The Cat is Back; What is He Plotting?

After learning about the measures of central tendency in my math class, we were introduced to the Box and Whisker Plot. This type of plot uses numerical data and focuses around the median. It is also known as the Five Number Summary due to the five numbers used to create the plot:

1. minimum
2. maximum
3. median (or Q2)
4. Q1 (or first or lower quartile) 
5. Q3 (or third or upper quartile) 

I found this data display to be quite interesting; particularly, because it provides the data spread in terms of the median. As a future teacher, I think this plot would provide a better pulse of how my students are actually doing. Using just the median for example, would only provide me with the middle grade received on the quiz. The median isn't affected by outliers, so I could have a median of a "C", but I could also have a few "F's" and not be aware of them. You can see how this may be deceiving; enter the Box and Whisker Plot.

My previous post Mean, Median, and Mode...Oh My! provides the details involved in finding the median of a data set, therefore, I am not going to reiterate it here. Feel free to check that out first if you need clarification.

Work by me

The steps to creating a Box and Whisker Plot are as follows; please reference my plot above as well, it makes it easier:

1. Determine the minimum and maximum data points in your set, and create a number line accordingly.
2. Draw a dot above the number line where both the minimum and maximum fall.
3. Determine the median (or Q2) of your data set and draw a line above the number line.
4. Strike a line through your data set to signify where your median is; this has divided your data set into two halves.
5. Determine the median of the first half (Q1), and draw a line above the number line.
6. Determine the median of the second half (Q3), and draw a line above the number line.
7. Using the Q1 and Q3 lines as end points, draw a rectangle to represent the "box" portion of the Box and Whisker Plot.
8. Draw a line from each end of the box to connect it to the minimum/maximum dots; this represents the "whisker" portion of the Box and Whisker Plot.

That's all it takes to create it! I do suggest making your own. I looked for simulators for the purpose of this post, and found some at shodor.org or meta-chart.com. However, I did not like the feel of them, and they weren't very user friendly.

Now, for reading it; which brings me back to my statement earlier about how this plot would provide a better pulse of how my students are actually doing. Basically, each section of the plot depicts 25% of your data points (as you can see in my image above). This may be confusing, because they are not equal sections. Remember though, my data points are not equally spaced either. So, if my data set above represented quiz scores, I would be able to determine a few things just by reading the plot. To name a few, I would see: the minimum/maximum scores, that 50% of the class scored between 10-15 points, that 25% scored between 4-10 points, and that 25% scored between 15-19 points.

So, you can see how the Box and Whisker Plot can be much more useful than just the median.

Happy Teaching!

-Amanda

Mean, Median, and Mode...Oh My!

The most recent topic discussed, and probably the most easily confused, was the measure of central tendency; or mean, median, and mode. I hope to provide you with simple definitions for each, my interpretation that is, as well as an example of a situation where each would be the best representation of the data. I will use the following numbers in all examples below:

10, 11, 13, 13, 15, 19, 22, 23

Mean

All data points are added together, and their sum is divided by the total number of data points. The mean is also commonly referred to as the "average," and is used with numerical data.

Work by me

It is best to use the mean when there are no outliers, or data points that are far away from the other data (an example would be the number 50 using the above numbers). An example of when the mean would be the best representation of the data is a sprint race. A sprint race is a short footrace that is usually less than a quarter of a mile run at top speed. Generally, you would expect that all finish times are around the same time. So, if you were trying to determine an "average" of how long it took each runner to run the race, the mean would be a good way to determine it.

Median
The median splits the data points exactly in half, and is used with numerical data. If the data points are not already in ascending order, this must be done first before you can proceed. The easiest way to determine the median, is to cross off a data point from each end until you have one data point remaining in the center. If there are an odd number of data points, this is easy to do. However, if you have an even number of data points as I do in my data set, you will actually end up with two data points left in the center. In this case, you will need to find the mean of the two remaining data points in order to determine the median.

Work by me

It is best to use the median when there are outliers present, as they should not affect the outcome. An example of when the median would be the best representation of the data is with test scores. You may have some zeros for those who did not take the test, and you may have some that scored a high "A." However, to obtain a better idea of how the class performed overall, you would not want these outliers taken into account. Therefore, the median may be the best fit here.

Mode
The data point(s) that occurs most frequently in the set, and is used with both numerical and categorical data. There may be no mode, two modes (bimodal), three modes (trimodal), or greater than three modes (multimodal). Generally, however, more than two modes is uncommon.

Work by me

It is best to use the mode when you are looking for the most popular result. An example of when the mode would be the best representation of the data is when determining favorite kind of pizza using a bar graph (categorical data). You will be able to determine the mode simply by locating whichever bar is the largest.

Still confused?
I hope that this post helped clarify some of the muddy points for you. If not, I found this short video on Khan Academy's website that does a great job explaining what each measure represents. Perhaps hearing and seeing it worked out will help.

