Chapter 2 Exploring Data with Graphs and Numerical Summaries

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Chapter 2Exploring Data with Graphs and
Numerical Summaries

• Learn .
• The Different Types of Data
• The Use of Graphs to Describe
• Data
• The Numerical Methods of Summarizing Data

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Section 2.1

• What are the Types of Data?

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In Every Statistical Study

• Questions are posed
• Characteristics are observed

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Characteristics are Variables

• A Variable is any characteristic that is
  recorded for subjects in the study

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Variation in Data

• The terminology variable highlights the fact that
  data values vary.

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Example Students in a Statistics Class

• Variables
• Age
• GPA
• Major
• Smoking Status

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Data values are called observations

• Each observation can be
• Quantitative
• Categorical

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Categorical Variable

• Each observation belongs to one of a set of
  categories
• Examples
• Gender (Male or Female)
• Religious Affiliation (Catholic, Jewish, )
• Place of residence (Apt, Condo, )
• Belief in Life After Death (Yes or No)

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Quantitative Variable

• Observations take numerical values
• Examples
• Age
• Number of siblings
• Annual Income
• Number of years of education completed

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Graphs and Numerical Summaries

• Describe the main features of a variable
• For Quantitative variables key features are
  center and spread
• For Categorical variables key feature is the
  percentage in each of the categories

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Quantitative Variables

• Discrete Quantitative Variables
• and
• Continuous Quantitative Variables

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Discrete

• A quantitative variable is discrete if its
  possible values form a set of separate numbers
  such as 0, 1, 2, 3,

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Examples of discrete variables

• Number of pets in a household
• Number of children in a family
• Number of foreign languages spoken

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Continuous

• A quantitative variable is continuous if its
  possible values form an interval

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Examples of Continuous Variables

• Height
• Weight
• Age
• Amount of time it takes to complete an assignment

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Frequency Table

• A method of organizing data
• Lists all possible values for a variable along
  with the number of observations for each value

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Example Shark Attacks
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Example Shark Attacks
Example Shark Attacks

• What is the variable?
• Is it categorical or quantitative?
• How is the proportion for Florida calculated?
• How is the for Florida calculated?

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Example Shark Attacks

• Insights what the data tells us about shark
  attacks

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Identify the following variable as categorical or
quantitative

• Choice of diet
• (vegetarian or non-vegetarian)
• Categorical
• Quantitative

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Identify the following variable as categorical or
quantitative

• Number of people you have known who have been
  elected to political office
• Categorical
• Quantitative

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Identify the following variable as discrete or
continuous

• The number of people in line at a box office to
  purchase theater tickets
• Continuous
• Discrete

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Identify the following variable as discrete or
continuous

• The weight of a dog
• Continuous
• Discrete

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Section 2.2

• How Can We Describe Data Using Graphical
  Summaries?

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Graphs for Categorical Data

• Pie Chart A circle having a slice of pie for
  each category
• Bar Graph A graph that displays a vertical bar
  for each category

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Example Sources of Electricity Use in the U.S.
and Canada
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Pie Chart
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Bar Chart
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Pie Chart vs. Bar Chart

• Which graph do you prefer?
• Why?

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Graphs for Quantitative Data

• Dot Plot shows a dot for each observation
• Stem-and-Leaf Plot portrays the individual
  observations
• Histogram uses bars to portray the data

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Example Sodium and Sugar Amounts in Cereals
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Dotplot for Sodium in Cereals

• Sodium Data
• 0 210 260 125 220 290 210 140
  220 200 125 170 250 150 170 70
  230 200 290 180

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Stem-and-Leaf Plot for Sodium in Cereal

• Sodium Data 0 210
• 260 125
• 220 290
• 210 140
• 220 200
• 125 170
• 250 150
• 170 70
• 230 200
• 290 180

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Frequency Table

• Sodium Data
• 0 210
• 260 125
• 220 290
• 210 140
• 220 200
• 125 170
• 250 150
• 170 70
• 230 200
• 290 180

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Histogram for Sodium in Cereals
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Which Graph?

• Dot-plot and stem-and-leaf plot
• More useful for small data sets
• Data values are retained
• Histogram
• More useful for large data sets
• Most compact display
• More flexibility in defining intervals

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Shape of a Distribution

• Overall pattern
• Clusters?
• Outliers?
• Symmetric?
• Skewed?
• Unimodal?
• Bimodal?

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Symmetric or Skewed ?
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Example Hours of TV Watching
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• Identify the minimum and maximum sugar values

2 and 14 1 and 3
1 and 15 0 and 16
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Consider a data set containing IQ scores for the
general public

• What shape would you expect a histogram of this
  data set to have?
• Symmetric
• Skewed to the left
• Skewed to the right
• Bimodal

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Consider a data set of the scores of students on
a very easy exam in which most score very well
but a few score very poorly

• What shape would you expect a histogram of this
  data set to have?
• Symmetric
• Skewed to the left
• Skewed to the right
• Bimodal

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Section 2.3

• How Can We describe the Center of Quantitative
  Data?

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Mean

• The sum of the observations divided by the number
  of observations

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Median

• The midpoint of the observations when they are
  ordered from the smallest to the largest (or from
  the largest to the smallest)

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Find the mean and median

• CO2 Pollution levels in 8 largest nations
  measured in metric tons per person
• 2.3 1.1 19.7 9.8 1.8 1.2 0.7 0.2
• Mean 4.6 Median 1.5
• Mean 4.6 Median 5.8
• Mean 1.5 Median 4.6

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Outlier

• An observation that falls well above or below the
  overall set of data
• The mean can be highly influenced by an outlier
• The median is resistant not affected by an
  outlier

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Mode

• The value that occurs most frequently.
• The mode is most often used with categorical data

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Section 2.4

• How Can We Describe the Spread of Quantitative
  Data?

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Measuring Spread Range

• Range difference between the largest and
  smallest observations

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Measuring Spread Standard Deviation

• Creates a measure of variation by summarizing the
  deviations of each observation from the mean and
  calculating an adjusted average of these
  deviations

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Empirical Rule

• For bell-shaped data sets
• Approximately 68 of the observations fall within
  1 standard deviation of the mean
• Approximately 95 of the observations fall within
  2 standard deviations of the mean
• Approximately 100 of the observations fall
  within 3 standard deviations of the mean

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Parameter and Statistic

• A parameter is a numerical summary of the
  population
• A statistic is a numerical summary of a sample
  taken from a population