tl;dr model: A group of scholars helped design and perform an experiment to find out whether or not bowls of Fortunate Charms are equally “fortunate” over the course of a field of cereal. Seems, not a lot. We estimate a lower of roughly 2.7 complete charms per extra bowl on common. This corresponds to greater than a 50% drop-off in charms from the primary bowl to the final. The burden of cereal additionally seems to play a job, and for every 1g of cereal we estimate roughly 0.5 extra charms on common with bowl held fixed. The interplay between bowl and weight is just not statistically important.

See this GitHub repository for the info, code, images, and so forth.

Background

Within the early 2010’s there was a kerfuffle on the Web over an investigation into whether or not or not “Double Stuf” Oreos had been truly double-stuffed. (They’re not.) It was a fascinating concept, and a considerable quantity of fabric has been written about it since, see right here for starters. The dialogue brought on sufficient splash that some academics had been evidently repeating the experiment as an exercise of their lecture rooms, and native college students have reported performing related experiments at their very own faculties greater than 10 years later.

Introduction

One morning in the summertime of 2023 I used to be consuming a bowl of Fortunate Charms for breakfast. The field was practically empty and I sighed to myself, “Can’t wait till this field is finished so I can open up a brand new one…” Now, in case you’re something like me, or thousands and thousands of different folks, then you definitely love Fortunate Charms, and also you’ve cherished them for so long as you may keep in mind, being round for 60 years and all. They really are Magically Scrumptious. However sitting there on that summer season morning of discontent with a spoon in my hand it struck me that the bowl of cereal I used to be consuming simply didn’t fairly appear as magical as the sooner bowls had been. It was lacking one thing. (The charms, after all.) Was it my creativeness? Might this impact be actual? And in that case, may it’s measured?

I occurred to be instructing an undergraduate chance & statistics course on the time and the 4 college students and I had been decided to seek out out.

After some dialogue, the group selected the next supplies and strategies.

Supplies

• Six (6) bins of Household-size Fortunate Charms (18.6oz, 527g)
• Digital kitchen scale
• Two plastic “bowls”, Container A and Container B, measuring 40.125g and 28.375g respectively
• Giant bowl for discards, some trash luggage, different ancillaries

The Fortunate Charms had been bought from our native retailer—Wal-Mart. There was nothing particular about n = 6 bins, it was merely the variety of bins an individual may carry with two arms to the sixth flooring of Cafaro Corridor in a single journey. The kitchen scale was for measuring the load of cereal, which the group thought is likely to be vital, and the dimensions would additionally assist with knowledge assortment as a result of we didn’t wish to be overly preoccupied with sampling the very same quantity of cereal each time.

For the needs of this experiment, a “bowl” was taken to be roughly 1 serving of cereal as advisable by the field (1 cup or 36g), despite the fact that it’s ridiculous for anyone however a tiny magic leprechaun to get by on 36g of Fortunate Charms for breakfast. The group was not particularly choosy about sustaining bowl measurement consistency, something near 1 cup was thought of ok. We had been accounting for mass of cereal with the kitchen scale anyway and had been taking pictures for a wholesome vary of noticed weights.

Every bowl of cereal was poured immediately from the field into the plastic container, weighed, after which emptied onto the desk floor for counting. The toasted oats had been separated from the marshmallows and discarded. Subsequent the next eight (8) allure varieties had been acknowledged and their quantity recorded: Pink Hearts, Rainbows, Purple Horseshoes, Blue Moons, Inexperienced Clovers, Unicorns, Tasty Crimson Balloons, and Orange Stars.

Sometimes there have been little marshmallow bits within the bowl; not each allure was 100% intact. To take care of this, the group tried to categorise the bit into the kind of allure (Inexperienced Clover, Blue Moon, and so forth.), and if the sort might be decided, then that bit was counted as 1 within the respective class. If the bit was nondescript or too small for sort identification then it was discarded.

Knowledge had been collected throughout two separate class conferences. The scholars labored in pairs to pour and depend the charms. I helped with the dimensions and recording a tough copy of weight values as they had been referred to as out for entry into the pc. The group bought into an information assortment groove and by the top of the experiment all 4 college students had been pouring and counting charms independently.

