Rather than discuss limits of the human field of view, or extrasensory perception (I don't know anything about the first, and the second is a myth), I think we can look at this as a simple case of illusory correlation (wikipedia), which is both a psychological phenomenon, and something psychologists need to overcome to investigate other phenomena.
In a nutshell, people aren't automatically good at thinking about when events co-occur, and have an inbuilt tendency to see things that happen together, like looking at someone, and having them look back, as being related, rather than random coincidence.
In your example, there are four possible incidents:
- You look at someone, and they turn around.
You have a pretty good idea of how often this happens.
- You look at someone, and they don't turn around. You can find this out, but people are much worse at making use of this kind of information.
- You don't look at someone, but they turn around anyway.
- You don't look at someone, they don't turn around.
In these terms, your question is simple:
Is someone I'm looking at more likely to turn and look at me then someone I'm not looking at?
If somehow you could data all four occurrences above, you might run an experiment where you sit on a bus behind 5 other people, and look at each of them for 4 1-minute intervals each.
Looking at 5 people 4 times each, there's 20 opportunities for them to look back at (first column), but there's also 4 times as many opportunities (80; second column) for one of the people you're not looking at to be looking at you.
Let's also assume that during this experiment, you're looked back at on 10 of the trials (so not looked back at on the other 90).
Under the null hypothesis (the default you assume if there's no effect of looking at people), if people only look at you on 10% of the trials, you would expect them to look at you on 10% of the trials where you're looking at them (2 trials), and 10% of the trials where you're not looking (8 trials). You can tabulate this expectation like so.
| | I'm looking | I'm not looking | Total |
| They look at me | 2 | 8 | 10 |
| They don't look at me | 18 | 72 | 90 |
| Total | 20 | 80 | |
If, however, you're right, and looking at people does cause them to look back, you should find a significantly greater proportion of people looking back when you look at them, and so a higher value in the top left cell here. The technique we use for finding out if the proportions are different to what you'd expect by chance is called the Pearson's chi-square test.
The problem, to repeat, is that people aren't good at intuitively figuring out these proportions, and tend to overemphasise those few cases where both events happen (you look, and they look back), without appreciating the other 3 cells in the table. And this, my friend, is why we run experiments.