When a gas gauge is jumping around it can either be a faulty gas gauge itself or it could be the gas gauge sending unit that is located in the gasoline tank of your vehicle.
Your gas tank has a gas gauge sending unit that has a float on it that raises and lowers depending on the level of gasoline in the tank.
When you fill your gas tank up the gas gauge sending unit has the float that floats upwards which then sends an electrical signal to the gauge to show the level and as the level of gasoline drops in the tank the float lowers and sends a signal to the gauge again.
Sometimes the sending unit could be faulty or short circuiting causing the gas gauge to jump around.
Or you could have a bad ground to the sending unit.
If it's the sending unit then you'll need to replace the entire unit as a whole because the sending unit and fuel pump are built into one unit.
The sending unit and fuel pump can cost from $50.00 to $150.00 depending make and model of vehicle.
And if you have a shop replace it the cost can be $300.00 to $500.00 for parts and labor.