Tools for Teachers
I found some great sites that would be helpful in teaching measures of central tendency:

• Illuminations, which I have mentioned in previous posts, is an excellent tool for teachers. This link in particular brings you to a search I did on the site for mean, median, and mode. There are several lesson plans provided, which include grade level appropriateness and activity sheets.
• education.com is another site I found that provides lessons, worksheets, and etcetera. This link in particular brings you to a fun card game I found on the site for helping students master measures of central tendency. Plus, cards are a cheap manipulative you can also use in your probability lessons.
• Math Worksheets 4 Kids is a site that is just what its name says, math worksheets for kids! This link in particular brings you to central tendency worksheets. Different levels are provided, worksheets for each individual topic, and worksheets for multiple topics on one are also provided. What I liked most, is that they provide the answer key for you!

Hopefully, you will now be able to distinguish what the measures of central tendency are, and when best to use them.

-Amanda

Bars to Circles With Little Error

As we moved into the world of statistics, we began with graphs used for categorical data. We also learned that making a circle graph may not always be so easy. In other graphs, like a pictograph or a bar graph for example, representing the data is pretty straight forward.

Below is a picture of a handout we did in class the other day. In the pictograph on the top, a key is provided that shows each circle represents 6 m&m's. Each color is listed on the vertical access, and a row of circles is located on the horizontal access next to each color . Using the key, we represented how many of each color we had. For example, we had 33 green m&m's. So, we colored in 5 1/2 circles with green crayon (because 33/6 = 5.5).

In the bar graph on the bottom, each color is listed on the vertical access again. However, this time instead of a key, the horizontal access includes a scale with intervals of 6 (this made it easier to visually compare it to the pictograph above it). Additionally, instead of pictures, colored bars are drawn to show the amount of each color. Using green as the example again, you can see that the green bar extends to approximately where 33 is on the horizontal access (again, because there are 33 green m&m's total).

Work by me

Now to circle graphs. As I mentioned in the beginning of this post, making a circle graph may not always be so easy. The main problem being, that it isn't always easy to properly divide the pieces of the pie so that they accurately represent the portion sizes. On one hand, if you have easy numbers to work with, quarters for example, it can be. On the other hand, when you don't, it can become quite cumbersome. This leads me to the purpose of today's post - I know, finally.

Using the bar graph on the above worksheet, my professor taught us a super easy way to convert the information to a circle graph. To begin, she provided us with two pieces of blank, white paper. On the first sheet, she had us trace our bars, cut them out, and color them to match the original bars. She then instructed us to tape them end-to-end, as pictured below.

Work by me

Next, she had us tape the two ends together so that they formed a circle:

Work by me

We then took this circle and placed it on top of the second blank piece of paper. We traced a circle using our taped bars as the template. We then marked on the circle where each color began and ended, and made sure to make a note of which color each section should be. I made an approximate center point in the circle to use as a guide for where my section lines should cross each other. Using the points marked for each color's section, We drew our lines to create the pieces of the pie. We were then told to color in and label each section accordingly. Everyone labeled their circle graph sections with the color, but there were different methods used for displaying the quantity of each color. Most appeared to enter the quantity, as you can see I did below, but one girl converted the quantities to percentages. (Note: while this was another great way to display the data, she had to take care in ensuring her percentages totaled 100 percent.) Last, we added a title to the circle graph so anyone would be able to tell what the graph was displaying.

Work by me

While this exercise did take some time to complete, I did think it was a great way to show how a bar graph correlates to a circle graph. Additionally, not only would this be something that elementary students would be able to do, it was fun. Heck - we're college students and we had fun doing it.

Before signing off, I'd like to share another option for creating a graph if you would like to incorporate technology into your lesson. On the National Center for Education Statistics (NCES) website, there is a Kids' Zone that has a variety of tools available. In particular to this post, there is a "Create a Graph" section that allows you to plug in your data to create any of five graphs: bar, line, area, pie, or XY (scatterplot for example). Not only are you able to design the graph, it also allows you to download or print it; great for submission.

-Amanda

Experimental vs. Theoretical Probability: Flip a Coin

The comparison of experimental probability and theoretical probability was one of the topics we discussed in class; and for me, one of the easiest to grasp. I did find, however, that some of my classmates appeared to struggle with the concepts. So, this post will be my attempt to simplify the information so that anyone can distinguish between the two.

To begin, I offer basic definitions of some terms associated with this discussion (courtesy of my math professor):

• Theoretical Probability: the outcome under ideal conditions.
• Experimental (empirical) Probability: determined by observing outcomes of experiments.
• Sample Space: a set of all possible outcomes for an experiment.
• Equally Likely: when one outcome is as likely as the other.
• Mutually Exclusive Events: the events cannot be both at the same time.
• Complementary Events: two mutually exclusive events that total the sample space.
• Bernoulli's Theorem: if an experiment is repeated a large number of times, the experimental probability of a particular outcome approaches a fixed number as the number or repetitions increases (Law of Large Numbers.)

One simple way to demonstrate this, is by the use of something familiar to most of us - the flip of a coin. 

