Mathematical Plots

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Mathematical Plots

• By Amber Stanek

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Types of Plots

• There are 3 types of plots you will be learning
  about in this power point presentation. They
  are
• Line Plots
• Box and Whisker Plots
• Stem and Leaf Plots

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Line Plots
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What is a line plot?

• A line plot (sometimes called a dot plot) is a
  graph that uses a number line to show the
  frequency of data.
• It helps to get a clearer understanding of a
  small number of observations.

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Lets Do an Example Together

• Well use the data from the temperatures every
  day for the month of August. There are 31 days in
  August, and here are the temperatures for that
  month August 76 82 83 90 93 85 78 75 71 71
  72 69 70 75 77 83 85 82 80 81 77 76 78 74 72 73
  77 77 76 77 72

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Step One

• There are 31 days in August, and here are the
  temperatures for that month
• August 76 82 83 90 93 85 78 75 71 71 72 69 70
  75 77 83 85 82 80 81 77 76 78 74 72 73 77 77
  76 77 72
• The first step is to take these data values and
  put them in order from least to greatest. -
  Example
• 69 70 71 71 72 72 72 73 74 75 75 76 76 76 77 77
  77 77 77 78 78 80 81 82 82 83 83 85 85 90 93

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Step Two

• Create a number line that is equally spaced and
  contains all the data.
• A good scale to use for this problem is a number
  line that increases in increments of 5.
• Example

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Step Three

• Put dots above the numbers to show the data for
  the variable.
• -Example The first data point is 69 degrees
  Fahrenheit, so a dot is placed above the number
  line at 69. Continue until all the data for the
  daily August temperatures has been recorded.

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Final Touches

• You have just created a line plot, but there are
  still two details needed to complete it. They are
  a title and a key.
• Title
• You need to choose a title that will explain
  what your line plot is about. A good title for
  this plot might be "Temperatures for August."
• Key
• You also need to have a key for your line plot.
  This tell anyone who looks at your plot what each
  data point represents. For your plot, a good
  example might be 76 76 degrees Fahrenheit.

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Final Product

• You have just completed your own line plot!
  Here's what it should look like
• Can you figure out the mode? What about the
  coolest temperature in August and the warmest
  temperature?

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Answers

• Mode 77 degrees Fahrenheit
• Coolest Temperature 69 degrees Fahrenheit
• Warmest Temperature 93 degrees Fahrenheit

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Stem and Leaf Plots
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What is a stem and leaf plot?

• A stem and leaf plot is a type of graph that
  shows the shape of a set of data.
• It is arranged in rows of grouped scores.
• Often the stem is the leading or most significant
  digit(s) in the group. The leaves relate to the
  least significant digits in the group.
• All scores are displayed on the plot in order.

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Lets Do an Example Together

• Well use the data we used when making the line
  plot earlier the temperature every day for the
  month of August.

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Step One

• August 76 82 83 90 93 85 78 75 71 71 72 69 70
  75 77 83 85 82 80 81 77 76 78 74 72 73 77 77
  76 77 72
• The first step is to take these data values and
  put them in order from least to greatest. -
  Example
• 69 70 71 71 72 72 72 73 74 75 75 76 76 76 77 77
  77 77 77 78 78 80 81 82 82 83 83 85 85 90 93

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Step Two

• The second step is to put the data values in sets
  of intervals. For this example, intervals of ten
  would be good to use. You'll put all the
  temperatures in the 60's together, and all the
  temperatures in the 70's together. Do the same
  thing for the temperatures in the 80's, and also
  for those in the 90's. - Example 69 70 71
  71 72 72 72 73 74 75 75 76 76 76 77 77 77 77 77
  78 78 80 81 82 82 83 83 85 85 90 93

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Step Three

• In step three you will create the stem of your
  stem and leaf plot. To do this, you will start
  with the lowest temperature (69). Take the tens
  digit of this temperature (6), and use this as
  your stem. Do the same thing for all the other
  temperatures by finding the tens digit of each
  one. (You don't need to do anything with the ones
  digit of each temperature at this point.) -
  Example 6 7 8 9

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Step Four

• The fourth step is to add the leaves to your stem
  and leaf plot. To do this, you use the digits in
  the ones place for each temperature. Put these
  leaves to the right of the stem they belong with.
  Be careful! There should be a number representing
  the temperature for every day of the month. Even
  if two or more temperatures are the same, you
  must put a leaf in for each value. Here's one
  last reminder before you try this step keep the
  leaves in order from least to greatest. -
  Example 6 9 7 0 1 1 2 2 2 3 4 5 5 6 6 6 7
  7 7 7 7 8 8 8 0 1 2 2 3 3 5 5 9 0 3

