Do height restrictions matter to safety on Roller Coasters?

The conversation started with an image on how to “outsmart” the roller coaster operators for kids who are not tall enough for a certain ride:

This sparked a large controversy amongst my Facebook friends.  There were certainly people who felt that this was a terrible lesson to teach kids; that one should cheat safety rules.  And, because this is a site about mathematics, I don’t want to go into the social or moral controversy here.  You can chose to parent however you chose!  But, what I do want to do is consider how dangerous it is to do something like this. Are you subjecting your child to a more dangerous experience by cheating a 1/2 inch on the height rule? The thread seemed to illicit a lot of responses over a long period of time (which is unusual for social media), so I wanted to write up my research here for anyone else who wanted to learn more about roller coaster safety.

Rates of roller coaster accidents and deaths, as it turns out, is a hard thing to research.  There is no centralized database which collects amusement park injuries and fatalities.  Wikipedia seems to have the most comprehensive view of injury tracking, but is missing data and missing reports. Additionally, the type of information that is available to the public changes year to year; there is no way to do a trend analysis.  I’m going to make a couple of egregious assumptions:

  1. I’m going to assume that the rates of injury have not changed over the last few decades. This way I can combine studies across several years to get to meaningful numbers. evidence: In 2004, the Consumer Product Safety Commission showed that mobile rides didn’t see a statistically significant trend, though inflatable rides did show an increasing trend (between 1997 and 2004). The study didn’t cover permanent rides. So, this assumption is “Plausible”. But not bullet-proof.
  2. I’m going to assume that all the studies are accurate and they received accurate coverage from the media. This assumption is plausible, as most media sites are copying numbers from reports, but we don’t know how much we trust the reports because most are not publicly available. Most are from oversight commissions and government agencies, so it’s reasonable to believe that these are pretty accurate.

Part 1: How dangerous are roller coasters in general?

First lets look at how many children get injured for any reason. NBC reported in 2013 that 4,400 children under the age of 18 get injured every year on roller coasters. Of that group, only 67 (1.5%) are injured badly enough to go to the hospital. This group is probably* about 30% carnival rides, 30% permanent rides and 12% mobile rides (like coin-op rides in a store or a mall), with the rest being unknown.  Thus, we can guess that around 1320 injuries took place at amusement parks for minors. And 2640 for amusement and fixed-park riders.

Let’s look at amusement park injuries in total. CNN reports in 2017 that there were around 30,000 injuries at amusement parks & fixed-park rides in 2016. If we combine this with the information above, that means 2640/30,000 = 8.8% of all injuries that happen at amusement parks are to minors.

How does this relate to percentage of park goers under the age of 18?  There are no direct numbers published on this, but the US census bureau says 22.8% of the US population are under the age of 18. The IAAPA reports that the demographic which visits parks are families with kids. So I’m assuming the rate of children is higher than the national average.  So, at least 23% of park-goers are under the age of 18.  Yet, only 8.8% of the injuries are coming from minors.  Therefore, one must conclude that parks are safer for kids than adults.

How scared should you be relative to other things? Compared to other dangerous activities, sharks injured just 13 people in a year in the US. (ref). So, this is more dangerous than swimming with sharks. However, the stats show that 8,000 – 16,000 kids are injured on school buses each year. (NBC article from 2006). Thus, amusement park accidents rank somewhere between shark attacks and school bus injuries.

All of the above was about park injuries. Let’s consider the more serious case of deaths. There have been 22 deaths since 2010, so 3.1 deaths/year based on CNN’s report.  In contrast, the Consumer Product Safety Commission report from 2005 shows only 2.5 deaths per year at fixed-location rides between 1987 and 2004. Let’s compare this to other deadly things. Spiders kill 6.5-7 people every year. And 330 people get struck by lightning. For other facts about dangerous things, check out our post on the dangers of ground beef). On the scale of deadly activities, amusement parks are less dangerous than spider bites.

Part 2: How dangerous is cheating the height requirement?

Now we are in really ambiguous territory. Because the parks certainly do some analysis to determine the height requirements. But we have zero visibility into what that analysis was. So we can’t conclusively say how quickly risk of injury changes with height.

