Kids mature both physically and emotionally at varying ages. Despite the lack of university thesis data you do need to evaluate your child before you hand them a firearm or drop them into an adrenaline-charged situation. It's up to you to make the best decision possible.
Kids aren't perfect, but living with them day to day gives you better insight on whether they have what it takes to be responsible with a firearm. Do they show respect for others? Do they complete chores? All of these clues and more tell you when your child may be ready for the range. Hopefully you planned on making a hunter out of your youth from the time of their birth. Taking them with you to the range and on hunts, even though they couldn't participate, shows them the responsibility and enjoyment of shooting sports.
Immersing them in shooting sports will automatically begin teaching them home-based respect for firearms. Nobody can do a better job of instilling basic firearm safety and respect than you with your actions. Always promote responsibility with your firearms and your children will mimic what they've learned.
If you feel your child is responsible, start them off with an air gun and work up. For even more involvement, peruse organizations that offer shooting training, like 4-H Shooting Sports. Kids 8 to 18 can get involved. The Boy Scouts, the National Wild Turkey Federation and others offer shooting programs. The National Shooting Sports Foundation has many links to get you started.
If hunting is your ultimate goal read your state's hunting regulations to give you more insight when your child can participate. Some states have no minimum age while many have a set, minimum age to hunt. If they've successfully completed a certified hunting safety course or a mentored hunting program they may be legally ready.
Follow these steps, use your parental intuition and you'll know when it's time to take your child to the range, and on to their first hunt.
For more info on NSSF's Project Child Safe go to ProjectChildSafe.org.