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Final Touches

• You have just created a beautiful stem and leaf
  plot, but there are still two details needed to
  complete it. They are a title and a key.
• Title
• You need to choose a title that will explain
  what your stem and leaf plot is about. A good
  title for this plot might be "Temperatures for
  August."
• Key
• You also need to have a key for your stem and
  leaf plot. This tells anyone who looks at your
  plot what each data point represents. You can
  pick any value you want to use as an example of
  what the stem and leaves represent. For your
  plot, a good example might be 7 6 76
  degrees Fahrenheit.

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Final Product

• You did it! You just completed your very own stem
  and leaf plot! Here's what it should look like
  Temperatures for August 6 9 7 0 1 1 2 2 2
  3 4 5 5 6 6 6 7 7 7 7 7 8 8 8 0 1 2 2 3 3 5 5
  9 0 3 Key 7 6 76 degrees Fahrenheit
• From this graph and your previous knowledge, can
  you determine the mean, median, and mode? (Go to
  the next slide to see the answers.)

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Answers

• Mean 77.65 degrees Fahrenheit
• Median 77 degrees Fahrenheit
• Mode 77 degrees Fahrenheit

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Box and Whisker Plots
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What is a box and whisker plot?

• A box and whisker plot is a visual representation
  of how data is spread out and how much variation
  there is.
• It doesnt show all the data values, but instead
  focuses on the median, extremes, and quartiles.

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What are the median, extremes, and quartiles?
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Median

• The median is the middle value of an ordered set
  of numbers. (If the numbers are not in order
  from least to greatest, do this first!)
• -Example 2 3 7 11 14
• -In this case, the median is 7. (There are 2
  values below it and 2 values above it).
• Note If there is an even amount of data
  values, the median is found by adding the 2
  middle values together and then dividing that
  number by 2.
• -Example 2 3 7 11 14 19
• -In this case, you do the following 7 11
  18 (Youre adding the 2 middle values). Then
  you divide 18/2 9. The median is 9. (There
  are 3 numbers below 9 and 3 numbers above it.)

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Extremes

• The extremes are the highest and lowest values of
  the data set.
• -Example 2 3 7 11 14
• -In this case, the lower extreme is 2 and the
  upper extreme is 14.

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Quartiles

• The quartiles are the median of the higher half
  of the data set and the median of the lower half
  of the data set.
• -Example 2 3 7 11 14
• -In this case, the lower quartile is 2.5 (2
  3 55/2 2.5). The upper quartile is 12.5
  (11 14 2525/2 12.5).

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Lets Do an Example Together

• Well use the data we used when making a line
  plot and a stem and leaf plot earlier the
  temperature every day for the month of August.

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Step One

• There are 31 days in August, and here are the
  temperatures for that month
• August 76 82 83 90 93 85 78 75 71 71 72 69 70
  75 77 83 85 82 80 81 77 76 78 74 72 73 77 77
  76 77 72
• The first step is to take these data values and
  put them in order from least to greatest. -
  Example
• 69 70 71 71 72 72 72 73 74 75 75 76 76 76 77 77
  77 77 77 78 78 80 81 82 82 83 83 85 85 90 93

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Step Two

• The second step is to find the extremes.
  Remember, the lower extreme is the lowest value
  and the upper extreme is the highest value.
• -Example
• 69 70 71 71 72 72 72 73 74 75 75 76 76 76 77 77
  77 77 77 78 78 80 81 82 82 83 83 85 85 90 93
• Lower Extreme 69
• Upper Extreme 93

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Step Three

• The third step is to find the median. Remember,
  the median is the middle value.
• -Example
• 69 70 71 71 72 72 72 73 74 75 75 76 76 76 77 77
  77 77 77 78 78 80 81 82 82 83 83 85 85 90 93
• -There are 31 values, therefore the median is
  the 16th value 77. (There are 15 values below
  it, and 15 values above it.)

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Step Four

• The fourth step is to find the lower quartile.
  Remember, this is the median of the lower half of
  the data.
• -Example
• 69 70 71 71 72 72 72 73 74 75 75 76 76 76 77
• -There are 15 values therefore the median is
  the 8th value 73. (There are 7 values below
  it, and 7 values above it.)

