Social Norms and Other Challenges

If you are drawn to a field that people say you can’t do, then keep pushing against those who seek to limit you. They don’t know what you can do. And, I bet, that you don’t know what you can do.

My senior year of high school, I was in a special program where I took all my classes at the junior college. The program was for people who knew what they wanted to do to. And I wanted to do theater. (spoiler, I now have a PhD in math and love it) But, I had taken my math and science classes thus far and I didn’t see any reason to stop. So I wanted to take Physics 101 as I would have done if I had stayed at my high school for senior year. But in this special program we had a special adviser. And my advisor noticed that the physics class had a “recommended” course of physics for non-physics majors. I didn’t want to take that class. I wanted the class with actual math in it.

My advisor and I got in a fight. She told me to “Go think about my options and let her know when I had reached a conclusion.” I came back and said, “I got an A in calculus. My dad is a computer scientist and he can help me with the homework if I get stuck. I know how to use my resources and I want to take the Physics 101 class.” She told me no. In addition she gave me a veiled threat that she would get me kicked out the program if I continued to try to get into the other class. I’ll never understand that decision of hers.

So I had no choice but to take the thought experiment based physics class where we thought about monkeys falling out to trees.

Classic physics problem about shooting a monkey out of a tree. You can complete this problem with or without math… or so they say!

The class was interesting, but not for me. I wanted the rigor. Because although high school me didn’t realize it at the time, I had an aptitude for math. But no one would tell me that. Probably because I couldn’t subtract numbers to save my life. (Luckily that’s not really what math is about.)

No one noticed that I never used to my calculator and said, “Hey, by not using your graphing calculator, you are forcing yourself to really learn how to graph and understand it. Thus making this class harder and not easier. I bet you secretly like math.” In fact it was the opposite. There were people activity pushing me away from the subjects I was interested in. Why had my otherwise nice adviser forced me to take the class she felt was appropriate? Was it because she had me pegged as a theater major and she couldn’t accept that I could do something else? Was it because I was female?

If you have faced challenges, then know that you are in good company. This morning a reader recommended this great clip of Neil deGrasse Tyson (below) which reminded me that acknowledging the struggle is valuable. I always appreciate when people who have “made it” admit to the challenges they faced. No one has succeeded without struggles. Succeeding without challenges is a probability zero event.

PS: If for whatever reason the clip below starts at the beginning then skip to time 1:01:30 to just hear the most relevant portion of the Q&A. Thanks!

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Hunger statistics and click bait rant.

I have a bit of a issue with an article came out a while ago on titled, “6 Insane True Statistics That Laugh In The Face of Logic”. While the title is total click-bait, and I recognize that, it still makes me angry. Statistics don’t laugh in the face of logic. The fact that statistics are logical is kind of the whole point of mathematics. Perhaps they are not intuitive and/or not obvious, but they are certainly logical.


The article has several examples of probability & statistics brain teasers that most students will see in an undergraduate Intro to Probability course. This article could have been titled, “6 unintuitive examples your professor will use to stump you on an exam”. I also think it could be called, “A statisticians take on Hunger Games”. Let’s look at the “Insane True Statistics” they cover through the lens of someone who has taken a few math classes.

1. Probability Dictates that “miracles” are routine.

The odds of me winning the lottery twice in a row are tiny (especially since I don’t buy tickets). But the odds of someone winning the lottery twice in a row are pretty good. The odds of Katniss’s little sister being pulled into the Hunger Games were small (she only had one token in the bowl after all). But the odds that someone from district 12 was going to the Hunger Games are 100%. Sample Size is important. With a large enough sample, lots of things with small probability will happen.


Did you ever hear the idea that if you put enough monkeys in a room with enough typewriters that eventually they will produce Hamlet? Theoretically, if one ignores all the tasks of training/feeding monkeys, would this be possible?

2. The odds of two people sharing a birthday in a small group is almost a certainty.

This is a feature that happens in groups due to the fact that there are only 365 days a year. It has to do with Independent vs Dependent variables. It has a wikipedia page. I have nothing else to add.

3. The probability that a man’s sibling is also a male is one in three (not 50-50)

The solution to this is Conditional Probability. If you know the definition of conditionally probability, you too can compute this. 