However, danger is rarely bimodal. If there is a risk from being short, it’s probably a curve and not a step function. Like, we can all agree that driving 66 mph in a 65 zone is probably not that much more unsafe than driving 65. If a cop gave you a ticket for driving 66, you’d be pissed (mph error gauge anyone?). So fudging a quarter inch on a kid might not be a huge deal. But how can we judge how quickly danger changes with height?

I’m going to hedge my bets that roller coaster safety is also a function of several other metrics which are not as easy or as feasible to collect as height. Based on all of my personal experience on indicator values of predictive functions, there are two rules for identifying indicators for various rules and restrictions.

  1. The indicator value, or input, must have some predictive power on the rate of the results. In this case, height must have some evidence for being predictive of injury.
  2. The indicator value must be accessible and easily captured. In this case, it’s possible to measure a kids height quickly and (relatively) accurately.

There are almost always other indicators you wish you had for your predictive algorithm, but can’t have.  For example, it’s not socially appropriate or feasible to give everyone a mental health test before letting them on the ride. (Though, there are many roller coaster injuries and deaths that seem to be related to mental health challenges). Similarly, it’s not socially appropriate to weigh children to see if they possess the requisite inertia to keep their butts on the seat of the coaster. Also infeasible, is to measure kids shoulder width to see if they have wide enough shoulders to not accidentally fit between the shoulder bars. I can imagine all of these metrics would provide a more predictive way to judge injury likelihood.

But based on the various explanations of roller coaster injuries I’ve read on Wikipedia and all my other research for this article, I would hedge my bets that roller coaster safety is also a function of weight and “wiggly-ness” of the kid (among other things). So, if your kid is skinny and wiggly, maybe it’s extra dangerous to cheat the height system. But if your kid can stay seated and doesn’t have a tiny frame- then maybe we have fewer concerns about cheating the height requirement. Height is probably the most easily accessible indicator for ride safety. At lot of injuries are caused by getting on and off rides (twisted ankles and wrist injuries) or by not “keeping your hands and arms inside the ride”. So, the ability of your child to follow instructions is definitely critical.  Notably, we can posit that kids are better at following instructions than adults (who over-index on amusement park injuries).

Sadly, I don’t have any numbers for anything in the above section. So, ultimately, I am unable to give a good answer to the height question. But, we can get closer: let’s look at the spread of height restrictions across rides. Presumably, the park would like their system to be as simple as possible for the ease of their guests. But, they don’t want to be unsafe. So I’m going to assume that they make as few different height restrictions as

possible.  Disneyland has height cut offs at 35″, 38″, 40″, 44″ and 52″.  If the amusement park is going to the trouble of identifying different cut-offs every 2-3″, this suggests that 2-3″ in a meaningful and important distinction, especially at smaller heights. So, let’s take Splash Mountain at 40″.  In terms of cheating the system, the danger scale starts at 0% dangerous when you are 40″, but is significantly more dangerous (perhaps not 100% dangerous, but certainly materially more dangerous) than if you are only 38″.  So, your kid has a 2 inch margin… ish.

Would I want to cheat that threshold by 1″? No, probably not. But how about by 1/4″? Or the width of a ponytail? Or the height difference of a pair of lightup sneakers versus a pair of thin-soled chucks? I don’t have data to answer that. That’s something you’ll have to decide for yourself.

Part 3: Summary

  • Riding roller coasters is less deadly than living around spiders and less dangerous for minors than riding the school bus.
  • Children under-index on injuries, sustaining less injuries than adults. (assuming as few key assumptions about the data).
  • The safe to not-safe threshold is probably 2-3″ wide, on average. So, it may be very unsafe to push the system by an inch, as suggested in the image at top.
  • I’m not making any claims on the morality of cheating the system or the parenting choice to model behavior which cheats the safety system at amusement parks.

*- Here only 25% of the 4,400 incidents per/year were location identified. So this is a generalization.

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One Line Proof

OneLineProof

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Science March Draws Controversy.

There is something about airing dirty laundry that makes everyone uncomfortable. And we scientists have some dirty laundry… And we are struggling to interface with the world about our feelings.