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Step Five

• The fifth step is to find the upper quartile.
  Remember, this is the median of the upper half of
  the data.
• -Example
• 77 77 77 78 78 80 81 82 82 83 83 85 85 90 93
• -There are 15 values therefore the median is
  the 8th value 82. (There are 7 values below
  it, and 7 values above it.)

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Step Six

• The sixth step is to plot the extremes (lower
  extreme 69, upper extreme 93), the quartiles
  (lower quartile 73 and upper quartile 82), and
  the median (77) on a number line.
• -Example

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Step Seven

• The seventh step is to draw a rectangular box
  extending from the lower quartile to the upper
  quartile. Indicate the median with a vertical
  line extending through the box.
• -Example

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Step Eight

• The eighth step is to connect the lower extreme
  to the lower quartile with a line (one "whisker")
  and the upper quartile to the upper extreme with
  another line (the other "whisker).
• -Example

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Final Touches

• You have just created a box and whisker plot, but
  there are still two details needed to complete
  it. They are a title and a key.
• Title
• You need to choose a title that will explain
  what your box and whisker plot is about. A good
  title for this plot might be "Temperatures for
  August."
• Key
• You also need to have a key for your box and
  whisker plot. This tells anyone who looks at your
  plot what each data point represents. For your
  plot, a good example might be 76 76 degrees
  Fahrenheit.

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Final Product

• You have just finished your very own box and
  whisker plot! Congratulations! Here is what
  your final product should look like

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What does the graph tell you?

• Well, you can see that the lowest temperature in
  August was 69 degrees Fahrenheit and the highest
  temperature was 93 degrees Fahrenheit. This
  gives you the range of the data 24. (93-69
  24)
• You also know that the median or middle value is
  77 degrees Fahrenheit.
• Since the medians (3 of them) represent the
  middle points, they split the data into 4 equal
  parts.
• One quarter of the data numbers are less than 73
• One quarter of the data numbers are between 73
  and 77
• One quarter of the data numbers are between 77
  and 82
• One quarter of the data numbers are greater than
  82

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A Special Case

• At some point you might see a box and whisker
  plot that has an asterisk like the example below.
  Sometimes there is one piece of data that falls
  way outside the range of the other values. This
  piece of data is called an outlier, and its
  shown by an asterisk in a box and whisker plot.
  If the outlier is included in the whisker, people
  might think that there are values dispersed
  throughout the whole range from the first
  quartile to the outlier. This would give them a
  false representation of the data.

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Practice

• Worksheet on Plots This worksheet gives you an
  opportunity to apply what youve learned about
  stem and leaf, box and whisker, and line plots.
  There is data provided for you to make these
  three plots.
• Answers to Stem and Leaf Plot Here are the
  answers to the stem and leaf plot from the
  worksheet above.
• Answers to Box and Whisker Plot Here are the
  answers to the box and whisker plot from the
  worksheet above.
• Answers to Line Plot - Here are the answers to
  the line plot from the worksheet above.

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Reference Websites on Mathematical Plots

• Im including the links to some websites that
  can be used as references for the plots youve
  learned about in this power point presentation.
• Intermath This site gives a description of the
  plots weve discussed in this power point. It
  will also give you an example of each.
• "MM's" Candies, Line Plots, and Graphing - This
  is a fun lesson plan on line plots and graphing
  for kids in grades 4-7. They use M M Candies
  for their data.
• Stem and Leaf Plot - This site gives a basic
  example of how to create a stem and leaf plot. It
  takes you through each step of the process.
• About Math - This site defines a stem and leaf
  plot in detail. It also gives you a few examples
  of how to create them.
• Box and Whisker Plot - This website will give you
  instructions on how to construct a box and
  whisker plot. There is also an example of one for
  you to look at.
• Statistics Canada - Here is the definition, step
  by step instructions, and an example of how to
  create a box and whisker plot.

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Statistical Websites to Create Your Own Plots

• Im included a couple links to statistical
  website. You can used the data to create your
  own plots.
• NFL Stats - This website gives the 2004 regular
  season National Football League statistics. I've
  chosen Brett Favre's statistics to be used as
  your data values. Create the three plots you've
  learned about using the statistics for his total
  touchdowns each year. They can be found in the
  eleventh column under the heading "TD.
• Basketball Reference - Here are Michael Jordan's
  statistics from his career in the National
  Basketball Association. Use his total points
  every season as your data values. Create a line,
  box and whisker, and stem and leaf plot using
  these statistics. His total point can be found in
  the seventh column under the heading "Pts."