Depending on how we define "saving lives," the average person could be said to save many a day (remember one person can be "saved" multiple times) because our society's rules, for all its flaws, are set up to encourage certain positive behaviors. Safety measures is a good example.
I suppose most people would say "saving" requires active prevention of catastrophe, in which case most public works projects are pretty good. Homeless shelters, public healthcare, etc. Those that are paid into by everyone, so everyone has a share of the "saving."
I still doubt most cynics would be satisfied with that, so if you want empathy and generosity, take a look at how many people open doors for others, give seats, are polite, etc etc etc. Small things I bet you don't notice. Most people are kind if they've no reason not to be.
In reality what I think people want are huge acts of bravery and self sacrifice. Judging people on lack of those is really missing the point since most people don't have the opportunity to do them or are in situations where they don't realize it's on their shoulders to do something (see: crowds staring instead of helping). Meanwhile everyone has the opportunity to do something horrific every minute of the day, and the vast majority don't. And no, it's not because of deterrent either (it's easier to get away with things than you might think).