Whenever I watch movies these days I tend to notice the little mathematical and statistical morsels that crop up. In fact, I’ve been noticing them for a while now such as the probability and game theory in The Hunger Games1 or the Chi-Square Test in Silver Linings Playbook. My latest excursion into math at the movies occurred when I recently saw the movie Divergent.

Divergent is based on a young adult novel by the same name that I haven’t actually read. My 12 year old daughter has read it, though, and was very excited to see the movie. Parents of excited 12 year olds often end up having to see such things, which is why I recently found myself sitting in a dark theater with my daughter waiting to see the adventures of Tris (one of the main characters in the movie).

The basic plot of the movie involves a society which is based on what appears to be a special kind of caste system. The castes are called factions and there are five of them:

• Amity – the peaceful ones
• Abnegation – the selfless ones
• Erudite – the intelligent ones
• Candor – the honest ones
• Dauntless – the brave ones

There’s also, in a sense, a sixth faction even though the movie calls them the 'factionless'. This is an 'underclass' of those who don’t belong to any of the five factions just mentioned.

As far as I could tell, people are initially born into the faction of their parents. At a certain age, though, each person is required to take an aptitude test. The test is designed to determine which faction they have the natural ability to be a member of. People are strongly urged to join the faction their test result has determined for them but they’re allowed to choose any faction they want to. Once that choice is made, though, they’re stuck with that faction and can’t choose another one later. If they chose a faction different from that of their parents and other family members, they must leave their family to go live with their newly chosen faction.

Now most people who take the test get a result which tells them they belong to one and only one faction. But rarely someone will get a result which indicates that they have a natural ability to be a member of more than one faction. Such people are called 'divergents' and are deemed a threat by public officials in this society – if caught they are killed by the authorities. It turns out that Tris is a Divergent although she chooses Dauntless as her faction. Most of the movie is about her trying to hide the fact that she’s a Divergent to avoid being killed.

Now you might be asking – where is the math or statistics in all this? Well for me the answer to this question is obvious. Because the first thing I wondered when watching Tris take the test is this: do public officials in this society allow for measurement error?

An assumption we make in the sciences is that all laboratory measurements are subject to measurement error. Mathematically we think of measurement in terms of the following equation: measured value = true value + error. Thus error = measured value – true value.

There is a whole area of statistics used by scientists to help them take error into account in their measurements. One of the most common ways of doing so is to measure the quantity of interest more than once and take the mean or average of those measurements as the best guess of the true value of the quantity.

For example, say you’re a physicist trying to measure the mass of some object in kilograms. You measure the object’s mass five times and get these:

• 1) 5.00kg
• 2) 5.20kg
• 3) 4.80kg
• 4) 5.02kg
• 5) 4.99kg

The mean of these measurements is 5.00kg to two decimal places so your best guess of the true mass of the object would be 5.00kg.

Now you might be asking how does calculating the average of a set of measurements tell us anything about the error in our measurements? That’s a great question and the answer requires me to say something about another statistical idea called the standard deviation. The standard deviation can be thought of as the average amount by which each value in a set of values differs from the mean of those values. That’s a mouthful so an example might help.

Instead of discussing the formula for the standard deviation, which I’ll leave for statistics textbooks, I’ll tell you what I got for the standard deviation of the mass measurements above using a statistical computing program called R. The value I got was 0.14 rounded to 2 decimal places. This means that on average our measurements are about 0.14kg from the mean of about 5.00kg. The smaller the standard deviation of a set of measurements the more bunched together or less dispersed those measurements are. When you’re trying to measure a quantity like mass, aptitude, or anything else a smaller standard deviation is better because it tells you that the values you got in repeated measurements are closer to the mean of those measurements, which is your best guess of the true value.

The way scientists typically account for error after calculating the mean of a set of measurements is to report the standard deviation of those measurements divided by the square root of the number of measurements taken. So in the case of our mass measurements this would be 0.14/√5 = 0.01. This value is sometimes called the standard error or standard error of the mean. Once this value is obtained a scientist would tell us that the mass of the object is 5.00 ± 0.01kg, meaning that it’s estimated to be anywhere between 4.99kg and 5.01kg2.

Now in the film Divergent it appeared that Tris’ aptitude was measured only once. But perhaps her aptitude should have been measured five, ten, or more times. The movie doesn’t make it clear what the unit of measurement for aptitude is or how authorities use aptitude measurements to determine what faction someone is most naturally fit for. So it’s not easy for me to know how an average of repeated measurements in this situation ought to be calculated. But remember one of the factions in this society is Erudite, the highly intelligent ones. Surely at least one of them would know enough statistics to appropriately take measurement error into account.

## Footnotes

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