I have seen this mentioned in various places that doing push ups is like bench pressing half your body weight.

Fact? Fiction? Myth? Urban Legend? Rumor?

well basically its true. If you can do more than 30 in a row, you should add some weight on your back or something cause otherwise its quite pointless

test how many pushups you can do, then test how many times you can bench half your BW.. get all your friends to do the same, and you got yourself a scientific study :P

A pushup would actually be easier than a bench pressing half your body weight because with the pushup you are not usiong the surrounding stabilizing muscles as you are with a bench press

you do use a lot of stabilization in a push up, thought not in the same way as with the bench press. In the bench press you're stabilizing the bar, in the pushup you're stabilizing the body. That's a bit of a gross generalization, but it's pretty much it. The bar is the main unstable thing in the bench press, since you're lying on your back. In the push up, though, your body is the unstable thing that needs to be stabilized.

On a final note, though, I don't really think we should compare the push up and bench press that much. They are different exercises.

It depends on many factors. The higher up you have your legs, the more weight you're pushing up. However, the lower your legs are to the ground, then the less body weight you're pushing up. But not only that, depending on the angle, you'll work different muscles of the body.

So in a way, it's not a bench press but you are using your own body weight as a means of exercise.