4. You can rig a game of coin flips just by going second

Here too, the solution is to use Conditional Probability.  As it turns out, conditional probability is a great source of non-intuitive puzzles. But thankfully, mathematics has rules for this kind of thing and statisticians figured this stuff out a long time ago. Back to de Moivre in in the 18th Century or perhaps even  Pascal and Fermat in in the 17th Century. So I’m not sure if this should really “laugh in the face of logic”. Perhaps it merely laughs in the face of people who haven’t seen conditional probability. But that seems like a far cry from fundamentally deposing logic! While these problems are not obvious, I believe that they are totally solvable with persistence.

5. Pi can be calculated by randomly dropping a bunch of paper clips.

Okay, I admit, this one is awesome and seemingly magical even when you understand what is happening. This is true and a very cool experiment called Buffon’s Needle!  In my Intro to Probability class, I did an equivalent experiment by dropping toothpicks onto a grid. I remember completing a big, complicated proof to show that it really was pi.  Advanced note: If you are working your way towards an undergraduate math major (or maybe you already have one) can prove the relationship using a probability density function and double integrals.

If you want to know and/or try your hand at a simulated version of Buffon’s Needle: Science Friday did a write up on it:

Science_Friday6. Lastly, when you shuffle a deck of cards, you’re creating a sequence that has never existed before.

This is a problem that you might consider at the beginning of a statistics course. Understanding how many combinations or permutations a certain set of objects can have is integral to doing probability and statistics. Unlike the odds in problem #1, the number of possibilities is really really big. So while you can’t truly “prove” that it has never existed before, the odds are ever in your favor.

So the next time you are stuck in the middle of a statistics or probability course and frustrated with the unintuitive nature of the problems. Don’t worry. The entire rest of the world is with you. And mathematicians have been thinking about this stuff for a long time. So while it may not feel intuitive, it’s definitely logical.


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Explaining Research in a Big Network of People

Now that I have finished graduate school and have my PhD, I find myself in a totally different dichotomy of “research”. After having several conversations with non-academic, industry-based colleagues, I know that my experience is not unique. Now, I have lots of conversations to learn what I used to learn by combing through peer-reviewed literature with a search engine. Now, instead of googling something, I must write emails. Lots of emails. Maybe you’ve written an email like mine?

“Hi! I was given your name by person P who suggested that you would be a good contact for my project… Do you have expertise in metric N? … If not, can you recommend someone for me to contact next?”

-sent repeatedly, January – May 2015

As a researcher, this is a challenging work environment. How do I describe my research process where there are no trusted written resources? There are no peer-reviewed papers or even white papers in many companies. I can’t just read a well-written text to understand a process. Often times I have to talk to a whole string of experts before I feel like I understand the process. And it takes sooo long! I think when you start working in industry you have to redefine your definition of a “source”.

Grad School: Peer reviewed paper.

Big Company Source: The person we hope never leaves because they know all the things about metric N.

As I was explaining my frustrating research process to my boss, he recommended that I at least document all the people I’ve been talking to. This seemed like a good idea. But how do I describe which experts I spoke with and who I got my information from?  A extensive bibliography with lots of “interview” references seems cumbersome. Surely there is a better solution. I don’t know what he was expecting, but I produced a diagram which resonated with everyone I showed it to at every level of my company. I want to convince you that this is a fabulous way to communicate why a research process may have taken months.

The people map diagram is a visualization of all the contacts I made for a particular project at work (see below). I anonymized the data, but changed nothing else. The ovals represent people who just recommended me to other people (Hubs) and the squares are people who actually had data for me (Warehouses). Take a look.

Click for larger version!

Click for larger version!

My favorite experience in living the reality of this chart was the week when Person 18 recommended I talk to Person 15, 16 and 17. ALL of whom recommended me to Person 14, who was my direct partner on the project… “Yes, thank you! I have talked to Person 14, in fact I talk to them daily. You see, I’m trying to validate their data, so I need a different table than the one Person 14 recommended…”


Sarcasm aside, I found that by creating a visual of all the conversations, my boss was much more understanding of the amount of time it took me to learn a new process. He easily grasped the breadth of research I had undertaken.  And the best part about this diagram is that everyone will see exactly what they want to see. In fact, I claim that diagram is a litmus test of what kind of person you are. Take our slightly satirical quiz and see if Social Math can guess your personality!

