Over the Skittles Rainbow A Statistical Analysis of 14 Bags of Candy

Over the Skittles Rainbow
A Statistical Analysis of 14 Bags of Candy
Cheryl L. Casazza
Salt Lake Community College
Math 1040
“Somewhere over the rainbow, way up high…” -Dorothy
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The bell-shaped curve illustrates a normal distribution as it rises “way up high” before falling
to create a symmetrical plot showing where data falls. There are many formulas for analyzing
and interpreting patterns in data. Learning to understand and apply these formulas will
improve one’s critical thinking skills.
“I could think of things I’d never thought before, if I only had a brain...” -Scarecrow
I can think of things I’ve never thought before, now that I have taken a statistics course. The
crowning achievement of the course is a project in which each member of the class analyzes a
2.17 oz. bag of Skittles candy, then contributes the data to create a simple random sample
representative of the entire population of bags of Skittles. Each member of the class goes
through the process of organizing categorical and quantitative data, generating charts and
graphs, creating confidence interval estimates and performing hypothesis using this collected
data. This low-cost, real world application progresses from the basic skill of counting to highlevel synthesis—performing hypothesis tests.
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“Because, because, because, because, because–Because of the wonderful things…”
– Wizard of Oz Lyrics
The most challenging part of a quantitative literacy project is using data to justify decisions.
For professional statisticians, applications rest on “because, because, because, because,
because…” Because life, limb and liability may be at stake. For beginners, it is preferable to run
a low-stakes game counting candy colors. No matter how far off the results may be, no one
gets hurt. Besides, who doesn’t enjoy working with candy? (This is a rhetorical question, not
begging a statistical response!)
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THE RAINBOW COLORS: ORGANIZING AND DISPLAYING CATEGORICAL DATA
PIE CHART
Proportions of Skittles Colors from Class Sample
Red
0.192
Purple
0.204
Orange
0.204
Green
0.189
Yellow
0.212
Red
Orange
Yellow
Green
n = 840 skittles (14 bags of candy)
Purple
PIE CHART
Proportions of Skittles Colors from My Sample
Red
0.175
Purple
0.286
Orange
0.254
Green
0.159
Yellow
0.127
Red
Orange
Yellow
Green
Purple
n = 63 skittles (1 bag of candy)
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PARETO CHART
Proportions of Skittles Colors from Class Sample
Proportion of Skittles Candies
0.215
0.212
0.210
0.204
0.205
0.204
0.200
0.195
0.192
0.190
0.189
0.185
0.180
0.175
Yellow
Orange
Purple
Red
Green
Skittles Colors
Yellow
Orange
Purple
Red
Green
n = 840 (14 bags of candy)
PARETO CHART
Proportions of Skittles Colors from My Sample
0.286
Proportions of Skittles Candies
0.3
0.254
0.25
0.2
0.175
0.159
0.15
0.127
0.1
0.05
0
Purple
Orange
Red
Green
Yellow
Skittles Colors
Purple
Orange
Red
Green
Yellow
n = 63 skittles (1 bag of candy)
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SAMPLE PROPORTIONS
TABLE
Proportions of Skittles Colors from Class Sample
Color
Red
Orange
Yellow
Green
Purple
TOTAL
Proportion
0.192
0.204
0.212
0.189
0.204
1.001
TABLE
Proportions of Skittles Colors from My Sample
Color
Red
Orange
Yellow
Green
Purple
TOTAL
Proportion
0.175
0.254
0.127
0.159
0.286
1.001
I observed inconsistency in the proportions. I was not surprised that the data from one bag
did not match the data from 14 bags of candy. For example, yellow candies took the lead in the
class sample, yet they were last in my sample. After viewing these colorful charts, I decided to
assess the standard deviations in the colors.
Standard Deviation of the Proportions
Color
Red
Orange
Yellow
Green
Purple
Standard
Deviation
1.787
2.547
4.479
2.620
3.167
The mean of these five values = 2.920. That is almost 3 standard deviations away from the
mean, a measure of center. A casual inference might be that proportions of Skittles colors vary
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greatly, but somehow each bag ends up with all of the promised colors. Wm. Wrigley Jr.