The plastic container + cereal had been weighed collectively every spherical, and the load of the container (measured firstly of the experiment) was subtracted from the noticed complete weight. The charms had been entered into their respective columns and totaled.

• Field: the field quantity (1 by 6)
• Bowl: the sequential bowl for every field (ranges from 1 to 13)
• Remark: the noticed order of bowls throughout bins (1 to 69)
• Totweight: weight of the plastic container + cereal, in grams
• Weight: of cereal in grams, after subtracting the load of the container
• Hearts, Stars, and so forth: what number of of that allure in that bowl
• Totcharms: sum complete of the numerous charms

Right here is a few R code to learn in and present the highest of the dataset (first 6 rows). The information and all code are shared in this GitHub repository.

library(readxl)
Fortunate <- read_excel("Fortunate.xlsx")
Fortunate$Field <- as.issue(Fortunate$Field)
head(Fortunate)

Outfitted with these knowledge we will report issues just like the imply noticed Weight was roughly 46.3g, the utmost variety of a selected allure in anyone bowl was 15 (Pink Hearts tied with Purple Horseshoes), and so forth. Certainly, we may spend all day computing statistics on this dataset to our Pink Coronary heart’s content material, however for the time being we’re primarily targeted on Totcharms and the way it pertains to Bowl and possibly Weight to a lesser extent.

Here’s a graph of Totcharms by Bowl, coloured by Field:

Fortunate |> ggplot(aes(x = Bowl, y = Totcharms, coloration = Field)) + 
geom_point(measurement = 3) +
labs(y = '# Charms') -> p1
p1

Right here we see a transparent reducing development in Totcharms as Bowl will increase, and the sample is surprisingly linear. There could also be a slight curvature. The colours are tough to pick, so let’s make a line plot and spotlight a few collection:

sizes <- c(2, 1, 2, 1, 1, 1)
alphas <- c(1, 0.2, 1, 0.2, 0.2, 0.2)
Fortunate |> ggplot(aes(x = Bowl, y = Totcharms)) +
geom_line(aes(color = Field, linewidth = Field, alpha = Field)) +
scale_discrete_manual("linewidth", values = sizes) +
scale_alpha_manual(values = alphas, information = "none")

There’s a normal development downward for all collection, however the path to get there varies for particular person bins. Discover how Field 3 begins excessive and stays excessive for a number of bowls earlier than dropping off easily to Bowl 10, crashing promptly afterwards. Have a look at how Field 1 begins the bottom of the gang, will increase after Bowl 5, peaks at Bowl 8, then nosedives right down to Bowl 12. The information would recommend that charms had been extra concentrated close to the highest of Field 3, however had been extra clustered in a pocket close to the center of Field 1. Some bins bounce round, different bins drift in additional of a straight line downward. Put all of it collectively, although, and the general development is reducing and linear. Notice that each field made it no less than to Bow1 = 11, however solely two bins had 12 bowls, and a single field (Field 4) lasted to Bow1 = 13.

Now let’s check out Totcharms versus Weight:

Fortunate |> ggplot(aes(x = Weight, y = Totcharms, coloration = Field)) + 
geom_point(measurement = 3) +
labs(x = 'Weight (g)', y = '# Charms') -> p2
p2

This plot is noisy as we’d have guessed. We have now a pleasant vary of weights, from a minimal below 30g to a most close to 70g. Discover there was one bowl that clocked-in terribly heavy. The outlier doesn’t have any apparent rationalization, but when we dig a bit of deeper and plot Weight versus Bowl we could acquire some perception:

Fortunate |> ggplot(aes(x = Bowl, y = Weight, coloration = Field)) + 
geom_point(measurement = 3) + ylim(5, 75) +
labs(y = 'Weight (g)') -> p3
p3

We see that the extra-heavy bowl was the final Bowl = 12 of Field = 1. The origin of that individual knowledge level has sadly been misplaced to the sands of time, however taking into consideration that it was the primary field the group had ever completed, close to the top it might have been troublesome to guage how a lot cereal was left, and maybe all the rest was dumped into that ultimate bowl—I do the identical factor at breakfast on a regular basis once I get near the top of a field of cereal. If that twelfth bowl of 70g had been break up into (say) two bowls of 40g and 30g, respectively, then there would have been two bins that made all of it the way in which to 13 bowls as a substitute of only one, and possibly the fashions beneath would have match the info barely higher. Alas! We’ll by no means know. Such is the scientific enterprise.

