Chapter 11 Displaying Distributions with Graphs

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Chapter 11Displaying Distributions with Graphs
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Histogram

• The most common graph of the distribution of a
  quantitative variable
• To create a histogram,
• Divide the range of the data into classes of
  equal size. Be sure that each individual fall
  into exactly one class.
• Count the number of individuals in each class.
• Draw the histogram. Each bar represents a class.
  The base of the bar covers the class, and the bar
  height is the class count (or percentage).

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Example

• ACT Scores Groups Counts
• 31 26 23 15-18 3
• 32 24 25 19-22 6
• 25 22 30 23-26 7
• 17 18 19 27-30 3
• 15 21 20 31-34 2
• 20 28 24
• 22 29 23

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Histogram

• Unlike the bars of the bar graph, these bars
  should have no space in between them (The only
  time there should be a space is if there is no
  observations in one of the classes).
• is not one right choice of the class size, but if
  you pick too many or too few, it is hard to tell
  the shape of the distribution

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Stemplot

• Only use with a small data set.
• To make a stemplot
• separate the numbers into stems and leaves. The
  stems can have as many numbers as needed, but the
  leaves can only have one number (the final
  number).
• Write the stems in a vertical column on the left
  (starting with the lowest value) and draw a line
  to separate them from the leaves.
• Write the leaves ascending to the right of the
  appropriate stem.

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Stemplot

• If there are equal numbers, then make sure you
  account for each one in your plot
• Make sure you put equal spacing in between the
  numbers.
• Once you begin numbering your stems, you cannot
  skip a stem

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Example

• 90, 87, 12, 45, 67, 34, 38, 54, 61, 92, 33, 21,
  7, 38, 26
• Stem Leaves
• 0 7
• 1 2
• 2 1 6
• 3 3 4 8 8
• 4 5
• 5 4
• 6 1 7
• 7
• 8 7
• 9 0 2

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Stemplot

• If you have a small spread of numbers, you may
  want to divide them into smaller groups (ex.
  Instead of having all the 30s together, put
  30-34 together and 35-39 together) so that you
  can see the shape of the distribution more
  clearly.
• If you have large numbers you may need to round
  so that you can limit your number of stems. If
  this is done, you need to make sure you label
  your graph with the correct information. (ex.
  Is your data in hundreds, thousands, millions)
  

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Example a small spread of numbers

• 20, 21, 22, 21, 20, 23, 20, 25, 26, 28, 29, 28,
  20, 25, 26, 22, 23
• Groups Stem Leaves
• 20-21 2 0 0 0 0 1 1
  
• 22-23 2 2 2 3 3
• 24-25 2 5 5
• 26-27 2 6 6
• 28-29 2 8 8 9

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Example larger numbers

• 15765 ? 15800 ? stem 15 leaf 8
• 6423 ? 6400 ? stem 6 leaf 4
• 19,333 19,300 ? stem 19 leaf 3
• 842 800 ? stem 0 leaf 8
• You can also use decimals as the leaf. Example
  8.6 ? stem 8 leaf 6

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Histogram and Stemplot

• A stemplot looks like a histogram turned on end.
• You can choose the classes in a histogram. The
  classes of a stemplot are given to you.
• The chief advantage of a stemplot is that it
  displays the actual values of the observations.
  Stemplot are also faster to draw than histograms.
• Stemplots do not work well with large data sets,
  because the stems then have too may leaves.

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Interpreting Distributions

• In any graph of data, look for an overall
  pattern and for striking deviations from this
  pattern
• Outlier - individual observations that falls
  outside the overall pattern of the graph.
• Whether an observation is an outlier is to some
  extent a matter of judgment.
• Many outliers are due to mistakes. Other outliers
  point to the special nature of some observations.

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Outlier
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Overall Pattern

• To see the overall pattern, Ignore any outliers.
• See if the Distribution has a simple shape
• Center and Spread

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Shape of a Distribution

• Symmetric right and left sides of the histogram
  are approximately mirror images of each other

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Skewness

• Skewed to the right if the right side of the
  histogram extends much farther out than the left
  side .(the histogram has a long right tail)

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Skewness

• Skewed to the left if the left side of the
  histogram extends much farther out than the right
  side (the histogram has a long left tail)

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• Center where the middle of the distribution
  seems to be (half of the values are below this
  point and half are above)
• Spread the lowest and highest numbers obtained
  by the graph on the x-axis (usually excluding the
  outliers)
• We will describe center and spread numerically in
  chapter 12.