I am neither a quant nor someone who enjoys statistics, yet I observe a strange statistical phenomenon almost every day on my commute. Before I complain about Ankara’s public transit, I need to dig out my old statistics notes and reintroduce you to the Central Limit Theorem (CLT).
In my own words: under the right conditions, if you add up enough independent random variables, the properly normalized sum tends toward a normal distribution. Hence the famous bell curve, the most basic-looking graph you will ever see.
What set this off is the Galton board, also known as the bean machine. You have seen it in a science classroom or in a low-quality Gen X Facebook video, because it works like magic. Beans drop from a funnel, bounce chaotically off a grid of pegs, and somehow stack into a neat bell curve at the bottom.
Like many scientists of the 1800s, Francis Galton did not care about beans. He was obsessed with heredity and eugenics, and hoped the board would support his theories about inherited traits. Still, he left us a good line in Natural Inheritance (1889):
“Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.”
So that’s how chaotic beans form a bell curve. Now let’s talk about why Ankaray looks like the epitome of anti-statistics, and how it almost made me give up on the CLT.
As for the Ankaray, is the oldest rail line in Ankara and the second oldest in Turkey. “Ankaray” is a fun portmanteau word (Ankara + ray). Riding it between 17:00 and 18:00 is not fun tho. It is suffocating. Being a pioneer also left it with some quirky architecture. Unlike newer, cookie-cutter lines like Istanbul’s M6 Levent–Boğaziçi, Ankaray’s stations are not identical. Some have escalators, some only have stairs at the far ends, some funnel you into the middle, and platform widths vary wildly.
I know this sounds mundane but you are reading a statistics blog about Ankaray, so lower your expectations. But in my head, these station quirks act like the pegs on Galton’s board, scattering commuters along the platform. When I studied statistics, I assumed the result would be a textbook bell curve: packed in the middle of the train, pleasantly empty at the tails. That is the rush-hour rule everywhere, even in Istanbul.
It isn’t here. The exact middle of the train is weirdly emptier than the stretches on either side of it. Plot crowd density along the platform and you don’t get a bell, you get a giant “M”. Ankaralılar collectively prefer to cram into the zones between the center and the ends, while the middle doors and the far ends stay relatively breathable. To be clear, this is purely "qualitative". My method is staring at people on platforms, which I have done many, many times. But the queues form an M before the train even arrives.
My best theory is that Ankaray is a mash of two Galton boards. A classic board makes one bell curve because every bean drops from a single funnel. On Ankaray, commuters seem to arrive through two separate entry points, or “pre-walk” the platform to line up with the exit stairs at big transfer hubs like Kızılay. Overlap two normal distributions and you get a bimodal one: our glorious M. Add what pedestrian researchers like Dirk Helbing have modeled in his Social Force Model, that people tend to minimize effort and won’t walk three extra steps once they’re off the stairs, and the mystery starts to look solvable.
There’s one more catch tho, and it’s the sharpest point here. The CLT is about sums of independent, similar random variables. Commuters are neither independent (we follow each other, we avoid each other) nor identical (a person with a bag and a person with a stroller are not the same bean). So Ankaray isn’t breaking the CLT. It just never met its conditions...
Also please note that my only failed course in my undergrad is statistics 1 (I passed the second part flawlessly) and I passed the first part on my second take! (Did I just waste everyone's time?)
PRO TIP: If you ever have to ride Ankaray and want a slightly less claustrophobic trip, aim for the middle of the center car or the far ends, and avoid the peaks of the M. And if it’s between 17:00 and 18:00? Just walk. It smells better and the scenery is much nicer.