Whereas there isn’t a lot of a linear affiliation between Totcharms and Weight by themselves, there’s a hidden relationship between Totcharms, Bowl, and Weight which might finest be explored with a 3D visualization:

library(plotly)
fig <- plot_ly(Fortunate, x = ~Bowl, y = ~Weight, z = ~Totcharms, coloration = ~Field) |>
add_markers() |>
format(scene = checklist(xaxis = checklist(title = 'Bowl'),
yaxis = checklist(title = 'Weight (g)'),
zaxis = checklist(title = '# Charms')),
legend=checklist(title=checklist(textual content='Field')))
fig

Plots in 3D are super-cool, however the above static show doesn’t do the info justice. I’ve arrange an interactive model of the plot on the following hyperlink which ought to work in most cell/desktop browsers:

Please go there, spin the info round, zoom, pan—test it out. In case you spin it round good you will notice that the dots scatter loosely a couple of flat airplane in 3D-space. That is precisely the form of relationship we’re in search of in a a number of linear regression mannequin (we’ll get to that in a minute).

Now let’s attempt to quantify the linear relationship between these variables. We’ll begin with a easy linear regression mannequin relating Totcharms to Bowl.

Right here is the mannequin:

mod1 <- lm(Totcharms ~ Bowl, knowledge = Fortunate)
abstract(mod1)

## 
## Name:
## lm(components = Totcharms ~ Bowl, knowledge = Fortunate)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -16.7629  -5.7629  -0.4327   6.2277  22.2277 
## 
## Coefficients:
##             Estimate Std. Error t worth Pr(>|t|)    
## (Intercept)  55.1309     2.1237  25.960  < 2e-16 ***
## Bowl         -2.6698     0.2985  -8.945 4.81e-13 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual customary error: 8.313 on 67 levels of freedom
## A number of R-squared:  0.5442, Adjusted R-squared:  0.5374 
## F-statistic: 80.01 on 1 and 67 DF,  p-value: 4.807e-13

We see that Bowl is strongly linearly related to Totcharms. The slope on Bowl is roughly −2.7, in different phrases, for every extra bowl of Fortunate Charms eaten we estimate the common Totcharms to lower by 2.7 charms. Our coefficient of willpower is R² = 0.5442, that’s, roughly 54% of the variance in Totcharms is defined by the regression mannequin with Bowl as a predictor. Subsequent we actually ought to incorporate a correct residual evaluation however we’re going to skip it. Suffice it to say that the residual plots are comparatively well-behaved. Let’s try a fitted line plot with confidence bands for the regression line (the default):

p1 + geom_smooth(technique = "lm", aes(group=1), color="black")

That’s a pleasant relationship with a transparent reducing development.

We’ll do the identical factor for Weight, ignoring Bowl in the interim. Right here we go:

mod2 <- lm(Totcharms ~ Weight, knowledge = Fortunate)
abstract(mod2)

## 
## Name:
## lm(components = Totcharms ~ Weight, knowledge = Fortunate)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -27.0151  -8.7745   0.6901   7.8328  24.4701 
## 
## Coefficients:
##             Estimate Std. Error t worth Pr(>|t|)  
## (Intercept)  22.1370    10.5650   2.095   0.0399 *
## Weight        0.3502     0.2256   1.552   0.1254  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual customary error: 12.1 on 67 levels of freedom
## A number of R-squared:  0.0347, Adjusted R-squared:  0.02029 
## F-statistic: 2.409 on 1 and 67 DF,  p-value: 0.1254

