Don’t Be Deceived by Brownian Motion

This post has no direct connection to transportation, however the phenomenon described in this post exist in any large, random system, including transportation. After reading this post, I encourage you to read Jarrett Walker’s post about confusing trendlines, and see if you can spot the Brownian motion argument, approached in a slightly different way.

Lately, I’ve had an increasing realization that understanding risk and probability is one of the most important numeracy tools we as human beings can learn and benefit from. Humans are terrible at estimating risk (for a number of reasons), and much of this stems from poor intuition and understanding of probability.

One of the biggest dangers of this is that there are ways in which people’s misunderstanding of probability can lead to being misled about information, regardless of the intent of the person communicating that information. Today I want to talk about a very important statistical and physical phenomenon known as Brownian motion.

Brownian Motion is named after a botanist, Robert Brown, who observed that particles of pollen grains moved through water, but wasn’t able to determine why they were moving. It turns out, as explained later by Einstein, that this motion helped definitively confirm that atoms and molecules actually exist.

But what does this have to do with probability, and why is it important to us?

The mathematical application of this phenomenon can be explained by a very simple game that we can play. Suppose we flip a coin. If it’s heads, we add one to our score, if it’s tails, we remove one from our score. We start the game with a score of zero. If we play this game for 200 flips and record our score every turn, the game might look something like this:


This is one of many possible outcomes, but I want you to notice two very important things about this graph:

  1. There is constant change in the graph
  2. The chart has fluctuations that are both large and small

The last point is the big one. This process was generated by one of the most fair and classical probability examples: flipping a coin. There is no outside influence. There are no government decisions, or natural disasters. There is no politics in this graphAnd yet it contains large periods of upward and downward “trends”. In fact, a graph like this looks familiar, doesn’t it? At least the shape and feel does. Here, for example, is the stock price of computer-chip manufacturer AMD for the past year:

or Alberta’s unemployment rate since 1980alberta1

While they are all certainly different (as would any other play of our game would be), they all have similar features; there is constant change, and both large and small fluctuations. It is no coincidence that Brownian motion has been used extensively to study stock market price and economic trends. If you knew nothing about the economic history of Alberta (or AMD, for that matter), you might consider this a fairly random process.

Long Term Trends

When mathematicians and probabilists talk about Brownian motion, they talk about long term trends. They talk about the average value of the process, which mathematically requires adding up an infinite amount of coin flips (which, incidentally is zero for our game). Because the process is so random, there is no use looking at short term trends for processes that follow Brownian motion, because the system is in constant fluctuation.

You might be inclined to argue that the graph of Alberta’s unemployment cannot possibly be as simple and random as flipping a coin, and you may very well be right. There are things the government can do to influence unemployment, and there are things that are out of its control that happen to, but are certainly somebody’s fault. That being said, is it really far-fetched to think that if the current choice in government is in some way random, then the “trends” the government creates is also random?

There is a concept in probability theory known as total ignorance, which argues that the less you know about a random situation, the less certain you are about a specific outcome, and the less certain you are about an outcome, the closer you get to “flipping a coin”. When we start to think about probabilities, we must start with this basic, totally ignorant assumption. Only once we have more information can we change our minds.

Here’s where you can get misled: If the information you are given looks like Brownian motion, be very wary of how much data you are looking atRemember, Brownian motion has meaningful long-term trends, not short ones. If I showed you the same Alberta unemployment data, but only in the past year, it tells a very different story:


The first question I ask when I’m shown this is “The data looks like it fluctuates frequently. Is there any more of it I can see? Have there ever been any big jumps?”. If there’s no good answer, I’m very wary of the information. Who’s to say we’ve not just been flipping a coin the whole time?


Before I sign off, I want to mention that Brownian motion comes in many flavours, including with drift. This system has similar properties (small and large trends, constant motion) but there is an underlying “force” or trend nudging the graph steadily up or down. In our coin-flip example, if we replaced our coin with a weighted one, so that heads was more likely, we would have added drift to our Brownian motion. The random processes above certainly could have trends in them that constitute drift. They likely do! These types of things are important to study and discuss, but the statement remains: we must look only at the long term.

So next time you read an article, or are shown an infographic, take a look at the graph from the eye of probability, and remember to ask yourself these questions:

  1. Is the data in constant motion?
  2. Am I being shown enough data to expect large and small fluctuations?
  3. Why am I being shown this specific piece of the data?
  4. Is there drift?

Happy coin-flipping!

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