Displaying Distributions with Graphs

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Displaying Distributions with Graphs

• Chapter 11

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Where does my school stand?

• A student at Chicago State University pays 6,834
  in tuition, and she is curious how this compares
  to what students at other universities in
  Illinois pay?
• There are 121 colleges and universities in
  Illinois!
• They charge between 1,536 and 30,729 per year
  in tuition
• Comparing Chicago State to the other 120 in a big
  list would not be practical, so we use a histogram

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How much does my school cost?

• We can also look at the same data using another
  type of graph the stemplot
• To make a stemplot
• Round the tuition charges to the nearest 100
  6,834 becomes 68
• Then put the thousands digit to the left of a
  vertical line (from smallest to largest
  top-to-bottom)
• And finally hang the hundreds digits one-by-one
  to the right of the line on the row that
  corresponds to their thousands digit
• This modified histogram contains more detail
• Chicago State is the 58th most expensive school

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Histograms

• Remember that categorical variables record group
  membership, and we used pie charts and bar
  graphs to display the distributions of
  populations according to categorical variables
• This works fine for categorical variables with
  relatively few categories
• What if we have a quantitative variable (like SAT
  scores) that can (and does) take many different
  values?
• In this case we display the distribution of the
  quantitative variable using a histogram that
  counts values that are similar to each other
  together in one class

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Example 1 How to make a histogram

• Divide the range of the data into classes of
  equal width. The data in Table 11.1 range from
  6.3 to 17.0 so we choose classes
• 6.0 percentage over 65 lt 7.0
• 7.0 percentage over 65 lt 8.0
• 17.0 percentage over 65 lt 18.0
• Count the number of individuals in each class

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Example 1 How to make a histogram

• Draw the histogram
• Label the horizontal axis with the scale for the
  variable whose distribution you are displaying
  percentage of residents ages 65 and over
• Label the classes or bins on your scale from the
  lowest to highest values 4 to 20 in our case
• Label the vertical axis with the scale of the
  counts number of states
• Label the vertical axis tick marks with the count
  values 0 to 15 in our case
• Place bars on the histogram for each class whose
  height corresponds to the count in each class,
  leaving no space between the bars

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Example 1 How to make a histogram

• Keep the bars the same width and dont leave
  space between them unless a class is empty
• Choosing the number of classes or bins is up to
  you
• Too many and there will be lots of empty classes
• Too few and you wont see enough detail
• Depends on your distribution

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Interpreting histograms

• Making an accurate, pretty graph is an
  accomplishment in itself, but its not the final
  step
• After you have the graph you need to examine it
  and interpret it see what it tells you
• First, look for patterns and deviations from the
  patterns

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Interpreting histograms

• In our example histogram, lets begin with the
  obvious deviations at the low and high end of the
  histogram. There are two states that are
  separated from the bulk in the middle, one with
  6.3 and one with 17.0 people 65 and older
• These two outlier states are Alaska and Florida

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Interpreting histograms

• Once we have seen these on the histogram, its
  easy to pick them out of the list as well
• What about a state like Utah with 8.5 65 and
  older?
• This is a matter of judgment, Utah is certainly
  unusually low according to this histogram, but it
  may not qualify as an outlier in the same sense
  as Alaska
• Once we have identified the outliers, the next
  step is to examine them closely and try to figure
  out why
• Commonly they are due to data problems, typing
  errors like 40 instead of 4.0
• Once the data problems are eliminated we need
  more information

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Interpreting histograms

• In this case, we know that Florida is a
  destination for retirees so it makes sense that
  Florida has a large elderly population
• We probably know less about Alaska, but it is a
  northern frontier area that is generally not a
  retirement destination so perhaps this makes
  sense too

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Interpreting histograms

• Now we move on to see the overall pattern of the
  histogram
• To do this we ignore the outliers and focus on
  the bulk of the histogram

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Interpreting histograms

• The center of the distribution is the point on
  the horizontal axis at which roughly half the
  observations are to the left and half the
  observations are to the right
• We can describe the spread of the distribution by
  giving the smallest and largest values ignoring
  the outliers

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Example 2 Describing distributions

• Lets describe our example histogram
• Shape the histogram is roughly symmetric and has
  a single peak
• Center the midpoint of the distribution is close
  to the peak at about 13
• Spread if we ignore outliers, the distribution
  runs from about 8 to about 16
• If we compare this to the histogram of the
  distribution of tuition at Illinois colleges, we
  see that the tuition distribution is quite
  different it is skewed

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Interpreting histograms

• To describe a distribution, look for major
  features
• Get an idea of the midpoint of the distribution
  and its spread
• Look for symmetry and skewness

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Example 3 Sampling again

• The values a statistic takes in many random
  samples from the same population takes a regular
  pattern
• The next histogram displays a distribution we
  went over in Chapter 3
• Take a SRS of 2,527 adults
• Ask each whether they favor a constitutional
  amendment that would define marriage as between a
  man and woman
• The proportion who say yes is the sample
  proportion p-hat
• Do this 1,000 times and collect the 1,000 sample
  proportions p-hat from the 1,000 samples
• Make a histogram next slide

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Example 4 Shakespeares words

• The next slide has a histogram that shows the
  distribution of lengths of words used in
  Shakespeares plays
• It has a single peak and is skewed to the right
• Lots of shorter words and a few longer ones
• The center is about 4 (4 letters)
• The spread is from 1 to 12 letters
• Notice the vertical scale
• Not the count of the words, but the percentage of
  all the words
• This is convenient when the counts are very large
  and when we want to compare distributions with
  different counts

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Interpreting histograms

• The overall shape of a distribution is important
  information about a variable
• Some types of data regularly produce
  distributions with a similar shape, for example
• Sizes of living things (like height of humans)
  tend to produce symmetric distributions
• Data on incomes tends to be skewed right many
  moderate incomes and few very large incomes
• It is common for data to be skewed right when
  there is a hard minimum like 1 for letters in
  words or 0 for income

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Interpreting histograms

• It is also common for distributions to be skewed
  left when there is a hard maximum like test
  scores
• There are many other shapes for distributions as
  well, that are neither skewed nor symmetric
• There may be double-peaked data
• Evenly distributed data, etc.
• Use your eyes and describe exactly what you see
• Heres another example

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Stemplots

• For small data sets a stemplot may be more
  informative than a histogram

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Example 5 Stemplot of the 65 and over data

• For the 65 and older data, the whole number part
  of the percentage is the stem, and the tenths
  digit is the leaf
• Stems can have as many digits as needed, but
  leaves must have only one digit

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Stemplots

• The advantage of stemplots is that the display
  the actual values of the variable in the
  distribution
• You can choose the classes for a histogram, but
  he stems are given to you in a stemplot
• If the data have too many digits and would
  produce a huge number of stems, you can round to
  the nearest hundreds or thousands or whatever to
  reduce the number of digits
• Because there is no choice of classes, stemplots
  are easier to make but can quickly become more
  clumsy and generally dont work well for large
  sets of data

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Summary

• The distribution of a variable tells us what
  values the variable takes, and how often each
  value is taken
• Use a histogram or stemplot to display the
  distribution of a quantitative variable
• When looking at a graph, look for
• The overall pattern,
• Deviations from the overall pattern, and
• Outliers

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Summary

• To describe the overall pattern of a histogram or
  stemplot, describe the
• Shape,
• Center, and
• Spread
• Some distributions have simple shapes that are
  either symmetric or skewed, others have more
  complicated shapes