Chapter 1 Looking at Data

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Chapter 3 Looking at DataDistributions
Chapter Three Looking At Data Distributions
Introduction 3.1 Displaying Distributions with
Graphs
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3.1 Displaying Distributions with Graphs

• Variables
• Examining Distributions of Variables
• Graphs for Categorical Variables
• Bar graphs
• Pie charts
• Graphs for Quantitative Variables
• Histograms
• Stemplots
• Time plots

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Statistics
Statistics is the science of learning from
data. The first step in dealing with data is to
organize your thinking about the data. An
exploratory data analysis is the process of using
statistical tools and ideas to examine data in
order to describe their main features.

• Exploring Data
• Begin by examining each variable by itself. Then
  move on to study the relationships among the
  variables.
• Begin with a graph or graphs. Then add numerical
  summaries of specific aspects of the data.

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Variables
We construct a set of data by first deciding
which cases or observations or individuals or
units we want to study. For each case, we record
information about characteristics that we call
variables.
Individual An object described by data
Categorical variable Places individual into one
of several groups or categories.
Variable Characteristic of the individual
Quantitative variable Takes numerical values for
which arithmetic operations make sense.
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Quantitative Variables

• Quantitative variables can either be counts or
  measurements or rates.
• SEE EXAMPLE 3.6 ON PAGE 84 (IN THE CHAPTER 3
  INTRODUCTION) FOR WHY RATES ARE IMPORTANT
• RATE of occurrences of the event per X in the
  population of all possible occurrences (where X
  is a large number (10,000 100,000 e.g.)
• Murder rate in NH County (murders in
  NHC/possible murders(i.e., population) large
  number (like 100,000)
• (in 2012, 9 murders, population estimate209234
  so the murder rate in NH County in 2012 is
  9/209234 0.00004301. Multiply by 100,000 to get
  the rate per 100,000. 0.00004301 100000
  4.3 murders per 100,000 people (or per capita)
  Guilford County 26/500879 100000 5.2 etc.

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Distribution of a Variable
To examine a single variable, we graphically
display its distribution.

• The distribution of a variable tells us what
  values it takes and with what frequency it takes
  on these values.
• Distributions can be displayed using a variety of
  graphical tools. The proper choice of graph
  depends on the kind of the variable and how easy
  it is to draw them. JMP makes easy work of
  graphing!

Categorical variable Pie chart I dont
recommend pie charts! Bar graph these are fine!
Quantitative variable Histogram Stemplot
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Categorical Variables

• The distribution of a categorical variable lists
  the categories and gives the count or percent of
  individuals who fall into that category.
• Pie charts show the distribution of a categorical
  variable as a pie whose slices are sized by the
  counts or percents for the categories hard to
  draw and hard to interpret!
• Bar graphs represent each category as a bar whose
  heights show the category counts or percents.

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Pie Charts and Bar Graphs
Material Weight (million tons) Percent of total
Food scraps 25.9 11.2
Glass 12.8 5.5
Metals 18.0 7.8
Paper, paperboard 86.7 37.4
Plastics 24.7 10.7
Rubber, leather, textiles 15.8 6.8
Wood 12.7 5.5
Yard trimmings 27.7 11.9
Other 7.5 3.2
Total 231.9 100.0
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Quantitative Variables

• The distribution of a quantitative variable tells
  us what values the variable takes on and the
  frequency with which it takes on those values.
• Histograms show the distribution of a
  quantitative variable by using bars whose height
  represents the number of individuals who take on
  a value within a particular class.
• Stemplots separate each observation into a stem
  and a leaf that are then plotted to display the
  distribution while maintaining the original
  values of the variable original values are not
  hidden as in the histogram.
• Time plots plot each observation (on the vertical
  axis) against the time at which it was measured
  (on the horizontal axis).

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Stemplots

• To construct a stemplot
• Separate each observation into a stem (first part
  of the number) and a leaf (the remaining part of
  the number).
• 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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Stemplots
Example Weight data?Introductory Statistics class
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Stemplots

• If there are very few stems (when the data cover
  only a very small range of values), then we may
  want to create more stems by splitting the
  original stems.
• Example If all of the data values were between
  150 and 179, then we may choose to use the
  following stems

Leaves 04 would go on each upper stem (first
15), and leaves 59 would go on each lower stem
(second 15).
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Histograms

• For quantitative variables that take many values
  and/or large datasets
• Divide the possible values into classes (equal
  widths).
• Count how many observations fall into each
  interval (may change to percents).
• Draw picture representing the distribution?bar
  heights are equivalent to the number (percent) of
  observations in each interval.
• JMP does all three of the above with a couple of
  clicks Analyze -gt Distribution -gt choose the
  variable to be plotted.

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Histograms
Example Weight data?Introductory Statistics class
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Examining Distributions

• In any graph of data, look for the overall
  pattern and for striking deviations from that
  pattern.
• You can describe the overall pattern by its
  shape, center, and spread.
• An important kind of deviation is an outlier, an
  individual that falls outside the overall pattern.

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Examining Distributions - Shape

• A distribution is symmetric if the right and left
  sides of the graph are approximately mirror
  images of each other.
• A distribution is skewed to the right
  (right-skewed) if the right side of the graph
  (containing the half of the observations with
  larger values) is much longer than the left side.
• A distribution is skewed to the left
  (left-skewed) if the left side of the graph is
  much longer than the right side.

Symmetric
Skewed-left
Skewed-right
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Outliers

• An important kind of deviation is an outlier.
  Outliers are observations that lie outside the
  overall pattern of a distribution. Always look
  for outliers and try to explain them.

This overall pattern is fairly symmetrical,
except for two states that clearly do not belong
to the main trend. Alaska and Florida have
unusual representation of the elderly in their
populations. Large gaps in the distribution are
places to look for outliers.
Alaska
Florida
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Time Plots

• A time plot shows behavior of a quantitative
  variable over time.
• Time is always on the horizontal axis, and the
  variable being plotted is on the vertical axis.
• Look for an overall pattern (trend), and
  deviations from this trend. Connecting the data
  points by lines may emphasize this trend.
• Look for patterns that repeat at known regular
  intervals (seasonal variations)
• Go over the US Regular Retail Gas Prices in JMP

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Time Plots
Look at the gas price data
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HW use JMP whenever possible to draw the graphs
HW Begin reading Intro. to Ch.3 and section 3.1
work through the Examples in 3.1 do the
Exercises 3.7, 3.10-3.14, 3.21, 3.24, 3.25,
3.27, 3.32, 3.33-3.36, 3.38 (JMP), 3.39