• Discrete quantitative data can be presented in
  tables in several of the same ways as qualitative
  data
• Values listed in a table
• By a frequency table
• By a relative frequency table
• We use the discrete values instead of the
  category names

38
Chapter 2 Section 2

• Consider the following data
• We would like to compute the frequencies and the
  relative frequencies

39
Chapter 2 Section 2

• The resulting frequencies and the relative
  frequencies

40
Chapter 2 Section 2

• Learning objectives
• Organize discrete data in tables
• Construct histograms of discrete data
• Organize continuous data in tables
• Construct histograms of continuous data
• Draw stem-and-leaf plots
• Draw dot plots
• Identify the shape of a distribution

41
Chapter 2 Section 2

• Discrete quantitative data can be presented in
  bar graphs in several of the same ways as
  qualitative data
• We use the discrete values instead of the
  category names
• We arrange the values in ascending order
• For discrete data, these are called histograms

42
Chapter 2 Section 2

• Example of histograms for discrete data
• Frequencies
• Relative frequencies

43
Chapter 2 Section 2

• Learning objectives
• Organize discrete data in tables
• Construct histograms of discrete data
• Organize continuous data in tables
• Construct histograms of continuous data
• Draw stem-and-leaf plots
• Draw dot plots
• Identify the shape of a distribution

44
Chapter 2 Section 2

• Continuous data cannot be put directly into
  frequency tables since they do not have any
  obvious categories
• Categories are created using classes, or
  intervals of numbers
• The continuous data is then put into the classes

45
Chapter 2 Section 2

• For ages of adults, a possible set of classes is
• 20 29
• 30 39
• 40 49
• 50 59
• 60 and older
• For the class 30 39
• 30 is the lower class limit
• 39 is the upper class limit

46
Chapter 2 Section 2

• The class width is the difference between the
  upper class limit and the lower class limit
• For the class 30 39, the class width is
• 40 30 10

• The class width is the difference between the
  upper class limit and the lower class limit
• For the class 30 39, the class width is
• 40 30 10
• Why isnt the class width 39 30 9?
• The class 30 39 years old actually is 30 years
  to 39 years 364 days old or 30 years to just
  less than 40 years old
• The class width is 10 years, all adults in their
  30s

47
Chapter 2 Section 2

• All the classes (20 29, 30 39, 40 49, 50
  59) all have the same widths, except for the last
  class

• All the classes (20 29, 30 39, 40 49, 50
  59) all have the same widths, except for the last
  class
• The class 60 and above is an open-ended class
  because it has no upper limit

• All the classes (20 29, 30 39, 40 49, 50
  59) all have the same widths, except for the last
  class
• The class 60 and above is an open-ended class
  because it has no upper limit
• Classes with no lower limits are also called
  open-ended classes

48
Chapter 2 Section 2

• The classes and the number of values in each can
  be put into a frequency table
• In this table, there are 1147 subjects between 30
  and 39 years old

49
Chapter 2 Section 2

• Good practices for constructing tables for
  continuous variables
• The classes should not overlap

• Good practices for constructing tables for
  continuous variables
• The classes should not overlap
• The classes should not have any gaps between them

• Good practices for constructing tables for
  continuous variables
• The classes should not overlap
• The classes should not have any gaps between them
• The classes should have the same width (except
  for possible open-ended classes at the extreme
  low or extreme high ends)

• Good practices for constructing tables for
  continuous variables
• The classes should not overlap
• The classes should not have any gaps between them
• The classes should have the same width (except
  for possible open-ended classes at the extreme
  low or extreme high ends)
• The class boundaries should be reasonable
  numbers

• Good practices for constructing tables for
  continuous variables
• The classes should not overlap
• The classes should not have any gaps between them
• The classes should have the same width (except
  for possible open-ended classes at the extreme
  low or extreme high ends)
• The class boundaries should be reasonable
  numbers
• The class width should be a reasonable number

50
Chapter 2 Section 2

• Learning objectives
• Organize discrete data in tables
• Construct histograms of discrete data
• Organize continuous data in tables
• Construct histograms of continuous data
• Draw stem-and-leaf plots
• Draw dot plots
• Identify the shape of a distribution

51
Chapter 2 Section 2

• Just as for discrete data, a histogram can be
  created from the frequency table
• Instead of individual data values, the categories
  are the classes the intervals of data

52
Chapter 2 Section 2

• Learning objectives
• Organize discrete data in tables
• Construct histograms of discrete data
• Organize continuous data in tables
• Construct histograms of continuous data
• Draw stem-and-leaf plots
• Draw dot plots
• Identify the shape of a distribution

