Monty Hall

Monty Hall

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Pitfalls and Traps: Research


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Happy Easter!

Happy Bunny Day from Social Mathematics!    


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Mathy Friendships

On a scale of one to ten, how much do you like each of your friends? We don’t often have reason to carefully evaluate the strengths of our relationships. However, as a fan of quantification, I’ve created a visualization of the way my Friend_Feetfriendships change over time. It’s a joy when an acquaintance becomes a BFF. There’s also beauty in allowing a friendship to fade or blow up. The strengths of our relationships are not based fully on logic. Bonds are forged and broken through chemistry, physicality, proximity, and circumstance. By tracking the strengths of our relationships, we can gain insight into those enigmatic social forces. Also, it makes for some cool graphs.

How to quantify your friendships!

Over the 6 years I lived in Minneapolis for graduate school, I met a lot of people. Some of them became great friends! Some didn’t. I tracked the story of those friendships by rating the strength of each bond over time. To begin this project, I thought a lot about how to quantify a friendship. I didn’t want to rate each person on how much time we spent together, or how heartily we laughed. Those were too tied to circumstance. I didn’t want to rate each person on how much I LIKED them, or how much they seemed to like me, because that didn’t really address how close we were.

Friend Ships. Sketched by our guest Author!

Different types of Friend Ships. Sketched by our guest Author!

Instead, I use a scale of “bondedness,” a measure which (admittedly abstractly) describes the amount of closeness, comfort, trust, and love in a relationship. This scale goes from -1 to 10. 10 is the most bonded I can be with another person. We have each other’s backs no matter what, we can spend time together without discomfort or awkwardness, we can be open about our thoughts and feelings, and we are invested in each other’s well-being. The amount of bonding decreases as the value approaches 0. A 5 is someone who I can spend some enjoyable time with but I wouldn’t be too upset if they missed my birthday party. A 1 is someone I am very slightly bonded to; we occasionally chat but not about anything of interest. A 0 is someone with whom I feel no bond. Either we have met but not connected, or our relationship has dwindled to the point where there is no gravity between us. A -1 refers to bonds that are actively negative – someone who I get upset thinking about. I considered extending the scale down to -10 to capture the full range of bitterness, but this seemed unnecessary. I tend to remove myself from negative relationships, and they eventually become zeros on the bondedness scale. For ease of discussion, I’ll call the unit of bondedness a “bondie.”

By rating each of my Minneapolis-derived relationships four times a year, I was able to visualize the strength of those relationships over time. Those graphs are below, but before you look too closely, be warned: this data is fake. In examining my real friendship data and considering its practical meaning, I stumbled upon a paradox: we can estimate the strength of bonding between any two people simply by observing their interactions, but despite the obviousness of these bonds, it’s not socially acceptable to publicly rank the members of your social group (unless you’re a contestant on Survivor)! In fact, there are probably people out there who find this exercise of quantifying friendships to be horribly callous and uncouth. I disagree; this data is for science, not evil! To keep it that way, I’ve chosen to protect the identities of all involved and created a fake dataset that replicates the spirit and trends of the original.

This is the graph of friendship strength over time, where the x axis is true months.BondiesByTime

This is the same data, where the beginning of each relationship is aligned to zero on the x axis.


The bonds are born

In my dataset, I identified relationships borne of three different scenarios: necessity, choice, and romance. Relationships of necessity are with coworkers, classmates, and roommates – people I was forced to be in close proximity to. I’ll call these relationships “requisites.” Relationships of choice are with people who would be easy to avoid. We chose whether or not to see each other, and therefore growing these relationships typically took conscious effort. I’ll call these relationships “electives.” Romantic relationships had a romantic component that was either one-sided (eep) or mutual.

The requisites

First, let’s take a closer look at the requisites. We’ve all heard that shared experiences bond people together. What I observed is that proximity was polarizing!


Some very strong and long-lasting bonds formed in the requisite group, while an equal number eventually depreciated (or crashed and burned). Despite the fact that, on paper, I had a lot in common with the requisites, many of those relationships didn’t translate to long-term friendships after the period of forced proximity ended. This could reflect my personal preference for maintaining a relatively small group of very close friends. Further, most of the requisite relationships took place within larger social groups where complex interpersonal forces were at play. I suspect this particular bondedness pattern reflects the social and temporal pressures of graduate school and is not generalizable to new social situations.

The electives

The electives looked a lot different. In this group, less than 20% of relationships ever hit bottom – the rest were pretty strong and continued growing over time.


After 36 months, the average requisite bond was 4.3 and the average elective bond was 6.8! That said, data selection skewed these results since I only included people who I bonded with to some degree. People in the requisite group whose bondedness rose and then precipitously fell might never have been included in the elective data set since, without an external force, we may have never bonded at all.

Bonding speed

I was interested in seeing if the speed of bond formation differed between the requisite and elective groups. However, each group’s average rate of bond growth during the first year wasn’t very informative (an average of ~0.47 bondies/month for both groups combined, if you were curious), since the distributions for both groups were so different and the sample size is so low. What WAS interesting was comparing those slopes to the bondedness rating after 3 years.


