Sometimes when a train arrives in a platform it's really busy at one end and not very busy at the other. This is generally because, further back down the line, more passengers piled aboard at one end of the train than the other. And this in turn is because most passengers don't walk very far down the platform when they arrive.
I see this regularly at Bow Road station, heading westbound into central London. The stairs deposit everyone on the platform adjacent to where the rear of the train will be when it stops, and most people don't walk very far away from this point. Specifically it deposits them adjacent to carriage 6 of a 7 carriage train, and most people wait within the range 5-7. Some people walk down to 1-4, particularly if that gets them closer to the exit at the end of their journey, but most people seem to stay within a range of plus or minus one carriage from the bottom of the steps.
When a District line train arrives, the back of the train is usually busier than the front. If it's the shoulder of the rush hour, or if there hasn't been a train along for several minutes, the last carriage of the train is often standing room only. There can still be several empty seats down the front, where those waiting could have had a more comfortable journey, but because the doors are about to go beep people hop in adjacent to where they're standing, and so the rear of the train gets even busier.
I wondered if there was a good reason why District line trains arrive at Bow Road rear-loaded, and there is, which is a peculiarity of the way the stations further east are laid out. At Upminster the main entrance is at the front of the train, but at the next eight stations the main entrance is at the rear (mostly accessed down a long ramp from the street). That's eight consecutive stations where passengers arrive on the platform at the very back of a westbound train, plus three out of five of the remaining stations, so it's no wonder the back is busier than the front.
Here's a diagram to show you what I mean.
At three of the stations prior to Bow Road, the entrance to the platform is at the front of the train, indeed right at the front. But at the other eleven it's right at the back of the train, and it's never anywhere in the middle. People do spread down the platform, obviously, which is one reason the middle of the train isn't empty. Another reason is that the new S Stock trains are essentially one long central corridor with seats, so it's easy to shuffle down if the part of the train you're in is overcrowded. Oh, and of course some stations are busier than others...
The busiest stations at the eastern end of the District line, in terms of passengers entering and leaving, are Barking (16m), East Ham (15m) and Upton Park (10m). They each have at least twice as many passengers as the remaining stations, and considerably more than the runts of the litter, which are Upminster Bridge (1m) and Hornchurch (2m). Barking and West Ham are also interchanges, which drives considerably more traffic, but still not enough to make the front of the train more popular than the back. If I add up the number of passengers annually at each of the 'front' stations, and compare it to the total at the 'back' stations, the back comes out around three times busier than the front.
To model this properly, I'd also need to take into account the fact that not every passenger is going all the way. A heck of a lot of people pour out of the train at West Ham to change to the Jubilee line, so never make it as far as Bow Road. Few people starting their journey at Upminster ride the District line all the way into town when they could have taken the fast c2c train instead. The further away someone began their journey, the more likely they never made it this far. But the position of each preceding station's entrance does give a rough and ready explanation of why the back of a District line train pulling into Bow Road is busier than the front.
On the Hammersmith & City line (and short-running Districts) it's different.
These trains are less busy overall because they haven't been to any of the eight stations before Barking, so you're more likely to get a seat. They're also more balanced, with the busy station at East Ham and the busy interchange at West Ham having more of an impact up front, relatively speaking. But in the morning rush hour H&C trains do still arrive with a standing crush at the back and room to breathe at the front, and so the pattern persists.
Something similar happens on the Central line at Mile End. I've noticed that when I try to catch a westbound train (outside the morning rush hour), the last carriage often has numerous empty seats even when the front of the train is packed. I wondered if a similar analysis might explain why.
The Central line's a lot more complicated, with numerous routes and starting points, and entrances that aren't all at the front and back. But if I simplify each train to front, middle and back, rather than individual carriages, here's what I get for services arriving from Epping.
Stratford's a special case because it has platforms on either side of the train, and passengers arrive all along their length, which spreads things out more. But what stands out here is the triple whammy of Leytonstone, Leyton and Stratford, not just the three busiest stations on the line, but also the last three before Mile End, hence the significant front-loading of an arriving train.
Here's my diagram for trains arriving from Hainault.
Again, several stations immediately before Mile End have entrances near the front of the train, while the rear-entrance stations are further away and they're less busy. So it doesn't matter whether a Central line train is arriving from Hainault or from Epping, or from anywhere else along either line, you should be more likely to find space at the back than at the front. That's my experience anyway.
It's possible to apply this kind of analysis to any station on the tube, although the results won't usually be so clear cut. I used the Stationmaster app to tell me where all the entrances were, rather than deliberately traipsing out to all of them to check. And you can of course do this analysis for any station on any line anywhere, not just on the tube, should you be so minded. Most trains departing London mainline termini will be fullest at the back, for example, which is why I usually make the effort to walk all the way up to the front for a quieter life.
TfL and other train companies must model this kind of thing, taking into account precise entrance locations and actual passenger numbers, because it can be a key contributor to overcrowding. But what they can't easily do is change whereabouts each station entrance dumps us on a platform, only nudge us to move along to make everyone's life a little easier. At least Crossrail has been engineered with multiple entrances to balance us all out a little better, but I predict most people will still try to board the train close to the point where they arrive on the platform, rather than understanding the secrets of where might be better.