Displaying Quantitative Data

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Chapter 4

• Displaying Quantitative Data

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Dealing With a Lot of Numbers...

• When looking at large sets of quantitative data,
  it can be difficult to get a sense of what the
  numbers are telling us without summarizing the
  numbers in some way.
• In this chapter, we will concentrate on graphical
  displays of quantitative data.

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Percent of Population over 65 per state (1996)

• 13.0 14.3 12.5 13.9 13.8 12.5 15.8 12.1
• 5.2 12.8 12.6 11.4 11.4 14.5 12.1 11.2
• 13.2 18.5 15.2 14.1 12.0 13.4 14.4 11.6
• 14.4 9.9 13.7 12.4 13.8 13.5 12.5 15.2
• 10.5 12.9 12.6 12.4 11.0 13.4 10.2 13.3
• 11.0 11.4 11.4 12.3 13.4 15.9 8.8 11.2
• 13.8 13.2

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What do these data tell us?

• Make a picture
• Histogram
• Stem-and-Leaf Display
• Dot plot
• First three things to do with data
• Make a picture
• Make a picture
• Make a picture

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Displaying Quantitative Data

• Histogram
• Give each graph a title
• Give each one of the axes a label
• Make as neat as possible
• Computer
• Grid paper

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Displaying Quantitative Data

• Histogram
• Divide data values into equal-width piles (called
  bins)
• Count number of values in each bin
• Plot the bins on x-axis
• Plot the bin counts on y-axis

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Example Population Over 65

• Decide on bin values
• Low value is 5.2 and high value is 18.5
• Bins are 5.0 up to 6.0, 6.0 up to 7.0, etc.
• Written as 5.0 X lt 6.0, 6.0 X lt 7.0
• Count number of values in each bin
• Bin 5.0 X lt 6.0 has 1 value
• Bin 6.0 X lt 7.0 has 0 values
• Bin 7.0 X lt 8.0 has 0 values
• Bin 8.0 X lt 9.0 has 1 value
• Continue counting values in each bin

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Example Population Over 65

• Plot bins on x-axis
• 14 bins from 5.0 X lt 6.0 to 18.0 X lt 19.0
• Plot bin counts on y-axis
• Bin counts are 1, 0, 0, 1, 1, 2, 9, 13, 13, 5,
  4, 0, 0, 1

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Displaying Quantitative Data

• Stem and Leaf Display
• Picture of Distribution
• Generally used for smaller data sets
• Group data like histograms
• Still have original values (unlike histograms)
• Two columns
• Left column Stem
• Right column Leaf

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Displaying Quantitative Data

• Stem and Leaf Display
• Leaf
• Contains the last digit of the values
• Arranged in increasing order away from stem
• Stem
• Contains the rest of the values
• Arranged in increasing order from top to bottom

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Example Population Over 65

• Leaf tenths digit
• Stem tens and ones digits
• Ex. 5 2
• Ex. 10 2 5
• Ex. 14 1 3 4 4 5

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Percent of Population over Age 65 (by state) in
1996
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Example Frank Thomas

• Career Home Runs (1990-2004)
• 4 7 15 18 24 28 29 32 35 38 40 40
  41 42 43

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Displaying Quantitative Data

• Back-to-back Stem-and-Leaf Display
• Used to compare two variables
• Stems in center column
• Leafs for one variable right side
• Leafs for other variable left side
• Arrange leafs in increasing order,
• AWAY FROM STEM!

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Example Compare Frank Thomas to Ryne Sandberg

• Career Home Runs for Ryne Sandberg (1981-1997)
• 0 5 7 8 9 12 14 16 19 19 25
  26 26 26 30 40

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Displaying Quantitative Data

• If there are a large number of observations in
  only a few stems, we can split stems.
• Split the stems into two stems
• First stem is 0 4.
• Second stem is 5 9.
• If you choose to split one stem you MUST split
  them all!

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Example Population Over 65
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Looking at Distributions

• Always report 3 things when describing a
  distribution
• Shape
• Center
• Spread

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Looking at Distributions

• Shape
• How many humps (called modes)?
• None uniform
• One unimodal
• Two bimodal
• Three or more multimodal

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Unimodal vs Bimodal
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Looking at Distributions

• Shape
• Is it symmetric?
• Symmetric roughly equal on both sides
• Skewed more values on one side
• Right Tail stretches to large values
• Left Tail stretches to small values
• Are there any outliers?
• Interesting observations in data
• Can impact statistical methods

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Examples of Skewness
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Looking at Distributions

• Center
• A single number to describe the data
• Can calculate different numbers for center

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Looking at Distributions

• Spread
• Variation in the data values
• Smallest observation to the largest observation
• May take into account any outliers
• Later, spread will be a single number

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Example Population Over 65

• Shape
• Unimodal
• Symmetric
• Two Outliers (5 and 18)
• Center - 12
• Spread - Almost all observations are between 8
  and 16

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Example Frank Thomas
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Example Compare Frank Thomas to Ryne Sandberg
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What Do We Know?

• Histograms, Stem-and-Leaf Displays, Back-to-Back
  Stem-and-Leaf Displays
• When describing a display, always mention
• Shape number of modes, symmetric or skewed
• Spread
• Center
• Outliers (mention them if they exist otherwise,
  say there are no outliers)

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What Do We Know? (cont.)

• A graph is either symmetric or skewed, not both!
• If a graph is skewed, be sure to specify the
  direction
• Skewed left or skewed right