The Root recently published a wonderful article which highlights the racist side of some of the leaderships teams for March for Science. It’s firing up a new hastag: #MarginSci. I am a white woman in STEM who often writes about how mathematicians interact with the everyday world. And if there is one thing that I truly believe, it’s that mathematicians and scientists don’t know how to get political.

Now, I’m going to spend a couple minutes talking about a white guy, but don’t let that put you off. White guys have had all the fun for the last 100 years (well, for way longer than that! but I digress). This particular white guy gives a good example of what happens when scientists spend years trying to influence politics. I present James Hansen.  No not the puppeteer Jim Henson, this is Jim Hansen. He is wildly famous in climate science circles for putting climate change on the map and into politics. He is also pretty famous as a scientist who, maybe, perhaps overstepped his bounds of a scientists into politics?

A colleague of Jim’s wrote, “I think he thought, as did I, If we get this set of facts out in front of everybody, they’re so powerful—overwhelming—that people will do what needs to be done. Of course, that was naive on both our parts.” Politicians didn’t respond to the facts. What’s worse, Jim started get a bad reputation in science circles. As New York Times author, Elizabeth Kolbert, wrote “Hansen argues that politicians willfully misunderstand climate science; it could be argued that Hansen just as willfully misunderstands politics.” [NYTimes]. His trouble is so famous, it’s included in his wikipedia page.

This is not limited to climate science.  I have PhD-wearing friends, astrophysicists and others, who feel that Neil Degrasse Tyson shouldn’t refer to himself as a scientist anymore because, well, he’s not doing science.  He is mostly doing science outreach. In the eyes of our serious academic partners, outreach doesn’t cut it.

We, as scientists, politely, yet inexorably, push the “less serious” out of our ranks. We excommunicate them from our inner circles because they want to influence the public (and, perhaps, the politics) of science. The old boys club is strong. We kick out some white people. We push away many black people and LGBTQ folks and a lot of others besides. We, as a group, quietly fill our plenary speaker spots with white men. There are several famous articles about the plenary speaker problem, the most beautiful by Lauren Bacon. Our women in science problems are large enough and public enough that major newspapers are talking about women in Silicon Valley.  The Atlantic article is particularly compelling about this. And, one cannot write an article about this without mentioning Susan Fowler’s review of Uber. And, I’m pleased to add #MarginSci to the list of public airing of science’s dirty failings.

I attended the women’s march in my city a few months ago. I was inspired and awed by the variety of issues that came to the table at that march. There were amazing posters about all kinds of issues. And there were some amazing posters about science.

And that march was organized fast! But what happens when a bunch of scientists take a few months to organize a science march?

Scientists (and I’m including Mathematicians in this too), are a different breed. We filter, classify, organize, and sort all of life into bins. We’d like to believe our sorting systems are unbiased. In fact, one could argue that’s the entire job; to create unbiased interpretations of the world around us.  But our biases are implicit. Sometimes we see them in advance. Like with Boston’s pothole app, where they purposely set out to get the app into every vehicle maintained by the city to get better coverage than just in the rich, cell-phone having areas. Harvard Business Review talks about these hidden biases in big data science.

But sometimes we don’t see them in advance. Sometimes we are human and we do stupid things. But now we, the scientists, are all trying to get into the public sphere to raise our issues. And people are starting to look more closely at our culture. Glass houses and all that. And people are starting to see that we, maybe, do not always live up to the ivory tower we’ve built for ourselves. And people are starting to wonder, “Maybe I don’t want to march for science, because science is just giving money to rich white guys.” To these people, I urge you to reconsider.

Science funding is key to our future. Science funding got us to space.  Science funding taught us about large prime numbers for internet security, about photovoltaic cells for solar panels, about cancer treatments, and our coral reefs. The list goes on and I, for one, will march for science and people’s equality. Even though there isn’t always people’s equality in science. The science is vital to our future. Despite our dirty laundry, we must persist and resist. We must talk about our issues, all of them, and bring them forward and solve them. Scientists are great at solving problems. But not always great at people. Let’s work to teach our scientific community about how to be inclusive and how to stand up for ourselves and each other.

Just as I saw posters for science at the women’s march, I hope I will see posters about everything else at the science march.  Because all of our issues need love. And all of our issues need science. They don’t seem mutually exclusive to me.

