How many times bigger is the Earth than the Moon?
This is true, however the Earth is approximately 6 times more massive.
We think the Earth-Moon system formed from a collision several billion years ago. The moon took a disproportionately large quantity of the surface of the earth, which is less dense, and the collision left us with an unusually large, hot and active core. As a result the moon is less dense on average than the Earth, so whilst it is true the Earth is 3.7 times larger, it is significantly more massive. As a result our value of g is 9.81