About 20 yrs ago I briefly possessed a collection of science fiction short stories by various authors. It began with a story titled Quietly. It was about a girl named Quietly who is being raised by her father. After only reading the first page or two of the story, the book passed out of my hands. All I know of the story is that the girl and her father live apart from society. Her father is very strict and unyielding. When Quietly is a teenager her father sends her out into the wilderness/world alone so that she may prove herself. That is as far as I read. I have been searching for this story, or at least the author, for 20yrs so that I might finish reading it.
This must surely be the short story Quietly by Theodore Sturgeon.
The plot fits perfectly with that recalled by the OP. Quietly is the name of a girl, who is raised by her father, after her mother dies in childbirth. When she is 18, her father locks her out of the house, "hungry and naked", to make her own way in the world.
Although it was probably written in 1947, the story was unpublished until 1998, when it was included in an anthology of Sturgeon's short stories, The Perfect Host. This is the only point which does not match with the question - as far as I can tell it has not been published in a collection of stories from various writers. The fit to the plot is so good, this may just be "false memory effect" though.