We don’t discover Weight to be a really helpful predictor of Totcharms by itself, which jives with the scatterplot we noticed earlier. We notice for reference that the slope on Weight is estimated at 0.3502, that’s, every extra 1g of Fortunate Charms corresponds to a mean Totcharms enhance of 0.35 charms. This sounds cheap: extra cereal, extra charms. The coefficient of willpower is fairly dangerous: R² = 0.0347, in different phrases, roughly NONE% of the variance in Totcharms is defined by the regression mannequin with Weight as a predictor. That’s okay; Weight was extra of a supplementary machine to assist management for variability within the cereal quantities. The residual evaluation right here seems to be not as dangerous because it may have been, which is reassuring, and we should always count on a number of issues anyway given the intense observations on the excessive/low ends of the load scale. For the sake of completeness we embrace one other fitted line plot:

p2 + geom_smooth(technique = "lm", aes(group=1), color="black")

I initially deliberate to make use of the ggpubr package deal to place these fitted-line plots collectively and check out to avoid wasting area within the dialogue, however the plots had been cramped and never very informative. Anyway, that is what I used to be going to do:

library(ggpubr)
ggarrange(p1 + geom_smooth(technique = "lm", aes(group=1), color="black"),
p2 + geom_smooth(technique = "lm", aes(group=1), color="black"),
align = 'h', labels=c('A', 'B'), legend = "proper",
widespread.legend = TRUE)

Now for the enjoyable half: we’ve explored the relationships Totcharms ~ Bowl and Totcharms ~ Weight individually, however what occurs if we put them collectively? Let’s discover out:

mod3 <- lm(Totcharms ~ Bowl + Weight, knowledge = Fortunate)
abstract(mod3)

## 
## Name:
## lm(components = Totcharms ~ Bowl + Weight, knowledge = Fortunate)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -12.8825  -5.4425  -0.9975   5.2475  26.5304 
## 
## Coefficients:
##             Estimate Std. Error t worth Pr(>|t|)    
## (Intercept)  33.3168     6.8655   4.853 7.78e-06 ***
## Bowl         -2.7552     0.2796  -9.855 1.35e-14 ***
## Weight        0.4819     0.1452   3.318  0.00148 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual customary error: 7.754 on 66 levels of freedom
## A number of R-squared:  0.6094, Adjusted R-squared:  0.5976 
## F-statistic: 51.49 on 2 and 66 DF,  p-value: 3.363e-14

Test it out! Now Bowl and Weight are each strongly linearly related to Totcharms. The slope on Bowl is sort of equivalent to what it was earlier than, −2.7, however the estimated slope on Weight has now elevated to just about 0.5 charms for every extra 1g of cereal. Our (adjusted) A number of R² has jumped to virtually 60%—that is outstanding contemplating the small pattern measurement (n = 6), the overall noise degree of the dataset, and possibly some questionable design selections (each little marshmallow bit counts as 1, and so forth.). Looking back, it’s form of wonderful that the info didn’t prove quite a bit worse. Actual knowledge collected by hand within the wild are seldom so good-natured.

The code for this visualization is a little more concerned than the opposite examples and has been omitted for brevity, however you may test all of it out on this GitHub Gist. Let’s get on with the plot:

Once more: a super-cool plot however the static model doesn’t do the info justice. Try the interactive model as a substitute:

Interactive 3D-plots are quite a lot of enjoyable. I hope you take pleasure in taking part in with the graph as a lot as I’ve. As a ultimate comment, within the tl;dr assertion we claimed that the interplay between Bowl and Weight is just not important. The reader can affirm it isn’t with the next (output omitted):

abstract(lm(Totcharms ~ Bowl * Weight, knowledge = Fortunate))

I initially thought that both the entire thing would develop into a figment of my creativeness or the impact could be too small to detect with out LOTS AND LOTS of Fortunate Charms. I used to be mistaken on each counts. The impact is actual, and it’s large enough to detect with a handful of bins, actually two arms full.

The complete mannequin leads us rapidly to some startling conclusions. For instance, what number of charms can we estimate within the first bowl of a field of Fortunate Charms? We noticed earlier that the common Weight on this examine was 46.3g. When Field = 1 the mannequin estimates the common Totcharms to be

33.3168 + (-2.7552)*1 + 0.4819*46.3

## [1] 52.87357

That’s, round 53 charms within the first bowl of cereal—Mmm, mouth’s watering already. What in regards to the final bowl? Okay, not each field made it to Bowl 13, however all of them made it to Bowl = 11. What number of charms?

