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The bystander effect is the sociological effect where people will sit and do nothing in the face of an event, even dangerous or tragic ones. Although everyone would like to think that they'd do something in the face of a problem, it's been shown both experimentally and in real life.
In 1964, Kitty Genovese was murdered in an attack that lasted over half an hour and attracted numerous witnesses. Because the multiple witnesses were perceived to have done nothing to halt the attack or call for help, the case prompted a lot of research into the bystander effect and it is still used as a parable to describe it despite the gross exaggerations of the media.
However, the attack and the behaviour of the witnesses has been subject to massive exaggeration, making it almost mythical in scale and scope. According to newspaper reports, 38 witnesses "saw" the event and did nothing - but in reality all but a handful weren't, in fact, eyewitnesses or first-hand witnesses to the attack. Further, and despite the popular claims, the police were called and several of the witnesses did move to help.
There are a few effects that contribute to the overall bystander effect.
If one person doesn't know something, it's quite likely that everyone doesn't know it too. This is obvious when you think about it, but people often forget it, which is what causes the bystander effect. People take their cues of what to do in a situation (whether it be dealing with someone who appears to be injured or if the building is on fire) from others. We often are clueless as to key information. If the crowd in general seem nonplussed, then someone is unlikely to react. The trouble with this is that this applies to everyone - and so the people in that unreacting crowd are likely to be thinking the same thing, trying to take their cues from others while trying to look as unflustered as possible.
Diffuse responsibility refers to how, in a crowd siutation, people are happy to assume that a task is someone else's responsibility. When everyone in a crowd begins to think like this, absolutely nothing will get done; everyone assumes that someone else will call the police, or help. This is one of the more serious causes of the bystander effect but it is also the easiest to overcome: with a little determination, you can overcome it and take the first move. Another option is to point to someone and tell them to sort it. It may be unfair, but this breaks the diffusion of responsibility pretty quick. 
The smallest number a crowd needs to be for diffuse responsibility to apply is, remarkably, 3.
- Debunking the myth of Kitty Genovese
- Darley JM, Latané B. "Bystander Intervention in Emergencies: Diffusion of Responsibility". Journal of Personality and Social Psychology, 1968, 8(4): 377-383.