Company, the maker of Skittles, markets a wide variety of flavors, including Banana Berry,
Mango Tangelo, Passion Fruit, and Cherry Lemonade. Labels such as “Tropical”, “Wild Berry”,
“Tart’n’Tangy”, “Crazy Cores”, “Skittles Confused” and “Smoothie Mix” adorn the small bags of
coveted candy. The simplistic approach to Skittles belongs to the past. It is logical to assume
that the Wrigley Company has a way of sorting theses millions of individual candies. I tried to
research how they do it, but most online sources refer to amateurs making candy sorting
machines for a hobby.
My conclusion is that it is not a priority for the company to closely control the exact
proportions of Skittles in the millions of bags of candy they sell. It is a very profitable business,
they are not monitored by government agencies on this point, so why would they bother to put
money into more accurate sorting methods? This comes under the category of “practical
significance” as opposed to “statistical significance”.
SAMPLE STATISTICS:
Mean: 60.0
Standard Deviation: 2.69
5-Number Summary:
Min: 54.0
Q1: 59.0
Med: 61.0
Q3: 61.0
Max: 63.0
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ORGANIZING & DISPLAYING QUANTITATIVE DATA: THE NUMBER OF CANDIES PER BAG
Histogram
Frequency of Number of Candies per Bag in Class Sample
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Frequency
5
4
3
2
1
0
54
55
56
57
58
59
60
61
62
63
Number of Candies per Bag
In quantitative literacy, analyzing a five-number summary often leads to creating a boxplot.
It is easy to spot potential outliers on a boxplot as shown here.
IQR = 2.0
IQR*1.5 = 3.0
Q1=59.0, and 59.0-3.0=56.0, so anything < 56 is an outlier
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“And my head I’d be scratchin’ while my thoughts were busy hatchin’ if I only had a brain…”
Scarecrow
QUANTITATIVE DATA
Fortunately for us, we have calculators and computers to “hatch” our numbers as we
perform analyses of data. One of the first values calculated is usually the mean, a measure of
center and an unbiased estimator of where data falls. The mean takes into account all of the
data, but it is sensitive to outliers. The lay term, “average”, calls to mind synonyms such as
typical, usual, likely, moderate, regular, normal, and middle of the road.
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“[Dorothy’s house]…landed on the Wicked Witch in the middle of the road…”
-Munchkins
With a sample mean of 60.0 and a sample median of 61.0, the spread of the distribution
appears relatively narrow. The data is skewed to the left since ten of the fourteen values are
greater than or equal to 60.0, the mean. The presence of two outliers, 54and 56, has
influenced the mean here. Using the interquartile range, 2.0, times 1.5 equals 3.0. The Q1
value of 59, minus 3.0 equals 56, thus 54 is definitely an outlier. The value of 56 is just on the
border. Technically, it is not an outlier but it is interesting to see what happens to the data
when an outlier and a borderline value are removed. After removing these two values, the
mean shifts to 60.83 and the standard deviation decreases to equal 1.75. The shape of the
distribution is not so skewed. In a sample of only 14 bags, removing two values can make a
discernible difference in the statistics.
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This project calls for analyzing the entire sample of 14 bags of candy. My expectations
before doing the number crunching was that there would be some level of variety within
certain reasonable parameters. In terms of practical significance, the difference between 54
and 63 pieces of candy in a bag of Skittles would probably not disturb many customers. I have
never seen anyone weighing bags at the store prior to purchasing candy. I have been known to
weigh prepackaged bags of vegetables, such as celery, to find the heaviest one. Counting pieces
of candy such as M& M’s, Skittles and other small piece products, is typically a casual source of
entertainment pursued by people who are interested in numbers, comparisons and consuming
the candy immediately after counting it! That’s exactly what I did. My bag of candy was on the
high end, containing the maximum value of 63 pieces of candy. Therefore, I was one of the
“lucky” ones. Based on our relatively small sample of 14 bags, there was only a 0 .214
probability of getting 63 candies in one’s bag.
