Ivars Peterson's MathTrek

March 10, 1997

# A Taste for M&M's

The package of M&M's chocolate candies (plain) I had just opened contained 56 colorful, little oblate spheroids. It was obvious from a quick glance that certain colors were more common than others in this sample of M&M's. I counted up each color and confirmed my suspicion.

Then I opened another packet to see if I would get comparable results. There were some interesting similarities and differences in the data, including the fact that the second package had three fewer M&M's!

First Package Second Package Combined Official
Brown 14 25.0% 1426.4% 25.7%30%
Red 1425.0% 611.3% 18.3%20%
Blue 1017.9% 713.2% 15.6% 10%
Orange 712.5% 4 7.5% 10.0% 10%
Green 610.7% 9 17.0% 13.8% 10%
Yellow 5 8.9% 1324.5% 16.6% 20%

Number of each color in two packages of M&M's plain chocolate candies.

I was tempted to open a third package to investigate further. Luckily for both my diet and pocketbook, however, Ronald D. Fricker Jr., a statistician with TRW, Inc. and a graduate student at Yale University, had already gone through a similar experience, which he described in a recent issue of Chance.

Soon after the official introduction of blue M&M's in September 1995, Fricker was grading homework assignments, with a bowl of the candies at his side. He noticed that there seemed to be far fewer of the new blue variety than of the other colors.

"This led me to think about how estimating the proportions would make an interesting class experiment," he says. "Ever on [the] lookout for an excuse to avoid grading papers . . . I decided to conduct my own experiment right then and there."

However, given that he had eaten most of the data, he first had to go shopping to replenish his supply. With the aim of getting samples from different manufacturing lots, he bought three pounds of candy in bags of various sizes from two different stores.

Counting 1,527 M&M's by color, Fricker found that brown M&M's were the most populous, followed by yellow and red, then orange, green, and blue. Intrigued by the uneven distribution, he investigated further by conducting a NEXIS search. He found a July 1, 1995, newspaper article in the Austin American Statesman that purported to report the true distribution of colors.

When he compared his data to the alleged "true" distribution, four of the colors matched up fairly well, but the brown and blue percentages were significantly off. It was particularly striking that the underrepresentation of blue nearly matched the overrepresentation of brown.

"I could come to only one of two conclusions," Fricker remarks. "Either I had uncovered a large corporate conspiracy designed to dupe an unsuspecting public out of blue M&M's, or the newspaper was wrong."

Checking with Mars, Inc., the manufacturer of M&M's, Fricker determined that the official distribution of colors fits the observed data better than the distribution reported by the newspaper. Even then, there were small discrepancies. "We can rationalize the observed differences from the distribution reported by M&M/MARS by recognizing that true randomness must be at least slightly violated in the candy manufacturing and bagging process," Fricker notes.

Many classroom teachers have gone through similar exercises with their students. Indeed, Mrs. Lieber's third-grade class at the Santa Fe Christian School in Solana Beach, Calif., sampled a total of 3,512 candies for a virtual science and math fair project and obtained roughly the same results as Fricker. M&M/MARS Consumer Affairs in Hackettstown, N.J., also publishes a leaflet on suggested math tasks using the candies and a brochure on the percentage of each color in various M&M products, including details of how the colors are chosen.

"Several insights resulted from this study," Fricker concludes. "The first is that you can have your data and eat it too. . . . Second, skepticism is healthy when reading facts and statistics in the popular media."

My data are now gone, and I'm starting to wonder about animal crackers. Barnum's Animals Crackers, made by Nabisco, contain a mixture of animals, some considered predators and the others prey. Are the prey-type animals more likely to be found broken than the predator-type?

This may call for some serious sampling.

### References:

Fricker, Ronald D., Jr. 1996. The mysterious case of the blue M&M's. Chance 9(No. 4):19-22.

Johnson, R.W. 1993. Testing colour proportions of 'M&M's.' Teaching Statistics 15:2-4.

Check out the official M&M's Web site at http://www.m-ms.com.

The electronic publication CHANCE News featured the Austin American Statesman data on the distribution of M&M colors and comments on the predator-prey relationship in Barnum's Animals Crackers. Check issues 4.12, 4.13, and 6.01 at: http://www.geom.umn.edu/docs/education/chance/chance_news/news.html.

Ronald Fricker can be reached at fricker@stat.yale.edu

Bar chart created using Mathematica 3.0 ( http://www.wolfram.com).

"M&M's" is a registered trademark of Mars, Inc., Hackettstown, N.J. 07840-1503, and "Barnum's Animals" is a registered trademark of Nabisco Brands.