 

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Data Science Triple Threats

Everyone likes a good list. There are some articles working around which talk about why Data Science doesn’t add business value. As a data scientist who love people, I wanted to add my voice to conversation. Here are three things that make it hard to work with a team of Data Scientists (& Engineers).

  • Data Scientists are less likely to be mobilized on incomplete information.

A data scientist, or perhaps a PhD in general, wants to fully understand the problem before choosing a solution. This is best explained in an example: There may be one business rule which turns a problem from linear to non-linear.  But, the effect of that non-linear portion is small, so the business doesn’t bother to mention it until part way through the build process. The business doesn’t understand that the data scientists might literally need to start over to incorporate that new feature. As a result, data scientists have developed a healthy suspicion of project requests. No one wants to start over because the problem wasn’t accurately described first. So, the team stalls until they believe they understand the full problem. And that can take a long time. It’s a big challenge to begin a problem fast and get quick wins while simultaneously going slowly enough to protect from future disruptions.

  • Data Scientists will not believe something until they see it with their own eyes.

This personality quirk is very important to their job.  It means they question everything, validate unknowns and solve “unsolvable” problems. (After all, if you believe your colleague who says that the problem is unsolvable… then you aren’t going to be the one to solve it!).  However, it’s challenging to have a team that won’t accept second hand knowledge. Teams are forced to include the DS in every meeting, in order to build the requisite business knowledge.  Meanwhile the DS might be pushing the meeting down a tangent which is not the main focus of the meeting.  This, in combination with #1, is a hard problem.

  • Data Scientists require leaders who are Triple Threats.

In the performing arts a Triple Threat is someone who can sing, dance and act. In Data Science, a triple threat is someone who can understand the Mathematics, the Business and the Communication necessary to be a liaison between the first two sets of people. And often these traits are negatively correlated. People who are good at Math are certainly perceived to be less good at people. Thus, Triple Threats are rare!

Incorporating mathematicians into the workplace is more valuable than ever. Finding and acquiring a triple threat can be a challenging prospect, but something which companies should not shy away from.

What can we do about these challenges? Have you made progress on solving any of these challenges? What do you think are the biggest challenges facing data scientists right now?

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Simon’s Cat

As a gamer, the holidays give me opportunities to play games with non-gamers. People like my family. Or, more relevantly to this article, people like my niece. Here’s what you need to know about my niece: she is 6 years old who really loves to play games, begot from gamer parents. This year we played Simon’s Cat. hedgehogsAnd because she is 6, we played Simon’s Cat a lot. (The hedgehogs were my favorite.) And, as a mathematician and a gamer, I have to say that Simon’s Cat has a fair amount of game for its simple rules system.

Playing games with a 6 year old, as an adult, can be challenging. Mostly because they aren’t playing games at this age. They are playing experiences. Chutes and Ladders is most definitely an experience, not a game. Despite all those fabulous ladders and exciting chutes, the game is just a very complicated randomizer. It’s basically the Rube Goldberg machine of coin flips. It hurts my brain when a child is sad because they “lost” experiences like this. I just want to say, “You only had 1/2 a chance. There was literally nothing you could do to avoid this fate.”

I think a game gets to be called a game if and only if the choices you make as a player influence your ability to win or lose.   So, the question of the hour is: Is Simon’s Cat a game or an experience? Let’s cover the components and the rules.  Here are all the cards in the deck arranged in a pleasing way:

all_the_cards

You are dealt a portion of the deck. Whomever has the Pink Cat 3 plays it to the center.  Next, you go around the table and if you have a card that is the same color or the same number, then you can play it. If more than one card is playable, then you pick which one to play.  If you don’t have any cards that you can play, you collect all the cards from the middle of the table. These cards all count as 1 mess that you had to clean up. If you collected a mess, then you play any card you want to the table.  Suffice to say, you play a card every time it’s your turn, but you want to collect as few messes as possible. Once all the cards are played, the player with the fewest messes wins! Got it? If you want the official rules, you can find those here.

A game has a game if and only if the choices you make as a player influence your ability to win or lose.

