Shot placement is always an interesting topic. Everybody probably has their own formula for the best way to down a deer. Mike mentioned the lungs as the biggest target which makes a lot of sense, up a little and you get the spine, down and you ruin the heart, back a little and there is the liver and forward in the neck just right and you get the spine again. Low in the neck can be a terrible disaster just as a low head shot resulting in a messed up jaw. This usually brings a slow death and a lost deer. I'm glad that here in Florida we can use dogs to both hunt and recover deer.
The best shot for me from a stand or a least with a rest, is about 6 inches below the top of the shoulder straight up the front leg on a standing deer. My experience has been dead before they hit the ground. Without a rest on a standing deer, I'll have to go with the lung shot, same on a moving deer. They will generally stop, just like on TV, when you make a small noise or whistle, they try to locate the source, it's time to shoot. In my 50 years of deer hunting I can probably count on one hand the number of times I've shot at a running deer. This doesn't count the times years ago when we hunted on horseback with hounds, that's a whole different ball game!
A long the lines of Jerry's comment about a deer standing dead probably has something to do with what kind of a bullet was used. I killed an emaciated buck several years ago that had a drooping jaw. I watched him for a good while with the binoculars to see what was wrong with him, just looked bad. It was a long shot for me, about 200 yards but from a rest with my old '06 we go it done. When I got to him he had been shot through the jaw with a bullet that apparently didn't expand. He'd been that way for several days as he was skin and bones. If that buck had been shot with expanding bullet he would probably have been in some one's cooler rather than stumbling around in the planted pines. As for bullet type I've shot several different ones over the years, especially when I was reloading. In the last 15 years all I have used is 100 gr. .243 Cor-Loct Remingtons. Here are two pictures of the shoulder blades from a deer I killed last season. The near side is obvious, but look at the devastation of the off shoulder blade and between the blades was the spine. It's a given, he fell on his nose and I didn't mess up the heart and liver which we enjoy eating!
Just an old man's opinions and thoughts, I know there are more opinions, lets hear 'em

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Treefarmer