SocialMathematics Quiz

Let us predict what kind of person you are!

Q: What underlying feature do you notice in this diagram? Is there any particular take-away that you see?


  1. I see frustration and hardship for the researcher dripping off the edges of this diagram. (go to C)
  2. I see hard work, dedication and resourcefulness of ME!’s work. (go to A)
  3. I see a social network which will identify the most useful employees in the company. (go to B)

A.  You are the perpetual optimist. You are eager to be convinced that nothing is wrong with your system. Instead, there is something wrong with all the other analysts who didn’t try hard enough!

B. You are the enthusiastic extrovert. What’s more, you are in the business of leading and reorganizing people. You are the clever person who couldn’t care less about the plight of your researchers, but rather see the larger picture about the social network of people.

C. You are the battle-worn data warrior. You are a detail-oriented researcher who dreams of organized self-serve data sets. You are used to being in data trenches of a big community.

I, myself, am a (C). But then, I was the person who had to talk to all the experts to learn the process…

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Measuring Boston coastlines with increasingly tiny measuring sticks…

OSH_in_BostonAs I toured Boston this past week, I seemed to be drawn inexorably towards the Old State House.  Historically the Old State House was built at the center of civic activity. The Old State House “was prominently sited at the juncture of King Street, Boston’s primary commercial thoroughfare, and Orange Street, the peninsula’s sole overland route to Roxbury and interior settlements,” says the Boston Landmarks Commission. What this means is that if you stood in front of the Old State House in 1700, you could see the bay only a few blocks away. Now, the bay is a ½ mile down the road and nowhere in sight.

Here’s two maps of Boston. Notice how the map on the left of Old Boston is almost an island (you can click on it to make it bigger)- Present-day Boston, on the right, does not look like that anymore. But in both cases, the Old State House is basically in the middle.


A lot has changed since 1630. Crazy!  Humans leveled the hills that existed in Boston to expand the shoreline. They did this without construction vehicles or steam engines. This blows my mind! Then, much later, Bostonians used landfill to fill more space in.  Here is a map of how things have changed over time.


This map is from who referenced which is amazing, but doesn’t seem to have the image on their site any longer.

Boston is a maritime city with lots of boat traffic. Because of this I would love to know how the length shoreline has changed over time. Can we fit more or less boats in Boston harbor than we used to? Did we get more shoreline out of this land expansion? Or less? Since we have a map of Boston, this should be an easy task right?

Shorelines are notoriously hard to measure. This is because shorelines are fractal-like. This is a mathematical feature which (loosely) means that at each level of zooming in there are lots of bumps and rough edges. You can zoom into a coast line over and over again and not know how close to the coast you are! How does this manifest itself when you try to measure the coast?

The length of your measuring stick directly affects the length of the coastline. This is called the Coastline Paradox. Here is an amazingly awesome video about measuring a coastline from science presenter Steve Mould.

But if you don’t have 8 minutes, that’s okay. The long and the short of it is


Thus we can get a shoreline of infinite length for any point over the history of Boston. If we picked a specific measuring stick length, we could compare the shoreline from one time to another. But we can never truly know the length of the shoreline of Boston at any time.

Despite the disappointing state of affairs of the shoreline, there does seem to be a link between the shoreline of Boston and the placement of the Old State House.  Somehow, I believe, it has continued to be the centroid of the peninsula. That is to say it is the center in some way. It is (approximately) the location of the weighted mean of all the points in the peninsula. Which seems pretty cool to me.

I believe this because of my limited experience in Boston. As I traveled back and forth across the peninsula, my trajectory through the city always included the Old State House. It was though the historic placement of the building drew me to it just as it drew travelers towards it 300 years ago. For travelers then and now, the Old State House holds a place of prominence even though Boston isn’t really an island and the shoreline is way further away. Not to mention that there is now a T-stop in the basement…

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Why do Hipsters look alike?

It was a hot day as I stepped into the air conditioned mall. It is 1995 or maybe 1996; my mall has slightly opaque sky lights so you could still see the bright midsummer sunlight outside. I would have preferred darkness. I wound my way around the teens who aren’t enlightened enough to make their own unique fashion choices as I made my way to the coolest store in the mall.


I may or may not have owned this exact pair of pants. I’m not admitting anything!