In order for this to be a real game, my choices have to influence my chances of winning.  Thus, there must to be some strategy to the order I play my cards that is better than another.  Is my chance of winning higher when I use a good strategy than it is for me to randomly chose a usable card?

A slightly different way to ask this question is: Are some cards more valuable than others? Can some cards be used in more situations? If there are cards which are more valuable, then I want to keep those in my hand as long as possible for increased flexibility in the late game.  However, if all cards are equally valuable, then there won’t be a ‘better’ or ‘worse’ strategy and Simon’s Cat isn’t a real game (as far as I’m concerned anyways!)

gnomesSince I can only play a card if the previous card shares a color or number, I want to know which card has the most other cards which share a number or color: the most similar cards.  Let’s consider the Gnomes. There are only two Gnomes in the game. At first, this may appear to be valuable. (Because you can block others perhaps?) But remember, your main goal is to not get messes. So, you are most concerned with whether a card is playable by you or not. The Green Gnome 1 is only playable off of the Green Gnome 2, Orange Mouse 1, and Purple Dog 1. This means there are three cards in the deck which can precede the Green Gnome 1.

What about my personal favorite, the Yellow Hedgehog 3? Yellow Hedgehog 3 can play off of any of the five other yellow cards or any of the four other 3s. Thus, there are nine cards that a Yellow Hedgehog 3 can play off of. This means that between Green Gnome 1, with three similar cards and Yellow Hedgehog 3, with nine similar cards, the Yellow Hedgehog 3 is a more valuable card. Except, these can’t be used in the same situation.  So, we’ve shown that cards have different valuable-ness, but we haven’t shown that this will ever matter.

Let’s consider an example.  Let’s say the card played before my turn was a Yellow Hedgehog 3.  I have a Blue Kitten 3 in my hand and an Orange Mouse 3.  Which one should I play? Well, the Blue Kitten 3 has eleven similar cards.  Orange Mouse 3 only has seven.  Assuming we don’t know any more information about what was played previously, it would be a better choice to play the Orange Mouse 3 because there are more cards that can trigger my Blue Kitten 3 than my Orange Mouse 3. So, in this moment of our investigation, we know that there will be a strategy which is better than random. That means Simon’s Cat is an actual game! I can make a better or worse decision. I can impact my fate! Thank you, Steve Jackson Games, for making a simple game (without reading!) that is still a game. Seriously. Thank you.

Before we go into the bonus lightning round, I have to insert an aside for the other gamer mathematicians out there: for surely you have already determined that which card is the most valuable is not just a function of the total cards in the deck, as I presented above.  Returning to my example: if all the other Blue cards had already been played, then (in that moment) the Orange Mouse 3 is actually a better play because, at the moment, Blue Kitten 3 is not at the top of her game.  All her friends have already been played to the table!  So, she is less valuable.  To those of you who thought of this, excellent work! Your insight also further proves that there is a real game to be had here.

Now for the first bonus round:

Bonus Question: Which card begins the game as the most valuable card? Can you figure it out by looking at all the available cards in the deck?  You can go look. Take a guess!

Answer: Hopefully you figured out it had to be a Pink Catsimon_cat3 because Pink has the most cards in its color.  And the most valuable card should also be a 3 or 4 so it shares the most colors across a single number. Thus, the 2 most valuable cards are the Pink Cat 3 and Pink Cat 4.  Except you will never get to play the Pink Cat 3 on another card, because it must be played first.  Thus, the most powerful card is the Pink Cat 4. Notably, Pink Cat 4 is the only Pink Cat who isn’t doing something crazy in the graphic on the card.  Therefore, I am left with no other option than to assume that Simon’s real cat is Pink Cat 4.

 

And the final bonus round:

Reader be warned! It is very possible to be dealt a hand of cards and preceding cards in a way which allows no choices. In this situation, we, the adults, suck it up and “play” our gaming experience. Simon’s Cat has more game than most of the games I played over the holidays, but it’s still a game which can be explained in 3 sentences. It’s bound to have some flaws. For a game which can be played with pre-readers, it gets high marks from me!

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Alternate Mathematics

Alternate Mathematics: sometimes things seem too good to be true. #AltMath

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Those Topologists…

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