Graphs, Good and Bad

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Chapter 10

• Graphs, Good and Bad

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Chapters 10 -11 Summarizing Data

• Presentation of data
• Frequency Tables
• Pictures of Data
• Numerical Summaries (Center and Variation)

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Summarizing Data

• Data is often collected in order to answer some
  question or
• address a specific issue.
• When analyzing a data set we should first
  consider whether the data comes from a complete
  population (remember this means taking a census -
  for example test scores of everyone in this
  class) or a sample.
• Methods of descriptive statistics are used to
  summarize the important characteristics of a set
  of data.

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Important Characteristics of Data

• The following characteristics of data are usually
  important
• Center An average value that indicates where the
  middle of the dataset is located.
• Variation/Spread A measure of the amount of
  variation in the data (average variation from the
  center).
• Distribution The shape of the distribution of
  the data (symmetric, uniform or skewed).
• Outliers Sample values that lie far away from
  the vast majority of the other values.
• Time Trend- changing characteristics of the data
  over time.

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Frequency Tables

• A frequency table lists data values (either
  individually or by groups of intervals called
  classes ), along with the number of items that
  fall into each class (frequency). Example
• Test Score Frequency
• 0- 4 3
• 5 - 9 10
• 10-14 12
• 15-19 35
• 20-24 20
• 25-29 15
• 30-34 5
• This frequency table has 7 classes
  (0-4,5-9,10-14,15-19,20-24,25-29,30-34). The
  frequency represents the number of students
  receiving that score.

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Example

• The heights (in inches) of 30 students are as
• follows
• 68 64 70 67 67 68 64 65 68 64 70 72
  71 69 72
• 64 63 70 71 63 68 67 67 65 69 65 67
  66 61 65
• Create a frequency table for the above data
• using the classes 60-61, 62-63, 64-65 etc.

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RELATIVE FREQUENCY TABLES

• Relative frequency frequency / total of items
  
• The relative frequency gives the percent of items
  in each class.
• A relative frequency table is a frequency table
  with a column for the relative frequencies. The
  relative frequencies might not add to 1 (100)
  due to rounding.
• Example Construct a relative frequency table for
  our last example.

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Distribution

• Tells what values a variable takes and how often
  it takes these values.
• Can be a table, graph, or function.

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Graphs for Categorical Data

• A picture (a good one) is worth a thousand
  words.
• Bar Graph
• Horizontal axis represents the categories.
• Vertical axis represents the frequencies.
• A bar whose height is proportional to the
  frequency is drawn
• centered at the category.

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Example

• The following table gives the grade distributions
  of a Math 161 Test
• Grade Frequency
• A 5
• B 7
• C 12
• D 5
• F 3
• Draw a bar graph for the data.

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Graphs for Categorical Data

• Double (Side-by-side) Bar Graphs
• Used to compare two different distributions.
• For each category, draw two adjacent bars (one
  for each
• distribution).

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Example

• Suppose we now have the grades for two sections
  of Math 161
• Grade Frequency
• Section 1 Section 2
• A 5 3
• B 7 5
• C 12 9
• D 5 4
• F 3 1
• Draw a double bar graph for the data.

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Example

• Are the frequency bar graphs the right way to
  compare the performance of the two sections? Note
  that the class sizes are not the same. What would
  be a better way to compare the two sections?
• Grade Frequency
• Section 1 Section 2
• A 5 3
• B 7 5
• C 12 9
• D 5 4
• F 3 1

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Graphs for Categorical Data

• Pie Chart
• Shows the whole group of categories in a circle.
• Shows the parts of some whole .
• The area of the sector representing a category
  is
• proportional to the frequency of the category.

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Example

• The following table gives the grade distributions
  of a Math 161 Test
• Grade Frequency
• A 5
• B 7
• C 12
• D 5
• F 3
• Draw a pie chart for the data.

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Pictographs

• A picture of a set of small
• figures or icons used to
• represent data, and often to
• represent trends.
• Usually, the icons are
• suggestively related to the
• data being represented.
• They can be misleading.

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Pictographs

• Double the length, width, and height of a
  cube, and the volume increases by a factor of
  eight.

