M&M Math

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Project Summary

Difficulty	3
Time required	Very Short (a day or less)
Prerequisites	None
Material Availability	Readily available
Cost	Very Low (under $20)
Safety	No issues

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Sponsor

Sponsored by a generous grant from Motorola

Objective

In this experiment you will do a simple statistical analysis of the frequency of colors of M&M's in a bag.

Introduction

"Life is like a box of chocolates—you never know what you're going to get." (Forrest Gump in Forrest Gump, 1994.)

The "Forrest Gump Chaos Theory" states that in a population of chocolate product (i.e.: a box of chocolates) the distribution and frequency of outcomes (i.e.: carmel filled, creme filled, nougat filled, etc.) are unknown and hence unpredictable. I BEG TO DIFFER! An alternative hypothesis is that the distribution and frequency of outcomes (i.e.: carmel filled, creme filled, nougat filled, etc.) in a population of chocolate product (i.e.: a box of chocolates) can be modeled and predicted by using simple statistical analysis (i.e.: math).

What are statistics? They are bits of numerical information gathered about a group that can be used to describe the group. For example, to describe your class (a group of students) you can count the number of boys and girls, measure the height of each student, or measure the weight of each student. These are statistics. Sometimes statistics can be unpredictable, meaning that they do not have any kind of pattern. However, many statistics do reveal patterns and can be used to make models, form hypotheses and make predictions about certain things.

One type of statistic is called frequency, which simply means the number of times something happens in a group. Frequency can be measured by counting the numbers of different types within a group (like the numbers of each different type of chocolate in a box of chocolates). In this experiment we will count the frequency of different colored M&M's in a bag of M&M candies (M&M's are much cheaper than a box of chocolates).

Will these statistics show a pattern? How will different types of graphs help reveal the patterns? What kind of comparisons will these statistics allow you to make? Is the "Forrest Gump Chaos Theory" credible?

Terms, Concepts and Questions to Start Background Research

To do this type of experiment you should know what the following terms mean. Have an adult help you search the Internet, or take you to your local library to find out more!

• statistics
• frequency
• population (group)
• hypothesis
• prediction

• How many M&M's are in each bag?
• How many of each color are there in a bag?
• Which color is there the most of or the least of in a bag?

Bibliography

• This site is a really good site for kids to review math skills, solve puzzles and read stories about math. They also have a section on geometry:
  Karen, 2005. "Cool Math 4 Kids," Coolmath.com, Inc. [accessed: 3/3/06] http://www.coolmath4kids.com/
• MacLeod, K., 2006. "Free Online Graph Paper / Grid Paper PDFs," Incompetech.com [accessed: 3/3/06] http://www.incompetech.com/beta/plainGraphPaper/
• NCES, 2006. "Create a Graph," National Center for Education Statistics (NCES) U.S. Dept. of Education. [accessed: 3/3/06] http://nces.ed.gov/nceskids/createagraph/
• This project is a simplified version of "The Great M&M Count", a lab exercise for the course "Introduction to Probability and Statistics for Engineers" (ENGR 315) by Dr. Jennifer Turns at the University of Washington:
  Turns, J., 2006. "The Great M&M Count," Dept. of Engineering, University of Washington. [accessed: 3/3/06] http://courses.washington.edu/~stats315/newpage1.htm

Materials and Equipment

• several packages of Plain M&M's
• 2 types of graph paper (both available at Incompetech.com)
  • quadrilateral (square)
  • polar (circular)
• colored pencils or markers

Experimental Procedure

1. In this experiment you will be counting the numbers of different colors of M&M's to calculate frequencies and distributions of the different colors. You will need a data table to keep track of your data:

   Number of M&M's of Each Color Counted in Each Package
   Package	1	2	3	4	5	TOTAL	AVERAGE	PERCENT
   Brown
   Blue
   Green
   Yellow
   Orange
   Red
   Whole Bag