There seemed to be a positive relationship between the ultimate strength of the bond and the speed with which that bond develops. Hey, that makes sense!

The romances!

The third type of relationship in my dataset was of a more… volatile nature. My romance-tinged relationships were characterized by tumultuous bondedness ratings.


More often than not, these relationships ended up at -1 bondies for some amount of time. I’ll note that these bondedness ratings are from MY perspective alone. I would be both interested and horrified to see the bondedness ratings from the opposite perspective! To what extent would they match up, and what would the mismatches reveal about our relationship’s twists and turns?

Fading away

During the time I lived in Minneapolis, many of my friends moved away to pursue exciting and prestigious opportunities elsewhere. In the absence of regular contact, these bonds sadly tended to fade.


It’s been my experience that relationships that start out with high bondedness will never reach zero, no matter how long we’ve been apart, and that they can easily bounce back up! But still, visualizing the wane of the bondies is upsetting (I’d like to think my friendships stay strong in perpetuity, not matter how much circumstance may try to tear us apart). In my dataset, separation caused bondedness to fall by 0.12 bondies per month.

There was a trend that relationships with higher bondedness prior to separation faded more slowly after the move, which makes sense. The people I am more bonded to are the people who I will probably continue to interact with on a regular basis.


What did I learn?

These data tell a story about what it was like for me to move to a new city where I knew no one, go back to school, and ultimately build a community. In retrospect, I highly value the pressure cooker of graduate school because interacting with a lot of really smart and interesting people allowed me to easily separate those people who were excellent friend matches for me (the lines rising to the top of the requisite graph) frFriend_Toesom those who, for whatever reason, just weren’t. Also, I learned that people who I bond with quickly tend to be long-lasting buddies. This data COULD be used for forecasting, allowing me to wisely invest my social energy in relationships with maximum “payoff.” For example, I could set a bonding speed cutoff, where if I haven’t reached, say, 3 bondies in 6 months (a slope of 0.5 bondies/month), then I should stop putting effort into the new relationship because most likely we’ll never reach the level of great friends. Don’t get me wrong: the idea of rejecting a relationship just because an equation recommended it makes me cringe! However, I think we all make similar calculations intrinsically as we navigate our complex social world.

It was particularly interesting to quantify the fade of bonding that occurs when people move away. I’m struck by the fact that I lose 0.12 bondies per month after the separation. That sounds pretty slow, but it adds up: after 2 years, the friendship has lost 3 bondies! This data has galvanized me to put more effort into my long-distance relationships to head off that drift.

Getting past the squeamishness I felt rating my friendships was hard. It’s fascinating that the social equations we constantly compute in our own brains must stay hidden there for fear of offending others. If you can put this squeamishness aside, should you track your own friendships? Sure, you might learn something about yourself! Everyone has different goals for their relationships, different categories of friends, different numbers of friends, different variables to consider, and different friendship-initiating factors in their lives. It’s very likely your graphs would look nothing like mine. And that’s what makes people, and data, beautiful!


Author’s bio: Tess Kornfield has a PhD in neuroscience and is currently a postdoc at UC Berkeley studying Alzheimer’s disease, neurodegeneration, and aging using cool imaging techniques like PET and MRI. She loves her friends and she loves quantifying things. How much? About a 10/10 and 9/10, respectively. Find her on Twitter @scyspy .

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An Unexpectedly Long Journey

THE HOBBIT: AN UNEXPECTED JOURNEYEvery book lover loves to talk about how horribly the filmmakers screwed up when they deleted/added/changed a scene from our favorite book. The desire stems from our deeply rooted need to compare the two mediums, as if it was somehow possible to fairly compare them. There are nuances of film which cannot be replicated in a novel and vice versa. But, it is our goal here at Social Mathematics to put a measuring stick up to every unfathomable relationship! And today we are starting down the dramatic path of book to film adaptations.


The wand chooses the wizard, Harry…

No adaptation article may ever again be written without referencing Harry Potter. Stretching over a decade, the Harry Potter film releases held many of us captive (or disgusted) for years on end. To watch the entire series you would be committing yourself to almost 20 hours of Harry Potter. (For scale that’s almost three complete seasons of Game of Thrones.) While the characters will forever remain in our hearts, some of the factual details have slipped our minds.


Because we have been taught to believe that J.K.Rowling grew the books to match the age of her primary readership, some people assume that the Harry Potter books get progressively longer, based on word count. However, Order of the Phoenix (book 5) is significantly longer than the rest. The purple area in the graph below shows book length in words. Book 5 is a clear peak in length. Ironically, Order of the Phoenix is the shortest film. The orange bars in the below graph below depict movie length.


Film length (in minutes) for Harry Potter franchise in orange with a background of purple which represents the word count for each book.

The next graph shows the same information in a different presentation. This is a Correlation Plot, a graph designed to help identify the relationship between two variables. Instead of plotting HP1: 77325 words in purple and HP1: 152 minutes in orange, we will plot both together in two dimensions as HP1 = (77325 , 152).  Doing this for all data points helps us identify clusters and patterns. If book length correlated well with film length then we would see a diagonal line from bottom left to upper right.


