It is a .177 pellet hole. The smooth edges of the hole connotate a wadcutter like projectile. A simple BB would probably only crack the window if fired from a close range. The hole really isn't a penetration hole, it's where the pellet contacted the window. You might find it between the screen and the window. Since it is a pellet, it rules out a Red-Rider type spring BB gun. We're now talking air rifle. Older kid.
To determine whether it was intentional or accidental?
Take a length of straight heavy gauge wire. Like the bottom piece of a coat hanger. Stick it through the pellet hole AND the hole in the screen. This will determine the direction the pellet came from. Since it was an air rifle, it would have more power and a flatter trajectory than a BB gun. (Closer to being directly pointed at by the wire) Due to loss of power, the wire will almost always point over where the pellet was fired from. (Not always. If it had enough power behind it, it may point directly at the firing point. It will never point BELOW thewhere it was fired from.)
Where does the wire point?Is there anything between your window and a good source of cover? If there is a tree or bush in that direction, chances are it was fired through the bush or missed the tree and might have been unintentional. If it points to open space such as a street or parking lot, it may have been randomly fired, therefore, unintentional.
If the wire points toward a neighbor kid's window, or there is nothing of any "target worthy value" between your window and a good firing platform, chances are it was an intentional shot at YOUR WINDOW.
I'd definately look for a teenager. Young children aren't usually allowed to keep posession of air rifles. I had one when I was six, but I wasn't allowed to shoot it alone. You are looking for a lone gunman. Probably from 12 - 15. Definately not a girl.
Now...BOOK 'EM DANO!