I got a lot of weird looks, probably because of my strap-covered purple pants and thick black eyeliner. I didn’t conform to the grey t-shirt and khaki cargo pants “uniform” of my peers. I was unique. I was a rebel. I confidently walked into Hot Topic knowing that I was anticonformist  and I didn’t think like the rest of the lemming-esk populous.

But on that hot day, I had a thought which would forever change the way I understood fashion: Why can the US have a nation wide store which sells anti-mainstream clothing to goths? Doesn’t this mean that my awesome pants aren’t actually that unique? Aren’t I actually following some trend by being anti-society and “goth” in my appearance?

Looking back at this time, I still have to laugh at my righteous beliefs of nonconformity. To add further irony, 1996 is the year that Hot Topic went public as a company. My realization couldn’t have been more true. How could there be a large enough, and predictable enough, section of the public to buy that many cookie-cutter goth styles? I think there will always be a cyclic behavior of people trying to be unique or trying to average which causes the population to always have people who will buy clothing at Hot Topic. My theory was recently supported by the work of Dr. Jonathan Touboul.

Dr. Jonathan Touboul, who, like me, is a mathematician interested in social constructions and mathematics. He and I actually have a lot in common because he also studies dynamical systems and his WordPress blog uses a theme that I used for years. (coincidence? I think NOT!).  He just posted a draft of his newest article on arXiv: The hipster effect: When anticonformists all look the same. The Washington Post recently covered the article as well. The Washington Post article does a nice job of explaining the overall choice algorithm of the model. At each moment in time, a single entity will stay with  her style or switch depending on if she wants to be in the majority or minority.

Dr. Touboul says:

Beyond the choice of the best suit to wear this winter, this study may have important implications in understanding dynamics of inhibitory networks of the brain or investment strategies finance, or the understanding of emergent dynamics in social science, domains in which delays of communication and the geometry of the systems are prominent.

Translated from I’m-writting-a-paper language, I believe that Touboul is saying that we may use this type of modeling to understand social decisions. By modeling our social interactions, we might be able predict if and when the hipster fashion will change. While we have used models for years to try to predict the stock market, he is saying that it may also be possible to predict the style choices of the anti-conformists at the same time as you predict the styles of the conformists. There will be some stable ratio and, if you can identify the important features (cut, sleeve length, fit, etc), then you could predict which styles will sell the biggest. Which is why it’s possible to make a graphic like this one:

Image Credit: The Guardian

Okay, let’s look at the math for a couple of minutes because it’s really cool and it’s alarmingly similar to the work that I did in my graduate research. Here’s a clipping from his paper where is shows the resonance that the hipster system has with the conformist system. An oscillation develops and is maintained after a certain point in time.


From The Hipster Effect: When anti conformists all look the same by Dr. Jonathan Touboul.

Touboul adds a delay in the system as well as some noise to simulate reality. He also uses a dynamic Hopf bifurcation. My dissertation focused on dynamic Hopf bifurcations in externally forced systems.  (Do we need any more evidence that he and I are secretly on the same intellectual wavelength. It’s almost creepy!) One can think of his hipster equation (red) as a forced oscillator. I show in my dissertation that if you add any noise to a dynamic Hopf bifurcation it is extremely likely that the system will lose any relationship with it’s initial condition and tend to latch onto the nearest stable trajectory within the solution space. Thus, it is not surprising to me that the anti-conformists are correlated and/or phase locked to the conformists. If you define your sense of self based of of someone else, even if your definition is “whatever they are not”, then you are dooming yourself to be forever influenced by their choices.

Thus, it is may not be so surprising that Hot Topic can predict the next biggest thing for goths to wear. In a similar way, American Apparel can predict hipster trends (Does American Apparel even count as hipster?). Maybe I shouldn’t be surprised that all hipsters look similar. But what does that mean about my style? I haven’t been in a Hot Topic in years. And I have made peace with the fact that I blend in more than I stand out. But I’m not sure that I can believe that mathematics will ever understand the nuances of fashion enough to predict what next year’s biggest thing is.


Posted in Communicating Math, Nature, Shopping, Social Mathematicians | Tagged , , | 1 Comment

Rate of Change of Change

This week Bloomberg published a great article about the change of change in the US. I couldn’t pass up on sharing this article with you all because it’s about math and about social. This article focuses on the bottoms up approach to government. Some big decisions are made at a local and state level before the central government weighs in on the issue.