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Line Graphs

• A line graph shows behavior over time.
• Time is always on the horizontal axis.
• Look for an overall pattern (trend).
• Look for patterns that repeat at known regular
  intervals (seasonal variations).
• Look for any striking deviations that might
  indicate unusual occurrences.

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Misleading Graphs

• Changing the scale of a line graph or a bar graph
  can make
• increases or decreases appear more rapid.
• Both graphs plot the same data. Which one makes
  the increase in cancer
• deaths appear more rapid? Which graph would a
  cancer advocate use?

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Salaries of People with Bachelors Degrees and
with High School Diplomas Which graph is
misleading?
40,500
40,500
40,000
40,000
30,000
35,000
24,400
30,000
20,000
24,400
25,000
10,000
20,000
0
Bachelor High School Degree Diploma
Bachelor High School Degree Diploma
(a)
(b)
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Important skills

• Computer programs will construct plots and
  calculate summary statistics automatically.
• The important skills for people are
• knowing what to use when.
• Interpretation.
• The tools used to analyze and summarize data
  depend upon the type of variable one is
  interested in.

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Principles for plots

• The way plots are used depends upon the purpose
  for which
• they are being used
• Exploration
• Principle Look at the data in as many different
  ways as possible searching for its important
  features.
• Communication to others (follows exploration)
  
• Principle Be selective. Choose the displays
  that best show to a reader features you have
  observed.

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Making Good Graphs

• Title your graph.
• Make sure labels and legends describe variables
  and their measurement units. Be careful with the
  scales used.
• Make the data stand out. Avoid distracting
  grids, artwork, etc.
• Pay attention to what the eye sees. Avoid
  pictograms and tacky effects.

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Key Concepts

• Categorical and Quantitative Variables
• Distributions
• Pie Charts
• Bar Graphs
• Line Graphs
• Techniques for Making Good Graphs

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Chapter 11

• Displaying Distributions with Graphs

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Stemplots(Stem-and-Leaf Plots)

• For quantitative variables.
• Separate each observation into a stem (first part
  of the number) and a leaf (the remaining part of
  the number).
• Usually, the last digit is used as the leaf and
  the remaining digits form the stem.
• If using the last digits as they are results in a
  lot of stem values, we could round the numbers to
  more convenient values.
• Write the stems in a vertical column draw a
  vertical line to the right of the stems.
• Write each leaf in the row to the right of its
  stem order leaves if desired.

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Weight DataWeights (in pounds) for a group of 40
students.
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Weight DataStemplot(Stem and Leaf Plot)
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 2
6
5
2
570
Key 203 means203 pounds Stems 10sLeaves
1s
2
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Stem-and-Leaf Plots

• Double stemmed (expanded) stem-and-leaf
• If there are a lot of leaves on one stem, we
  could break it up into two stems one for the
  digits 0-4 and the other for the digits 5-9.
• Back to back Used to compare two different sets
  of data.

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Histogram

• A histogram is a bar graph in which
• the horizontal axis represents the items or
  classes.
• the vertical axis represents the frequencies.
• the height of the bars are proportional to the
  frequencies.
• There are usually no gaps between the bars
  (unless some classes have 0
• frequencies).
• To draw a histogram, we first need to construct a
  frequency table.
• Example draw a histogram for our weights
  example.
• The number of classes can affect the shape of the
  histogram.
• http//www.stat.sc.edu/west/javahtml/Histogra
  m.html

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Weight Data Frequency Table
Left endpoint is included in the group, right
endpoint is not.
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Weight Data Histogram
Left endpoint is included in the group, right
endpoint is not.
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Shape of the Data

• Symmetric
• bell-shaped
• other symmetric shapes
• Asymmetric
• skewed to the right
• skewed to the left
• Unimodal, bimodal

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Symmetric Distributions
Bell-Shaped
Mound-Shaped
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Symmetric Distributions Uniform
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Asymmetric Distributions
Skewed to the Left
Skewed to the Right
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Number of Books Read for Pleasure
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Outliers

• Extreme values, far from the rest of the data.
• May occur naturally.
• May occur due to error in recording.
• May occur due to error in measuring.
• Observational unit may be fundamentally different.

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Key Concepts

• Displays (Stemplots Histograms)
• Graph Shapes
• Symmetric
• Skewed to the Right
• Skewed to the Left
• Outliers