By Dan Hales, STEM Teacher
I shared a casual observation with AP Statistics during the first week of school: Fun-Size packs of M&M’s feel like they have too many brown candies. This certainly felt true, but was it actually true? We decided to investigate.
The first step in getting a definitive answer was carefully phrasing our query, because definitive answers come from well-defined questions. A “maybe” would not be enough for me. We settled on: Is the percentage of brown candies higher in Fun-Size packs of M&M’s than in packages of other sizes?
In order to tell whether or not that percentage was unusually high, we recognized that we needed to find at least two pieces of data: the percentage for Fun-Size packages (the number that interests us), and the percentage for other packaging sizes (for comparison). Our class of fifteen split up into five groups of three: two groups counted Fun-Size packs, two groups counted Sharing-Size packs, and one group counted a Family-Size pack.
All groups had roughly the same number of candies to count, but not exactly the same number. To account for these differences, we defined some terms and found the conditional distribution of color for each packaging size. This led to a discussion about the different ways we could have crunched these numbers: Why wouldn’t it answer our question to find the conditional distribution of package size for each color? What question would that have answered?
Ultimately, we found numbers that appeared to confirm my suspicion: For Family-Size packs, 13.7% of M&M’s were brown, and for Sharing-Size packs, 14.0%. For Fun-Size packs, a whopping 14.5% of M&M’s were brown.
Our analysis is far from over, however. Does this mean that 14.5% of all Fun-Size packs will be brown, or was that number unique to this group of M&M’s? Would that number change if we counted more packs of M&M’s? Does the time of year we purchase M&M’s matter? Is there actually an answer to my initial question at all?
As we continue investigating topics that interest us throughout the year, we will use these real-world questions to motivate exploring many topics, such as confidence intervals, random variables, study design, and significance testing. As we refine our methods for collecting, analyzing, and presenting data, students will be given opportunities to define, investigate, and ultimately answer their own interesting questions. I’m excited to see what they come up with!