Unlike pianos, violins come in various sizes, from 1/16th to full size; depending on your child’s age and hand and arm sizes, you will need to buy a size that fits her.

Regarding quality of tone, don’t expect much of a sound to come out of a tiny 1/8th or a quarter one; starting from a half size violins start sounding half decent. Even for a ¾ size you need to dash out significant amount of money to get something that sounds fairly good. Apart from the size, many other factors influence quality of sound: the skills and care applied to manufacturing of the instrument, quality of the wood and other components used and the conditions under which the violin has been stored under.
 
Unfortunately there are no hard and fast rules in determining the quality of the violin and the fair price to pay. It’s best to ask around to find a reputable seller in your area and trust that he will sell you an instrument based on your expectations and budget.