2. Open the first bag of M&M's. Count the number of M&M's of each color and write the number in your data table. Don't eat the M&M's before you count them!
3. Repeat for each bag of M&M's. You should sample at least five bags, but you can do as many as your parents will let you open. The more samples you take, the better your data will be. However, too many samples and you might get a tummy ache!
4. Calculate the total number of M&M's in each bag by adding down each column. Write the answers in the "Whole Bag" box in the bottom row.
5. Calculate the total number of each color in all of the samples by adding across each row. Write the answers in the "TOTAL" column.
6. Calculate the average number of each colored M&M per bag by dividing the total number for each color (calculated in step 5) by the number of samples (equal to the number of bags of M&M's you counted, in this case, 5). Write the answers in the "AVERAGE" column.
7. Calculate the percentage of each colored M&M in the bag from the average data. Do this calculation by dividing the average number of each color by the average number of M&M's in the whole bag and then multiplying your answer by 100. For example, if there are an average of 5 red M&M's in each bag and an average of 50 M&M's in a whole bag, you will divide 5 by 50 (which equals 0.10) and then multiply by 100 (which gives 10%). Write each answer in the "PERCENTAGE" column. (notice that if you do the same calculation with your totals, you should get 100%)
8. After you have finished your calculations, it is time for you to make graphs to help you analyze your data and detect trends. Here I show you how to make the graphs by hand, but you can also use an online resource like the Create a Graph website for kids from the National Center for Education Statistics.
9. Now you are ready to make your first graph, the histogram. The histogram is a bar graph that will tell you the frequency (number of occurrences) of each color of M&M in the bag. It is useful for comparing the absolute frequencies of each individual group to each of the other other individual groups. You can answer questions like, "In the average bag of M&M's, which color are most of the M&M's (highest frequency)? Which color is the most rare (lowest frequency)? Do any of the colors have the same frequencies?"
10. You will use the average color data for this graph, so make sure you can locate this information in your data table. (for a more advanced project, you can count more bags and show a frequency histogram for each color separately, instead of using averages, see Variations below)
11. To make the Histogram, label the left side (Y-axis) with a scale representing the numbers of M&M's. The smallest number of the scale (minimum) will be zero and the largest number of the scale (maximum) will be set just above the largest number of M&M's in any group. For example, if the largest number of M&M's counted was 25 brown M&M's, then I should make my scale go up to 30.
12. Draw a bar for each color that goes up to the number on the scale corresponding to the average number of M&M's counted for that color. You may want to order the bars on your histogram from smallest to largest. Label each bar with the correct color (and even color in the bar with the matching color too). Label the histogram with a title.
13. Now you are ready to make your second graph, the pie graph. It will tell you which portion of the whole bag is of each color. It is useful for comparing the relative proportion of each individual group to the whole population. You can answer questions like, "What is the percentage of each color of M&M in the bag? Does any color make up more than half of a bag? More than a third of the bag?"
14. For this graph I recommend that you use polar graph paper. You can also use a protractor and a compass, but I think the polar graph paper is much easier because it divides the circle up for you into discrete units, and you simply count the number of slices for each group and color them in. You can buy this type of graph paper at an office supply or artist supply store, or you can make your own polar graph paper by going to Incompetech.com and selecting the polar graph paper link.
15. You will use the percentage data for each color to make this graph, so be sure that you can locate this information in your data table.
16. To make the pie graph color in a slice that is equal to the percentage of the data. For example, if blue M&M's are 15% of a bag, and if your polar graph paper has a bar for every 5 units, then color in three slices for the blue M&M's. It is nice in a pie graph to use a matching color for each section and to provide a color key to label your data. You can even write the percent data inside each slice. You also need to make a title for your graph.
17. After you have made your two graphs, you can interpret your data, answer some questions and make a conclusion.

Variations

• To really test the "Forrest Gump Chaos Theory" you could invest a bit more cash, and talk your parents into springing for 10 boxes of chocolates. For those investigators on a tight budget, you can do a similar statistical experiment with almost any other product that comes packaged with a mixture of colors, shapes, sizes or types. Here are more ideas: Skittles, jellybeans, a bag of chips, marbles, Pokemon cards, etc.
• You can use the same analysis for more advanced comparisons: the number of different languages spoken by students in your school, the different types of pets owned by different people, car ownership of families at your school, number of seeds sprouting per seed packet, how tall people are at your school, how many girls and boys in each class or grade, etc.
• Another more advanced way to look at frequencies is to look at a distribution of frequencies to see how much variation there is in your data set. You can do this by making a histogram for each group of your analysis separately, instead of making an average. You can perform this type of an analysis for your M&M data if you count the color frequencies from more bags of M&M's (more than 10 bags of M&M's, or maybe a bag of individually wrapped small bags of Halloween candy?). Then make a frequency table for each color by tallying up how many cases there were of each count. For example, if you counted blues in ten bags of M&M's, you might get 8 blues in one bag, 10 blues in 2 bags, 12 blues in five bags, and 14 blues in 2 bags. With this data you can make a histogram showing the number (frequency) of getting each count, and see that it was most common to count 12 blue M&M's.

• For more science project ideas in this area of science, see Pure Mathematics Project Ideas

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Credits

Sara Agee, Ph.D., Science Buddies

This project is a simplified version of "The Great M&M Count", a lab exercise for the course "Introduction to Probability and Statistics for Engineers" (ENGR 315) by Dr. Jennifer Turns at the University of Washington. http://courses.washington.edu/~stats315/newpage1.htm

Last edit date: 2006-04-20 00:06:13

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