The theoretical probability of flipping a coin is 1/2. There are two possible outcomes when you flip a coin: heads or tails. The probability of the two outcomes together equals one; they are complementary events and are equally likely.

Image by me

The experimental probability is not as straight forward, however, much more fun to find! I am an extremely visual person, so I searched for simulations on the internet that would allow me to also show you experimental results of flipping a coin. Rather than type a book, I will provide visuals. There are several websites that offer virtual coin tosses: Random.org, Virtualcointoss.com, Shodor.org, and Justflipacoin.com to name a few. However, of these only Shodor.org actually tracks the ratios for you. I wanted more, so I went outside the box here - and I am so glad I did.

I found an adjustable spinner simulator on The National Council of Teachers of Mathematics (NCTM) website that is A-mazing. Here are just some of the reasons why:

1. adjustable number of sectors. This allowed me to change it to only two for the purpose of a coin-flip experiment.
2. adjustable name in the "color" column. This allowed me to assign Heads to blue, and Tails to yellow (the default was just the color name).
3. adjustable number of spins per spin. This expedites the time it takes to flip a coin many times.
4. "Skip to End" button. If you enter 100 spins per spin, this button will skip to the end of the results, instead of making you wait for the spinner to spin all 100 times.
5. BONUS: Experimental % and Theoretical % columns!!! How better a way to put it into perspective? This one feature captures the essence of why I am writing this post in the first place!

Now for the experiment. This first image is how I started the experiment. Note that the theoretical probability currently shows as a 50/50 percent ratio. As mentioned above, the outcome of heads or tails is equally likely.

Work by me

This next image is after 20 spins, or coin flips. Now you are able to see the amount of times I flipped a head versus a tail, as well as the experimental probability showing a 60/40 percent ratio. While this ratio is not very far off from the theoretical ratio of 50/50, watch what happens when we increase the number of spins.

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This last image is after 500 spins, or coin flips. Again, you are able to see the amount of times I flipped a head versus a tail, as well as the experimental probability showing a 49/51 percent ratio. By increasing the number of spins, my experimental probability became much more comparable to the theoretical probability. In other words, an example of Bernoulli's Theorem.

Work by me

I hope I simplified this topic for you! By making this an interactive math lesson, I believe you can also make it easy and fun for your students to understand. Using my example, the 20 spins could reflect the spins one of your students would make, and the 500 spins could be your class total. Make it fun!

-Amanda

The Cat in the...Drawer?

Let me begin by saying, if you are ever presented with the "Cats and Socks" probability math problem...do not, I repeat, do NOT overthink it!

This story begins with my professor handing out our homework. As she was doing this, she began to explain how she knew that we were going to overthink this assignment. She urged us not to. She reminded us to take time to read the problem, and to just think on it for a while. We all agreed, of course, and class was dismissed.

After reading the problem carefully, which I actually always do, I began going through the first steps of the problem solving template. No problem, easy-peasy. I confidently jotted down my Goal, Givens, Strategy, and Conjecture (reference the photo below if you require clarification on what these are). As I tackled my Solution pathway, I immediately thought, "Ah, the probability tree diagram like we did in class!" I began sketching out my visual, carefully adding each branch's probability as I went, and of course making sure each draw's total probability equaled 1. As my tree began to become the size of a Redwood tree, my professor's voice rang in my head..."don't overthink it." Still, I was sure this was how to solve it. How else could I??

When I finished my tree, I was glad that I had persisted because it was going to make it SO much easier for me to solve the problem. This is when it hit me like a ton of bricks. I had just spent at least 30 minutes drawing my tree diagram, and now who knew how long it was going to take me to figure out the probability of every single one of these outcomes - no words.

This is when I emailed my professor crying for help; I was in too deep, and needed a lifesaver.

Now for your amusement; I attach a picture of my beautiful, yet useless, tree diagram:

 

photo and work by me

Unfortunately for me, my professor's email response never came (we later discovered it was sent but got stock in queue - lucky me). What made matters worse is that with the help of not one, but two tutors from the tutor center, we continued to overthink this. I am not even going into how...

Now for the amusing part. As I sat crying in my milk about this problem before Math class, a classmate joined me at my table. We began chatting about this problem, and within a few minutes I stopped short. My mouth dropped open - the light bulb had finally gone on. Just talking to her about it, out loud, made me realize why my professor said, "DO NOT OVERTHINK THIS." It was so simple!!

If I had only stopped and thought about what I had written as my prediction, I would have saved myself a LOT of time, work, and stress. Bottom line:

You have 12 socks total. There are only 3 colors to choose from, but you have to pull 4 out. What is the probability that you will pull 2 of the same color? 100%! If you only have 3 colors, there is no way to pull 4 without duplicating at least one color!

Sigh...I know - duh.

Well, at least I had some great practice in drawing a tree diagram. With that said, I will leave you with a website I found that has a pretty simple explanation about Probability Tree Diagrams. Enjoy!

-Amanda