## 2019

ONE GOOD IDEA

# A Sweet Way To Learn DoE

**by Tony Lin and
Matthew S. Sanders**

To learn the basic concepts of analysis of variance (ANOVA) and blocking design in entry level design of experiments (DoE) courses, all the students need is a 1.74oz bag of peanut M&M’s.

The main objective is to determine whether this particular size bag has the same number of colored M&M’s candies on average. Basically, the experiment illustrates a single factor (color of M&M’s candy) fixed effects model with six levels (colors).

This experiment also introduces the concept of the nuisance factor and blocking in the design. The nuisance factor is the bag that contains the M&M’s because it might impact the variability of the number of colored M&M’s in the bag, but we are not interested in the effects. Therefore, the bag is used as a blocking factor. A single factor randomized complete block design is introduced, and the structural model is given by the following:

yij = µ + τ i + βj + εij.

i = 1, 2, …, a (number of colored M&M’s).

j = 1, 2, …, b (14 bags or blocks).

Where µ = overall mean, τ i = ith
treatment (color) effect, βj = jth bag effect, and random
error εij ~ NID (0,σ^{2}).

### Data Collection and Analysis

Before the experiment was carried out, each
student purchased one 1.74-oz bag of peanut M&M’s. The
bags were collected and put in a box. Each student drew a bag of
M&M’s from the box and counted the number of colored
M&M’s in the bag. This information was recorded on a
data sheet (see Table 1).

Though there were many options available to
analyze the randomized complete block design, we chose to use
Minitab’s general linear model to analyze the data. The
ANOVA output is shown in Table 2.

Based on p-value of 0.000 for the colored M&M’s effects, we concluded the average number of each color of M&M’s in the 1.74 oz bags is significantly different. Interestingly, the bag effects (blocks) are not statistically significant because the total count of M&M’s candies in each bag is fairly consistent.

After we concluded the treatment effects
(colored M&M’s) were statistically significant, we
conducted pair-wise comparisons for the treatment means (means of
the colored M&M’s) using Tukey’s experiment-wise
comparisons procedure, which examines every pair of means to
determine which ones are statistically significant.^{1}

The results of the test using α = 0.05 are
shown in Table 3. We determined brown and blue are significantly
different from orange, green, yellow and red; brown and blue are
not significantly different from each other; and orange, green,
yellow and red are not significantly different from each
other.

This simple, sweet M&M’s experiment is a fun way to teach the basic concepts of DoE.

### REFERENCE

- D.C. Montgomery, Design and Analysis of Experiments, fifth edition, John Wiley & Sons, 2001.

**TONY LIN** is a professor
of industrial engineering at Kettering University, Flint, MI. He
earned a doctorate from Iowa State University. Lin is a certified
reliability engineer and a member of ASQ.

**MATTHEW S. SANDERS** is an
associate professor of industrial engineering at Kettering
University. He earned a doctorate from Texas Tech
University.

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GOOD INTRODUCTION TO DOE

--RAJI JOHNSON, 09-26-2015