“I’d be clever as a gizzard if the wizard is a wizard…” - Scarecrow
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It makes no sense to aim to be as “clever as a gizzard.” Making sense is very important
statistics. In order to make sense, one must have a clear understanding of the differences
between categorical and quantitative data. Categorical data encompasses groups or categories,
such as political affiliations, colors, professions, and pets. Other than in terms of frequency,
these groups do not translate into quantities—numbers—thus they cannot be analyzed in
terms of mathematical relationships such as mean, standard deviation and variance. Mode
could be labeled as the one appearing most frequently. The best way to display categorical data
is in a well-planned pie chart or bar graph that illustrates the frequency. A Pareto chart is
especially useful since its descending order immediately focuses the viewer on the most
frequent event, perhaps the most prominent or important part of something. One has to be
careful about misleading representations such as distorted 3 -dimensional charts and graphs as
well as non-zero axis graphs that exaggerate differences in values.
Quantitative data deals with numerical values and has a true zero. It can be compared in
terms of proportion, mean, standard deviation and variance. Obviously, to apply formulas
taught in statistics, one needs numbers. Quantitative data may include any number of variables.
In Math 1040 we have studied bivariate data in terms of the x and y axes and finding the
equation of the line of regression. Many types of graphs fit quantitative data, including
frequency polygons, line graphs, stem and leaf plots, box plots, scatter diagrams, bar charts and
histograms and Pareto charts. Pie charts may be used, but they are usually not the most
informative choice for displaying this type of data. Quantitative data can include units of
measurement such as centimeters, yards, hours, dollars, etc. Numbers that substitute for
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names, such as those on the jerseys of athletes, qualify as categorical data because the
numbers do not represent a mathematical relationship.
“I could change my habits, never more be scared of rabbits if I only had the nerve!”
-Cowardly Lion
Confidence is a wonderful thing; a confidence interval is a wonderful construct carefully
calculated using formulas involving probability and proportional relationships. A confidence
interval defines a range of values aiming to include the true but unknown value of a population
parameter, such as the mean height of all women in the United States. It is built around a point
estimate taken from a sample value. The level of confidence derives from the alpha, or amount
of area in the uncertain part of the range of values. It is possible for a true population
parameter to fall outside of the range, but depending on the level of confidence, it is relatively
unlikely for that to happen. Confidence levels are often fixed at 90, 95or 99%. The higher the
confidence level, the wider the range of values. Outside of the confidence interval there can be
a left-tail for “less than “ tests, a right tail for “greater than” tests or one tail on each extreme
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for “not equal to” tests. The tails contain the alpha value, the amount of uncertainty for a
particular test. The value of alpha chosen depends on the consequences of an error.
Confidence intervals can be used for making decisions ranging from marketing level significance
to life and death situations.
DISCUSSION OF THREE CONFIDENCE INTERVALS
One of the most common applications of statistics is using sample statistics to construct
confidence intervals that establish lower and upper limits for population parameters. The
degree of certainty about the accuracy of these limits is quantified by a percentage, such as
95% or 99%. The true value of the population parameter, perhaps for the proportion, mean or
standard deviation, may be impossible to ascertain. At best, it is not practical to do so. Even
with the best data and experienced professionals working, there is always the slight possibility
of error, but experienced statisticians know how to set confidence levels for specific real world
applications.
The first confidence interval defines lower and upper limits for the true proportion of purple
candies in the population of Skittles. The 95% confidence level indicates the 5% possibility that
the true population proportion of purple candies does not lie in the interval from 0.177 to
0.237. The sample statistic will always be exactly in the middle of this interval because the
interval is created by subtracting and adding E to the sample statistic.
The second confidence interval sets lower and upper limits for the true mean number of
candies per bag. This is a statistic that could hold meaning for true lovers of Skittles, who want
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to make sure they get their fair share in each bag. It was determined that with 99% confidence,
the true population parameter for the mean number of candies per bag lies within the interval
from 57.835 to 62.165. Of course, Skittles in the real world arrive in whole numbers, so
approximately 58 to 62 Skittles is a reasonable estimate for number of candies in most bags.
The sample mean, 60 per bag is exactly in the middle of these whole numbers.
STANDARD DEVIATION CONFIDENCE INTERVALS
“Which way do we go?” -Dorothy
“People do go both ways.” -Scarecrow
The third confidence interval addresses the question of variety, or standard deviation of the
mean number of candies per bag. Symbolically, there is variety inherent in the calculation of
variety. To mathematicians, this is perfectly logical, but for students it requires extra thinking.
