If the child is 15 years old, we can assume they know what a nucleus and an atom are. Imagine there is a small object orbiting this nucleus, called an electron. We can explain that the electron is in a 1s-orbital, which has the shape of a sphere. An electon being in an orbital means that the there is a 95% probability of finding the electron within this orbital, thus within a certain radius from the nucleus of atom. The electron will orbit around the nucleus and sometimes even pass through it, as long as it stays in the orbital.
Imagine that over a large period of time, we look at the electron and measure the distance between the electron and the center of the nucleus every second. The electron can be further or closer from us depending on the time we record the position. After multiple measurements, we calculate the average position of the electron with respect to the nucleus. This will be the time-averaged position of this electron.