Displaying the Observed Distribution of Quantitative Variables

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Displaying the Observed Distribution of
Quantitative Variables

• Histogram
• Divide the range of the variable into equally
  spaced intervals - called bins
• Determine the frequency of observations falling
  within each bin
• Form a histogram based on the bin frequencies
• The x axis is the intervals with the interval
  midpoint depicted.
• The y axis is the frequency or relative frequency
• Draw bars the height of frequency centered at the
  interval midpoint.

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Example

• Data frame giving the heights of singers in the
  New York Choral Society. Components are named
  height (inches) and voice.part.
• Cleveland, William S. (1993). Visualizing Data.
  Hobart Press, Summit, New Jersey.

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Example, cont.

• Range 60 to 76 inches
• Frequency distribution

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Height of Singers, Histogram
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What parameters affect the histogram?

• Starting Point
• Bin width
• Lets try the same example but altering these
  parameters.

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Height of Singers, Histogram
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Height of Singers, by Voice Part
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Histogram

• Graphical representation of the frequency
  distribution.
• Graphical representation of the observed values
  of the variable of interest.
• Provides a summary of the observed distribution.
• Shape changes with the interval definitions
  (starting point and interval width)

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Time Series Plots

• If we observe a variable over consecutive time
  points.
• X-axis is time
• Y-axis is the value of the observed variable
• Demonstrates the observed changes over time of
  the variable.
• Major trends
• Seasonal Variation

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Example

• Ozone
• 11 to 22 measurement sites throughout the Houston
  area.
• Hourly measurements (average of 5 minute
  observations for the given hour)
• Focus on one site at 1pm for the year, 1997.
• At what levels does ozone become a concern?

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1997 Ozone (ppm)Location - Downtown HoustonTime
- 1pm
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Bivariate/Multivariate Data?

• Measuring more than one variable at a time.
• How would you graphically describe the
  relationships between the variables?
• Scatterplot
• 2 dimensional histogram

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Example

• Measurements of daily ozone concentration (ppb),
  wind speed (mph), daily maximum temperature
  (degrees F), and solar radiation (langleys) on
  111 days from May to September 1973 in New York.
• Cleveland, William S. (1993). Visualizing Data.
  Hobart Press, Summit, New Jersey.

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Ozone, TemperatureNew York, May to Sept, 1973
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Histograms of Each Variable
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Bivariate Histogram
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Numerical Summaries of Data

• Measures of Central Tendency
• Mean
• Median
• Mode
• Measures of Variation
• Standard Deviation
• Interquartile Range
• Range

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5 Statistic Summary

• 5 Number Summary
• Minimum
• Q1
• Q2
• Q3
• Maximum

• Boxplot
• Box
• Q1
• Q2
• Q3
• Lines to last obs. within
• Lower extreme median - 1.5 x IQR
• Upper extreme median 1.5 x IQR
• Individual points
• Observations beyond the extremes
• Many variations on Boxplots.

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Boxplot