Photo: PTI
Photo: PTI

A tall woman and a bell

Why do the first compartments of a train during rush hour seem relatively empty, but the ones in the middle don’t?

This column is prompted by something that angered me no end.

Wandering through the recent Kala Ghoda festival, I saw a young woman in the crowd who must have been 6’6" tall, if not more.

She towered over us all. Nearby was a small group, looked like college students. They were pointing and sniggering, and I heard one say: “Seems like she’s standing on a table!"

More on that in a moment.

By now, it has happened so often that you would think I wouldn’t get fooled. But I do.

Perhaps you have seen it too: you are on the crowded platform during rush hour at, say, Lower Parel. A train pulls in. As the first few coaches rumble past, you peek in and notice that they aren’t overly crowded.

“Good!" you think, “I’ll get a place, maybe even to sit." Then the train halts and what do you know—the coach in front of you is so crammed that you can barely get a toehold.

What happened here? Why does a train that seems relatively empty turn out to be jampacked?

More correctly, why do the first compartments seem relatively empty, but the ones in the middle don’t? Or maybe the most correct question is: why is the train pretty full in the middle, but less so at its ends?

The best way to think about this is as an example of a bell curve.

In mathematics, what is called the “normal distribution", when plotted as a graph, is shaped like a bell. There are plenty of explanations of this distribution and the bell, but I have always found the easiest one to understand is this: most of us are average.

It is likely you have heard of bell curves, because they show up all over the place.

Heights and weights, for example. But also what a class of students scores in an exam—one or two are probably toppers, similarly at the bottom, and most of the rest cluster around the average. Or the time it takes to commute to work every day. Or the money you spend on food every month (well, probably adjusted for inflation). Or the number of visitors a museum takes in.

For each of these, you can imagine what an average means. And whenever you talk of an average, you are really talking about a bell curve.

Take the Museum of Modern Art in New York, for example. It gets about 8,000 visitors every weekday. What this means is that on most such days, that’s about how many people stroll into the Museum of Modern Art. But certainly there will be occasional days with as few as 4,000 visitors, or as many as 12,000. And it is certainly possible, if very unlikely, that there will be a day with only a dozen visitors. (A measure called the standard deviation better indicates all these numbers, but let that be). Graph visitor numbers against how many days see those numbers, and you will get a classic bell curve with its peak right at 8,000, effectively defining that as the average. That is exactly the story the curve tells.

So what is the link to local trains? Well, consider a station that has just one entrance, in the middle of the platform and so at the middle of the train. Every commuter enters the station through this entrance. What happens when the train arrives?

Let me suggest: the way it gets filled has something to do with bell curves. Many commuters who come to the station to catch the next train stay put about where they enter the platform, meaning near the centre of the train. Why walk more, they think. But some others will walk, wandering some distance right or left along the platform and stopping there to wait.

A few determined stragglers will wander all the way to either end of the platform.

So the train departs with most of its passengers crowded into the middle compartments. (That is, the average commuter boards in the middle). Few, at either end of the train.

Because most of us are average. The bell curve, next time you commute.

And if you are an inveterate train watcher, as I am, you will be able to see the curve skewed if, for example, the entrance is at one end of the platform. (How it is skewed, I will leave to you). Or if there is more than one entrance. Or by some other interesting factor.

Returning to Kala Ghoda (after all, a train took me there): if the average Indian woman’s height is 5’3", let us say, a bell curve says that most women cluster around that height.

The further from there you move on the curve, the fewer the women you will find at that particular height.

This is why you don’t see too many women who are 6 feet tall, or 4’6" short for that matter. And the occasional 6’6" ladies, like the one at the festival, are outliers whom you’ll see only once in a while.

Whether that allows a bunch of unthinking yahoos to point fingers and laugh is another question.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers will explore the joy of mathematics, with occasional forays into other sciences.

Comments are welcome at dilip@livemint.com. To read Dilip D’Souza’s previous columns, go to
www.livemint.com/dilipdsouza

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