53
Chapter 2 Section 2

• A stem-and-leaf plot is a different way to
  represent data that is similar to a histogram

• A stem-and-leaf plot is a different way to
  represent data that is similar to a histogram
• To draw a stem-and-leaf plot, each data value
  must be broken up into two components
• The stem consists of all the digits except for
  the right most one
• The leaf consists of the right most digit
• For the number 173, for example, the stem would
  be 17 and the leaf would be 3

54
Chapter 2 Section 2

• In the stem-and-leaf plot below
• The smallest value is 56
• The largest value is 180
• The second largest value is 178

55
Chapter 2 Section 2

• To read a stem-and-leaf plot
• Read the stem first
• Attach the leaf as the last digit of the stem
• The result is the original data value

• To read a stem-and-leaf plot
• Read the stem first
• Attach the leaf as the last digit of the stem
• The result is the original data value
• Stem-and-leaf plots
• Display the same visual patterns as histograms
• Contain more information than histograms
• Could be more difficult to interpret (including
  getting a sore neck)

56
Chapter 2 Section 2

• To draw a stem-and-leaf plot
• Write all the values in ascending order

• To draw a stem-and-leaf plot
• Write all the values in ascending order
• Find the stems and write them vertically in
  ascending order

• To draw a stem-and-leaf plot
• Write all the values in ascending order
• Find the stems and write them vertically in
  ascending order
• For each data value, write its leaf in the row
  next to its stem

• To draw a stem-and-leaf plot
• Write all the values in ascending order
• Find the stems and write them vertically in
  ascending order
• For each data value, write its leaf in the row
  next to its stem
• The resulting leaves will also be in ascending
  order

• To draw a stem-and-leaf plot
• Write all the values in ascending order
• Find the stems and write them vertically in
  ascending order
• For each data value, write its leaf in the row
  next to its stem
• The resulting leaves will also be in ascending
  order
• The list of stems with their corresponding leaves
  is the stem-and-leaf plot

57
Chapter 2 Section 2

• Modifications to stem-and-leaf plots
• Sometimes there are too many values with the same
  stem we would need to split the stems (such as
  having 10-14 in one stem and 15-19 in another)

• Modifications to stem-and-leaf plots
• Sometimes there are too many values with the same
  stem we would need to split the stems (such as
  having 10-14 in one stem and 15-19 in another)
• If we wanted to compare two sets of data, we
  could draw two stem-and-leaf plots using the same
  stem, with leaves going left (for one set of
  data) and right (for the other set)

• Modifications to stem-and-leaf plots
• Sometimes there are too many values with the same
  stem we would need to split the stems (such as
  having 10-14 in one stem and 15-19 in another)
• If we wanted to compare two sets of data, we
  could draw two stem-and-leaf plots using the same
  stem, with leaves going left (for one set of
  data) and right (for the other set)
• There are cases where constructing a descending
  stem-and-leaf plot could also be appropriate (for
  test scores, for example)

58
Chapter 2 Section 2

• Learning objectives
• Organize discrete data in tables
• Construct histograms of discrete data
• Organize continuous data in tables
• Construct histograms of continuous data
• Draw stem-and-leaf plots
• Draw dot plots
• Identify the shape of a distribution

59
Chapter 2 Section 2

• A dot plot is a graph where a dot is placed over
  the observation each time it is observed
• The following is an example of a dot plot

60
Chapter 2 Section 2

• Learning objectives
• Organize discrete data in tables
• Construct histograms of discrete data
• Organize continuous data in tables
• Construct histograms of continuous data
• Draw stem-and-leaf plots
• Draw dot plots
• Identify the shape of a distribution

61
Chapter 2 Section 2

• A useful way to describe a variable is by the
  shape of its distribution
• Some common distribution shapes are
• Uniform
• Bell-shaped (or normal)
• Skewed right
• Skewed left

62
Chapter 2 Section 2

• A variable has a uniform distribution when
• Each of the values tends to occur with the same
  frequency
• The histogram looks flat

63
Chapter 2 Section 2

• A variable has a bell-shaped distribution when
• Most of the values fall in the middle
• The frequencies tail off to the left and to the
  right
• It is symmetric

64
Chapter 2 Section 2

• A variable has a skewed right distribution when
• The distribution is not symmetric
• The tail to the right is longer than the tail to
  the left
• The arrow from the middle to the long tail points
  right

