At what age?

It depends entirely on the children in question and the situation of the environment. Leaving a 9yo alone in a home s/he knows the dangers of in a "safe" community is different than leaving a 15yo alone at home in a place you've just moved to, in a crime-ridden area.

You're a parent... use your judgement.

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Ignorance is the true enemy.

I've seen the village, and I don't want it raising my children.

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There isn't any one answer for this question. It depends on a lot of factors - maturity of the kids, location, neighbors, etc.
My oldest just turned 13 and the youngest is 11. Dad is away during the week and I work all over this end of the county in the summer. Frequently, I am unreachable by cell phone. We live on a dead-end gravel road with four other houses. Everyone else on the road also works during the day. I just don't feel comfortable leaving them alone all day. Weird things happen all the time and I am not just talking about perverts. One day, we were here at home and a wood duck fell down the chimney. The horse has been tangled up in a hog panel, the water softner started spewing water all over the garage, etc. They do get left home for a couple of hours while I run errands, etc. Maybe next summer they will be home during the day by themselves.

I am too paranoid about my children's safety to answer this directly. My 10 year old is extremely responsible, but accidents happen. I let her "babysit" her younger brothers (4 and 3) while I'm taking a shower or taking the dog out in the backyard for a potty break. If she were 16, I would ask her to watch them while I ran to the convenience store perhaps, or to the post office.

It has been proven that adults and teenagers get distracted with younger children. I'd personally recommend hiring someone to watch your children and not playing russian roulette with the possibilities. I don't have a teen in the house yet though and I don't know how mature my DD will be when she is 16/17 etc.

I don't know if my rambling was helpful at all lol.

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A closed mouth gathers no foot.

The oldest should be old enough to babysit other people's kids all day - 12 or 13. We live in a very rural area and I could leave the oldest with the youngers all day at 13. If the oldest had been one of my girls, maybe 12, they were more responsible. That said, we live about 6 miles from town, and I'd leave them alone here for an hour or so while I ran to town to run errands when the oldest was 11. Not all day, and not at night. We don't have neighbors, but our front gate locks.