Correlation graph of Harry Potter Franchise. Book length in words versus film length in minutes. The outliers are HP 5 (long book/short film) and HP7 (long book/ 2 films).

Harry Potter doesn’t line up nicely on a diagonal because there isn’t a clear relationship between book and movie length. As one might expect from our previous conversation about HP5 – it is an outlier from the cluster of the rest of the HP stories. The other outlier is HP7. HP7 is an outlier because, while the book is not particularly long, two separate movies were produced from it! For the sake of this article, we combined the two movies together into one supermovie. This supermovie is more than four and a half hours long.


Speaking of which, book to movie adaptations can get ridiculously long! Especially when the studio insists on splitting the last book into two movies to extend a successful franchise. Let’s look at two other franchises that did this: The Hunger Games and Twilight. (The last installment of The Hunger Games has not yet been released so, for the purposes of this analysis, we are assuming Mockingjay Part 2 will be the same length as Mockingjay Part 1 to get the supermovie runtime)

The last book of a series is often longer than the previous. In the case of Twilight, Breaking Dawn is 30% longer than Eclipse. However, the length of the Breaking Dawn supermovie is 87% longer than the Eclipse movie. That’s almost twice as long!


There and Back Again and Again…

Compared to other adaptions, the economy of the new Hobbit movies is seriously lacking. The book is relatively brief at only 95,022 words long. The 1977 animated TV movie is a standard 90 minutes and basically covers all the major points in the book. In contrast, the new trilogy clocks in at 474 minutes (almost 8 hours). That is 5x longer than the cartoon adaptation. The recent Hobbit series was originally going to be only 2 movies, but Director Peter Jackson decided it should be a trilogy, to mirror The Lord of the Rings trilogy, making a … hexology? Anyway, the combined total of all the extended blu-ray editions of The Hobbit is longer still: 542 minutes or 9 hours.


Perhaps it’s important to remind ourselves of how long it takes to read The Hobbit as a book. Since we all have different reading speeds, we’re going to use the average text to speech ratio of 150 words per minute. At this rate, The Hobbit could be audibly consumed in 633 minutes. So, in shear amount of time, one saves 159 minutes by watching the movie (only 91 for blu-ray). Choosing the movie over the book accounts for 25% off and 14% off of your time commitment of listening to the book audibly. And we all know you can read silently faster than you can read out loud. We bet that you can read at least 25% faster when you read silently. This brings us to the weirdest point of our investigation…


The grand collection:

By this point, we were in pretty deep. We collected a bunch of data about the biggest recent SF/F book to movie franchises. We looked at The Hobbit, The Lord of the Rings, Twilight, The Hunger Games and Harry Potter. Nothing about this data is nearly as interesting as the outliers we discussed already. Predictably, the Lord of the Rings books get shorter while the films get longer. Unsurprisingly, the Twilight books & movies are about the same length- except the last one (book & movie) which is longer than the others. If you want to look at the numbers for yourself, the data we collected is available here! In summary, here is a correlation plot of what we found when we considered the whole dang group:


Correlation graph of the major book to movie adaptation franchises. Book length in words versus film length in minutes. The outliers are HP 5 (long book vs short film) and the New Hobbit (short book vs 3 films).

While HP7 is the outlier of the Harry Potter series, it is but an insignificant shadow compared to the new Hobbit adaptation. Sure, they added some stuff from The Silmarillion, but The Hobbit supermovie is 198 minutes longer than the HP7 supermovie. That’s ridiculous! Especially when you consider that the HP7 book is over twice as long as The Hobbit book.

So, the next time you want to rant about the ridiculousness of the length of a movie adaptation, just know, if the movie isn’t longer than The Hobbit or shorter than a really short adaptation of the lengthy book (HP5), then you’re talking about an adaptation that is probably just par for the course.


Co-Author Bio: Jason P. Schumacher is an award winning independent filmmaker and freelance video producer in Minneapolis, MN. His most recent short film, Sad Clown, is currently screening in film festivals around the country.


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Author Interviewed!

More fun news from Social Mathematics!

This week, Social Mathematics’ founder, Samantha Oestreicher (that’s me!), published an interview at MathTango…  In the interview I talk about my path to becoming a mathematician as well as my favorite books, blog posts, and stories! The interview questions were insightful and I had a great time working with MathTango on the project!


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New things!

There are lots of new things happening here at Social Mathematics! This week I want to point you towards all the wonderful new aspects of Social Mathematics.

MathGuyWe have a newly designed banner to celebrate the cartoon mathematicians who have been gracing the site lately. Don’t they look great up there? Speaking of which, I’ve received more than a few questions about the art on the site! In response I put together a page about the original comics and art I create for Social Mathematics. I hope you’ll take a look!

If that wasn’t enough, Social Mathematics received some fabulously kind words from the AMS blogger, Evelyn Lamb in her article, The Social side of Mathematics.  Evelyn says, “I think the combination of a math-averse past and her experience in the arts give her a perspective not seen on many math blogs.” Thank you!

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