Right
65
Chapter 2 Section 2

• A variable has a skewed left distribution when
• The distribution is not symmetric
• The tail to the left is longer than the tail to
  the right
• The arrow from the middle to the long tail points
  left

Left
66
Summary Chapter 2 Section 2

• Quantitative data can be organized in several
  ways
• Histograms based on data values are good for
  discrete data
• Histograms based on classes (intervals) are good
  for continuous data
• The shape of a distribution describes a variable
  histograms are useful for identifying the shapes

67
Chapter 2Section 3

• Additional Displays of
• Quantitative Data

68
Chapter 2 Section 3

• Learning objectives
• Construct frequency polygons
• Create cumulative frequency and relative
  frequency tables
• Construct frequency and relative frequency ogives
• Draw time-series graphs

69
Chapter 2 Section 3

• Learning objectives
• Construct frequency polygons
• Create cumulative frequency and relative
  frequency tables
• Construct frequency and relative frequency ogives
• Draw time-series graphs

70
Chapter 2 Section 3

• We will cover other ways to display quantitative
  data
• These methods are somewhat more specialized not
  as general as the methods in the previous section

71
Chapter 2 Section 3

• Frequency polygons are similar to histograms
  except a polygon (a line graph) is used instead
  of rectangles (a bar graph)

• Frequency polygons are similar to histograms
  except a polygon (a line graph) is used instead
  of rectangles (a bar graph)
• To draw a frequency polygon
• Compute the class midpoints (the average of the
  upper and lower class limits)
• Plot the frequencies of each class above each
  class midpoint
• Connect the points

72
Chapter 2 Section 3

• The following is an example of a frequency
  polygon
• A histogram would be similar, just with bars at
  each point

73
Chapter 2 Section 3

• Learning objectives
• Construct frequency polygons
• Create cumulative frequency and relative
  frequency tables
• Construct frequency and relative frequency ogives
• Draw time-series graphs

74
Chapter 2 Section 3

• Cumulative frequencies are obtained by adding up
  all the frequencies for the smaller categories

• Cumulative frequencies are obtained by adding up
  all the frequencies for the smaller categories
• If the data is
• 1, 1, 1, 1, 2, 3, 3, 5, 5, 6
• then the cumulative frequency for the category
  2 is equal to 5
• There are 4 observations of 1
• There is 1 observation of 2
• Thus there are 5 observations of values less than
  or equal to 2

75
Chapter 2 Section 3

• Cumulative relative frequencies are obtained by
  adding up all the relative frequencies for the
  smaller categories

• Cumulative relative frequencies are obtained by
  adding up all the relative frequencies for the
  smaller categories
• If the data is
• 1, 1, 1, 1, 2, 3, 3, 5, 5, 6
• then the cumulative relative frequency for the
  category 2 is equal to .5
• The relative frequency of 1 is .4
• The relative frequency of 2 is .1
• The cumulative relative frequency of 2 is .4
  .1 .5

76
Chapter 2 Section 3

• A cumulative frequency table summarizes the
  cumulative frequencies in a table

• A cumulative frequency table summarizes the
  cumulative frequencies in a table
• A cumulative relative frequency table summarizes
  the cumulative relative frequencies in a table

77
Chapter 2 Section 3

• For the data set
• 1, 1, 1, 1, 2, 3, 3, 5, 5, 6
• the cumulative tables are

78
Chapter 2 Section 3

• Learning objectives
• Construct frequency polygons
• Create cumulative frequency and relative
  frequency tables
• Construct frequency and relative frequency ogives
• Draw time-series graphs

79
Chapter 2 Section 3

• Graphs of the cumulative frequencies and the
  relative cumulative frequencies are called ogives

• Graphs of the cumulative frequencies and the
  relative cumulative frequencies are called ogives
  
• To construct an ogive
• Compute the cumulative frequencies
• Plot the cumulative frequencies above the upper
  class limits
• Connect the points

• Graphs of the cumulative frequencies and the
  relative cumulative frequencies are called ogives
  
• To construct an ogive
• Compute the cumulative frequencies
• Plot the cumulative frequencies above the upper
  class limits
• Connect the points
• An ogive is an increasing function

80
Chapter 2 Section 3

• The following is an example of an ogive
• The curve is increasing because it is a
  cumulative calculation

81
Chapter 2 Section 3

• Learning objectives
• Construct frequency polygons
• Create cumulative frequency and relative
  frequency tables
• Construct frequency and relative frequency ogives
• Draw time-series graphs