The article highlights the beautiful connection between the emotion-driven social situations and the rationally-driven analysis/desire to understand. The thing I love most about this article is that the analytical results so clearly reconnect with emotion. This article gives me hope and makes me optimistic about the rate of emotional change of US citizens.

Compared to issues in the past, we are changing our minds so much faster than before. We no longer live in a wait-long-enough-and-everyone-who-disagrees-will-be-dead scenario. We are more pro-active and more responsive than ever before.  So, with that build up, I really hope you’ll go read about America’s rate of change, “This Is How Fast America Changes Its Mind.Rate_of_change_Bloomberg

Posted in Communicating Math | Tagged , , | 2 Comments

Popcorn Analytics

I’m sitting at my dining room table trying to discern the difference in texture, squeakiness, and flavor of the food in my mouth. Across from me, several of my friends are thoughtfully chewing on popcorn and making notes. We all survey the colors and various sizes of kernels in the bowls in front of us while we sip on our wine.  Then we determine that we don’t have enough data points and we need to run the blind taste test again with different numerical assignments. This must be done to determine if our judgments are reproducible. We are pretty sure our judgments are subjective and varying- so we probably won’t get a single best popcorn brand. This means that the only consistency we can hope for is that each person’s opinions are reproducible from one taste test to the next.

On an evening when there wasn’t much of a plan, we realized that we haBowl_of_popd 5 different kinds of popcorn in the apartment. Some expensive, and some not so. There have been lots of experiments about the quality and flavor of wine. When I ask Google, the first few results are impressive: Forbes, Wired, and BuzzFeed.  On the other hand, when I ask Google about taste tests for expensive popcorn I find none that are really about popcorn taste tests. (notable exception seen here.) But who needs Google when you can do the tests yourself?

5 Brands. 2 taste tests. 5 taste testers.

The contenders: Ancient Heirloom, PopSecret, Jolly Pop - White, Jolly Pop - Yellow, and Jack Rabbit.

The contenders: Ancient Heirloom, Pop-Secret, Jolly Pop: White, Jolly Pop: Yellow, and Jack Rabbit. Listed left to right by price- most expensive on the left to the convenience store brand on the right.

Each person gets two votes for each brand (one for each blind taste test we did). Only 3 out of 5 of us picked the same brand as our favorite from one test to the next. Thus, we are only 60% reproducible. Pretty dismal! But our sample size is small. So maybe reproducible isn’t the ideal. Perhaps we are smarter as a group than we are individually? When we compile the votes, something surprising happens. Here’s the big table of ranked votes by brand.

Ancient Heirloom PopSecret Jolly Pop – White Jolly Pop – Yellow Jack Rabbit
First 5 3 1 1
Second 3 3 3 1
Third 4 3 1 2
Fourth 1 2 2 6
Fifth 2 2 1 3 1

As you can see, most brands got lots of different scores. We should take a weighted average of each brand to understand the overall story. We’ll give 5 points to each 1st place vote, 4 points for a 2nd place, 3 for 3rd and so on. Then we divide by 10 to average the weighing. 

Ancient Heirloom PopSecret Jolly Pop – White Jolly Pop – Yellow Jack Rabbit
Weighted Average 3.9 3.1 3.1 2.7 2.3

We, as a collective, ranked the popcorn by cost! That’s kind of impressive and surprising. Based on this small study, I boldly propose that popcorn brand preference is more correlated by price than wine is. This is a great feature of popcorn because, unlike wine, the price difference between the cheapest and most expensive is a few bucks! Cheers to that!

By the way, did I mention that popcorn is my favorite food? At work last week, I had to describe how popcorn represents my personality: Explosive! Comforting… and Boring? I love popcorn so much that I even make cupcakes that look like popcorn for parties.

I made these. Well, not these exactly. But I used this recipe to make popcorn cupcakes that looked almost exactly like this. It’s an equivalence relation of popcorn!

My friends are fabulous. On a similar evening not too long ago, we were making measuring tape circlets as we were trying to determine whose head was larger. Or that other time when we were measuring shoulder width to try to decide how that effects rock climbing techniques… There are so many wonderful things in this world which can be seen more clearly through the lens of mathematics and data. I will happily do analysis on almost anything.  And apparently my friends will too!

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