Mathematical Plots By: Amber Stanek

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

~Line Plots~

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.

Let’s Do an Example Together… We’ll 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

~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

~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:

~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.

~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.

~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?

~Answers~ Mode = 77 degrees Fahrenheit Coolest Temperature = 69 degrees Fahrenheit Warmest Temperature = 93 degrees Fahrenheit

~Stem and Leaf Plots~

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.

Let’s Do an Example Together… We’ll use the data we used when making the line plot earlier: the temperature every day for the month of August.

~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

~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

~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 |

~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

~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.

~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.)

~Answers~ Mean = 77.65 degrees Fahrenheit Median = 77 degrees Fahrenheit Mode = 77 degrees Fahrenheit

~Box and Whisker Plots~

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 doesn’t show all the data values, but instead focuses on the median, extremes, and quartiles.

extremes, and quartiles? What are the median, extremes, and quartiles?

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 (You’re 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.)

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.

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 = 5…5/2 = 2.5). The upper quartile is 12.5 (11 + 14 = 25…25/2 = 12.5).

Let’s Do an Example Together… We’ll 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.

~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

~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

~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.)

~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.)

~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.)

~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:

~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:

~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:

~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.

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

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

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 it’s 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.

Practice Worksheet on Plots – This worksheet gives you an opportunity to apply what you’ve 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.

Reference Websites on Mathematical Plots I’m including the links to some websites that can be used as references for the plots you’ve learned about in this power point presentation. Intermath – This site gives a description of the plots we’ve discussed in this power point. It will also give you an example of each. "M&M'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.

Statistical Websites to Create Your Own Plots I’m 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."