82
Chapter 2 Section 3

• When the variable is measured at different points
  in time, the data is time-series data
• It is natural to plot time-series data against
  time
• Such a plot is a time-series plot
• Time series plots are used to
• Identify long term trends
• Identify regularly occurring trends
  (seasonality)

83
Chapter 2 Section 3

• The following is an example of a time-series
  graph
• The horizontal axis shows the passage of time

84
Summary Chapter 2 Section 3

• Additional displays of quantitative data can
• Show cumulative frequencies
• Show cumulative relative frequencies
• Show the time relationship of time-series data

85
Chapter 2Section 4

• Graphical Misrepresentations
• of Data

86
Chapter 2 Section 4

• Learning objectives
• Describe what can make a graph misleading or
  deceptive

87
Chapter 2 Section 4

• Lies, damn lies, and statistics
• Figures lie and liars figure

• Lies, damn lies, and statistics
• Figures lie and liars figure
• Statistics displays can distort the truth
• Unintentional distorting mislead
• Intentional distorting deceive

• Lies, damn lies, and statistics
• Figures lie and liars figure
• Statistics displays can distort the truth
• Unintentional distorting mislead
• Intentional distorting deceive
• There are several common errors, such as
• Charts where the visuals and the numbers do not
  correspond
• Charts that have an artificial base point to
  exaggerate the differences

88
Chapter 2 Section 4

• The classic short book that identifies
  misrepresentations (graphic and numeric) is How
  to Lie with Statistics by Huff
• The examples are out of date (the book was first
  published in 1954) but the invalid techniques
  are still used today

89
Chapter 2 Section 4

• Common errors are
• Inaccurate displays
• Unclear vertical scale
• Truncated vertical scale
• Misleading dimensions

90
Chapter 2 Section 4

• Despite advances in publishing and statistical
  software, there are still simple numeric mistakes
• Lengths of bars in bar charts and histograms that
  do not match their frequencies
• Slices of pie charts that are not proportional to
  their relative frequencies

91
Chapter 2 Section 4

• This pie graph is deliberately incorrect
• The red region is too small
• The aqua region is too large
• Pie charts are sometimes incorrect in this way

92
Chapter 2 Section 4

• Some charts have a vertical scale that is unclear
• The scale is possibly not labeled
• The zero point of the scale is unclear
• In these graphs, the order of the sizes is
  accurate, but the relative comparisons can be
  misleading

93
Chapter 2 Section 4

• In this graph, it is unclear
• Where the vertical scale begins (bottom of or top
  of the shirts) starts
• What the scale increments are

94
Chapter 2 Section 4

• The vertical scale is truncated when the vertical
  scale does not start at 0

• The vertical scale is truncated when the vertical
  scale does not start at 0
• When the vertical scale starts at a higher
  number, the differences between the bars is
  exaggerated
• For some data, magnifying the differences is
  important
• For some data, magnifying the differences is
  misleading

95
Chapter 2 Section 4

• The two graphs show the same data the
  difference seems larger for the graph on the left
• The vertical scale is truncated on the left

96
Chapter 2 Section 4

• When there is interest in small changes showing
  day to day stock prices for example then the
  graph on the left is appropriate

97
Chapter 2 Section 4

• When there is interest in the totals showing
  yearly salaries for example then the graph on
  the right is appropriate

98
Chapter 2 Section 4

• Some charts are made visually more attractive by
  using symbols and graphics instead of plain bars
  and lines

• Some charts are made visually more attractive by
  using symbols and graphics instead of plain bars
  and lines
• If one category has twice the frequency of
  another, that graphic is doubled in size

• Some charts are made visually more attractive by
  using symbols and graphics instead of plain bars
  and lines
• If one category has twice the frequency of
  another, that graphic is doubled in size
• If the graphic is a three dimensional graphic,
  then doubling each dimension increases the volume
  by eight times which is misleading

99
Chapter 2 Section 4

• The gazebo on the right is twice as large in each
  dimension as the one on the left
• However, it is much more than twice as large as
  the one on the left

Original
Twice as large
100
Summary Chapter 2 Section 4

• Displays are powerful for representing data
• Displays are also powerful for misrepresenting
  data
• Avoiding distortions in your own work is very
  important
• Recognizing distortions in other peoples work is
  very important also

101
Chapter 2

• Organizing and
• Summarizing Data
• Summary

102
Summary Chapter 2

• Summaries of qualitative data
• Frequency tables
• Bar graphs
• Summaries of quantitative data
• Frequency tables
• Histograms
• Pie graphs, time-series graphs, etc.
• Cumulative frequencies, ogives, etc.