SUMMARIZING AND GRAPHING DATA

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SUMMARIZING AND GRAPHING DATA
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Overview
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Planning and Conducting a Study

• Understand the Nature of the Problem
• Decide What to Measure and How to Measure It
• Data Collection
• Data Summarization and Preliminary Analysis
• Formal Data Analysis
• Interpretation of Results

Statistics The Exploration and analysis of Data,
4th ed. Devore/Peck
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Important Characteristics of Data

• Center A representative or average value that
  indicates where the middle of the data set is
  located.
• Variation A measure of the amount that the data
  values vary among themselves.
• Distribution The nature or shape of the
  distribution of the data.
• Outliers Sample values that lie very far away
  from the vast majority of the other sample
  values.
• Time Changing characteristics of the data over
  time.

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Frequency Distributions
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Definition

• A frequency distribution (or frequency table)
  lists data values (either individually or by
  groups of intervals), along with their
  corresponding frequencies (or counts).

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More Definitions

• Lower class limits are the smallest numbers that
  can belong to the different classes.
• Upper class limits are the largest numbers that
  can belong to the different classes.
• Class boundaries are the numbers used to separate
  classes, but without the gaps created by class
  limits.
• Class midpoints are the values in the middle of
  the classes. Each class midpoint can be found by
  adding the lower class limit to the upper class
  limit and dividing the sum by 2.
• Class width is the difference between two
  consecutive lower class limits or two consecutive
  lower class boundaries.

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Procedure for Constructing a Frequency
Distribution

• Decide on the number of classes needed.
• CalculateRound this result to get a convenient
  number.
• Starting point Begin by choosing a number for
  the lower limit of the first class.
• Using the lower limit of the first class and the
  class width, proceed to list the other lower
  class limits.
• List the lower class limits in a vertical column
  and proceed to enter the upper class limits.
• Go through the data set putting a tally in the
  appropriate class for each data value. Use the
  tally marks to find the total frequency for each
  class.

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Example

• Use Data Set 6 Bears, and construct a frequency
  distribution for the lengths of bears using 11
  classes.

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Example (continued)
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Example (continued)
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Relative Frequency Distribution

• A relative frequency distribution includes the
  same class limits as a frequency distribution,
  but relative frequencies (a relative frequency is
  found by dividing a class frequency by the total
  frequency) are used instead of actual frequencies.

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Example

• Use Data Set 6 Bears, and construct a relative
  frequency distribution for the lengths of bears
  using 11 classes.

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Example (continued)
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Cumulative Frequency Distribution

• A cumulative frequency distribution includes the
  same class limits as a frequency distribution,
  but cumulative frequencies (a cumulative
  frequency for a class is the sum of the
  frequencies for that class and all previous
  classes) are used instead of actual frequencies.

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Interpreting Frequency Distributions

• Is the distribution normal?
• Do the frequencies start low, then increase to
  some maximum frequency, then decrease to a low
  frequency?
• Is the distribution approximately symmetric? That
  is, are the frequencies evenly distributed on
  both sides of the maximum frequency?

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Histograms
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Histogram

• A histogram is a bar graph in which the
  horizontal scale represents classes of data
  values and the vertical scale represents
  frequencies. The heights of the bars correspond
  to the frequency values, and the bars are drawn
  adjacent to each other (without gaps).

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Example

• Use Data Set 6 Bears, and construct a histogram
  for the lengths of bears using 11 classes.

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Example (continued)
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Histogram

• A relative frequency histogram has the same shape
  and horizontal scale as a histogram, but the
  vertical scale is marked with relative
  frequencies instead of actual frequencies.

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Example

• Use Data Set 6 Bears, and construct a relative
  frequency histogram for the lengths of bears
  using 11 classes.

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Example (continued)
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Interpreting Histograms

• Is the distribution normal?
• Do the frequencies start low, then increase to
  some maximum frequency, then decrease to a low
  frequency?
• Is the distribution approximately symmetric? That
  is, are the frequencies evenly distributed on
  both sides of the maximum frequency?

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Statistical Graphics
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Frequency Polygons

• A frequency polygon uses line segments connected
  to points located directly above class midpoint
  values.
• A cumulative frequency polygon (or ogive) uses
  line segments connected to points located
  directly above class midpoint values.

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Dotplot

• A dotplot consists of a graph in which each data
  value is plotted as a point (or dot) along a
  scale of values. Dots representing equal values
  are stacked.

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Stemplot

• A stemplot (or stem-and-leaf-plot) represents
  data by separating each value into two parts
• the stem (such as the leftmost digit), and
• the leaf (such as the rightmost digit).

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Pareto Charts

• A Pareto chart is a bar graph for qualitative
  data, with the bars arranged in order according
  to frequency. Vertical scales in Pareto charts
  can represent frequencies or relative frequencies.

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Pie Charts

• A pie chart is a graph depicting qualitative data
  as slices of a pie.

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Scatterplots

• A scatterplot (or scatter diagram) is a plot of
  paired (x, y) data with a horizontal x-axis and a
  vertical y-axis. The data are paired in a way
  that matches each value from one data set with a
  corresponding value from a second data set.

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Time-Series Graph

• A time-series graph is graph of time-series data,
  which are data that have been collected at
  different points in time.

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Presenting Data Graphically

• Some important principles
• For small data sets of 20 values or fewer, use a
  table instead of a graph.
• A graph of data should make the viewer focus on
  the true nature of the data, not on other
  elements, such as eye-catching but distracting
  design features.
• Do not distort the data construct a graph to
  reveal the true nature of the data.
• Almost all of the ink in a graph should be used
  for the data, not for other design elements.

The Visual Display of Quantitative Information,
2nd ed. Tufte
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Presenting Data Graphically

• Some important principles
• Dont use screening consisting of features such
  as slanted lines, dots, or cross-hatching,
  because they create the uncomfortable illusion of
  movement.
• Dont use areas or volumes for data that are
  actually one-dimensional in nature.
• Never publish pie charts, because they waste ink
  on non-data components, and they lack an
  appropriate scale.

The Visual Display of Quantitative Information,
2nd ed. Tufte