There is no set distance. When you get in a self-defense shooting situation it all boils down to the individual event. That includes location, background objects, people and other things between you and the attacker, etc. Every shooting situation you might be presented with will have different obstacles or challenges. However, most self-defense shootings would more than likely be within that less than 15 yard distance as they will be a robber, mugger, rapist, etc that is trying to get to you and closing the distance or already on you.I was involved in two different shooting situations as LEO many years ago. On one I had drawn down on the person as they were running toward a 24 hour quick stop. I didn't immediately have a clear shot as my angle had him in the line of fire for the glassed area of the store. However, in about 1 second he would have had a brick wall in the background before he got inside and I would have taken a shot. Another officer had a clear line of fire immeadiately and shot the guy with his .357 from about 25 yards. The bullet hit him in the middle of the back and the guy dropped, so I never fired. Just remember, LEO's will be shooting at a fleeing felon, where you more than likely will only be shooting at someone who would be an aggressor and moving toward you.
I did have one scenario recently where I had a person come in from behind my left side to get in my face. I caught a glimpse of him about 1/2 second before he was on me. I was backed up against a truck, but did have time to quarter toward him as he got to me and push him away as he shoved against me. I also had my hand behind me and under my shirt ready to draw as I pushed him away in case he had a weapon or continued an attack. Luckily he had nothing in his hands, did not re-close the short distance, and was only verbally attacking me from that point on. I was able to get him away to move farther away and go on in the store. A call had been made to 911 to respond. The guy came back out of the store and shouted expletives at me as he left before police arrived. I didn't pursue it any farther than just telling the officer the story and giving him the license number of the vehicle. Had that guy wanted to, he would have easily been able to stab me before I could respond as he came at me from a blind area.