This interval is not constructed by subtracting and adding an E (margin of error) value. The fun
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part of this calculation is in using the chi square distribution table. On this table, the value from
the left is placed in the denominator on the right and vice versa. People-- and numbers --do go
both ways, as the scarecrow stated.
The sample standard deviation of 2.69 is not exactly in the middle of the confidence interval
because the chi square distribution is not symmetrical. Here, the lower limit equals 4.107 and
the upper limit equals 27.688. This is quite a wide interval, but the confidence level is very
high—98%. One can say with98% confidence that the true population parameter for the
standard deviation of mean number of candies per bag lies within the interval from 4.107 to
27.688.
“Somewhere over the rainbow skies are blue, and the dreams that you dare to dream really do
come true…”-Dorothy
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HYPOTHESIS TESTS
The purpose of a hypothesis test is to use quantitative analysis to weigh evidence, then make
decisions. These decisions include the limit for the number of people allowed in a particular
room due to fire safety considerations, whether or not to purchase a specific math program for
a school district and how much to charge for a movie ticket in a certain city. The applications
are practically unlimited! Without the use of numerical data and proper formulas, these
decisions would be made in an imprecise, inconsistent and unsafe manner.
There are several conditions for doing interval estimates and hypothesis tests for population
proportions:
1. The sample is a simple random sample
2. The conditions for a binomial distribution are satisfied. There are two mutually exclusive
outcomes possible (yes/no), a fixed number of independent trials and probability is
consistent throughout.
3. There are at least 5 successes and 5 failures.
These conditions are met by our sample although it is a small sample—only 14 bags of
candy. In the case of proportion of one color, yellow is success and not yellow is failure. The
bags were purchased at various places in at least two counties in Utah. There is really no
way to know if geographical location of purchase affects randomness here, but in general a
variety of locations improves randomness. It was not a convenience sample, with one
person buying all 14 bags at onetime in one place.
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THE EMERALD SKITTLES
Testing the Claim that 20% of all Skittles Candies are Green
The results of the test show test statistic Z= -0.7763237543 (technology). At alpha = .01,
the critical value for Z = ± 2.575. Since the test statistic is in the fail to reject region, much
less extreme than the critical values, we will fail to reject the hypothesis. There is not
sufficient evidence to warrant rejection of the claim that the true proportion of green Skittles
is 20%.
Testing the Claim that the mean number of candies = 56 per bag
Since the population mean (mu) is unknown, we will use the Student t distribution table.
Alpha = 0.05 puts us at a confidence level of 95%. The test statistic, t=5.57 (rounded) is
more extreme than the critical value of t= 2.160. Thus, we reject the null hypothesis. There
is sufficient evidence to warrant rejection of the claim that the mean number of candies per
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bag is 56. This could be good or bad: If we are getting less than the mean, we could use
Skittles as comfort food. If we’re getting more, we could gather samples of the many
different flavors of Skittles, have a taste test party and then start another delicious
statistical analysis…
“Skittles, taste the rainbow”™
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REFLECTION
What does the word “statistics” mean to you? Is it a heinous subject, full of confusing,
difficult tasks that must be completed in order to achieve an important goal? Or does the
sound of “statistics” conjure up a beautiful image of a kingdom in which the magical
language of mathematics rules and laws are based on true principles determined by precise,
quantitative calculations? A confidence interval estimate of my own positive feelings about
“statistics” would fall somewhere in the middle, higher than the mean but not high enough
to pursue a PhD in this challenging, fascinating subject!
There are several very practical reasons for educated people be literate about statistics.
We live in a world full of studies. Every day we hear statistics quoted as advertisers,
politicians, healthcare professionals and many other people try to persuade us to believe
their claims are true. With the skills learned in Math 1040, one is much better equipped to
evaluate and accept or reject these claims, if one wants to do the research and analysis.
Improved critical thinking skills always increase the quality of decision making.
The very practical skill of using the TI 84 Graphing Calculator may change my life for the
better. I am also grateful that I will be able to apply these skills in future classes, such as
Chemistry and Physiology. In the future, the knowledge I have gained will support my
chosen profession of nursing, as I will be reading and analyzing medical journal reports of
various studies. In the healthcare field, life and death are at stake. In this situation, more indepth knowledge is required than in other fields. Quantitative literacy applies more here
than almost anywhere else.
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Credits
Wizard of Oz Lyrics
www.lyricsmode.com