33.3168 + (-2.7552)*11 + 0.4819*46.3

## [1] 25.32157

WOW. 25.3 charms on common. This corresponds to a 52% discount in charms from the primary bowl to the eleventh. No, it was most positively not my creativeness. Overlook a number of linear regression fashions and fancy 3D plots, a hungry toddler may detect this distinction carrying a blindfold.

Subsequent query: Why is there a drop-off? An evaluation on bodily grounds would possibly go one thing like this: Take into account a field of Fortunate Charms to be a easy mechanical combination of frosted toasted oats and marshmallows. Many exterior forces agitate the field over the course of its lifetime, comparable to jostling throughout transport, placement on the shop shelf, and transit to the house, to not point out exercise in and across the cabinet. This inevitably results in a shifting of contents, with the marginally much less dense marshmallows migrating towards the highest of the field, and the denser toasty oats settling towards the underside.

This rationale is logical, anyway. Nevertheless it leaves some associated questions unanswered:

1. Does the identical sample maintain true for the person allure varieties? (A fast look suggests “No”.)
2. Is the affiliation actually linear, or would a extra difficult mannequin higher describe the connection?
3. What different vital components have we missed?
4. Are there methods an individual can use to decelerate the Fortunate Attraction drop-off?

• Can we cleverly shake the field (someway) to raised combine the marshmallows?
• What about storage practices? Does it assist if the field is saved upside-down?
• Or flat on its facet?
• and so forth.

These open questions should wait for one more day.

Because the authentic experiment in Summer season 2023, I’ve rerun the experiment a pair extra instances with different teams of scholars. The primary was in November 2023 with center schoolers on YSU MegaMath Day. I failed to present the MegaMath college students very particular directions and earlier than I knew all of it groups had eliminated the plastic luggage of cereal from the field and had been scooping from the center of the bag unfold flat on the tabletop. I couldn’t blame them; it’s simpler to scoop cereal from the center with the bag out within the open. Sadly, this strategy fully destroys any pure density sort-order which will have have been current, the important thing underlying factor we suspect is at play, which compromises the integrity of the experiment. Plus, I doubt anyone’s mother and father ever allow them to eat their Fortunate Charms that approach.

The second was in February 2024 with highschool college students at YSU MathFest in a sequence of two workshops. This time I used to be prepared for them. I put collectively and distributed an information assortment sheet (which you will discover right here) with extra detailed steering. You may try the additional datasets on GitHub within the extraData listing.

Shifting ahead, extra knowledge are wanted to raised estimate the Fortunate Attraction drop-off, and it could be fascinating to check methods for distributing the charms extra uniformly all through the field. If profitable, the primary bowl of the field may not be so magical, however however, possibly these ultimate bowls received’t really feel like such a chore ready to open the following model new field of Fortunate Charms!

This experiment and these outcomes wouldn’t have been doable with out the infectious enthusiasm and tireless consideration to element of all 4 STAT 3743 college students in Summer season 2023: Brenna Brocker, Kate Coppola, Gavin Duwe, and Haziq Rabbani. I thank them for mountain climbing down this statistical path with me. I’d additionally wish to thank the Division of Arithmetic and Statistics at Youngstown State College for supporting each this analysis and extra knowledge assortment at YSU MegaMath Day and YSU MathFest.

In case it isn’t already abundantly clear, the writer is a Fortunate Charms fan, so too are the 4 college students. The outcomes reported right here weren’t and will not be meant as a critique of Normal Mills, Inc., its subsidiaries, their manufacturing facility manufacturing requirements, nor the high-quality people and robots gainfully employed there. We’re all certain by the identical Legal guidelines of Physics, and that features bins of breakfast cereal.

And full disclosure: I’ve taken a peek on the further knowledge collected within the reruns of the experiment. From what I can inform the impact continues to be current, nevertheless it isn’t as dramatic. I don’t know if it’s because the impact is actually smaller than what we initially estimated, or whether it is someway associated to the info assortment protocol within the center/highschool setting. Solely time — and extra knowledge — will inform.

In placing collectively this text I attempted to maintain a document of the locations I went to seek out code to construct the plots that I wished and beneath is a principally full checklist, however possibly I missed some hyperlinks. In case you discover one thing I missed, then please alert me within the feedback and I’ll repair it.