Is it normal for the gas gauge needle to jump around? Oh it's a '78.

 

No. The most likely cause is the fuel sender wire or ground wire being almost broken completely through somewhere in the line (usually near the tank or a metal terminal); or the sender unit in the tank might have the rheostat (slide-wire) going bad. But, the good news is that it's NOT the gauge; nor is it likely to be any wiring in the gauge cluster area.

 

This is a easy test.

Open your gas cap lid, pull back the rubber cover to expose the sending unit.

Locate the wire that connects to the post colored yellow in my diagram.
Once you disconnect the wire note how fast the needle moves to the 3 o'clock position. If its instant or very quickly, then the gauge is faulty. If it moves slowly 6+ seconds then you have a problem somewhere else. Also if you connect the wire direct to ground you should see the gauge move towards empty. Again if it is instant then the gauge is faulty. From full to empty a good gauge will take 8-12 seconds.

 

The good news its a 78 and you can access the sending unit from the top as suggested by cargo and 7T1.

If you find the wire in order, the next step is to retrieve and clean the pot and its sprint contact. Over the decades, the gas causes a varnish like substance to build between the pot cover and contact. Needs to be cleaned off. It is also possible the spring contact has lost contact with the pot cover.

This article describes how to access, clean and test the sending unit.

Fuel Tank Sending Unit Repair

 

http://willcoxcorvette.com/instructi...nual_ver_5.pdf

If you need more technical help this covers everything you ever want to know and more.

I have 2 bad units. 1 gauge and 1 sending unit. and using this document I have verified both. Also if you need a gauge be careful on what you purchase most of the reproductions are not 100%. On the original car there is a 45 ohm resister across the terminals. This needs to be removed and use the supplied one. Also test the accuracy before you install in the car. The above document will give you the instructions on how to test. The best gauge I have found is this unit.
http://www.mamotorworks.com/Corvette...-1982-632003-1
Now there are others that may also do very well but if they make the original AC / Del reproduction part it should be a perfect fit. Waiting to see what they deliver, currently on backorder. I dont know how to tell who makes the other repo parts, but my repro part is not accurate. Reads ok on empty but is way off on anything else. 1/2 tank reads about 3/4 and full is at about 2 o'clock posistion. Tried using the supplied resister and the original one. Neither worked properly. My new sending unit is worse than my original unit so I repaired the original unit (cleaning). The repro unit has spots where it breakes connection. Never could return it due to my supplier giving 90 warrenty and I waited to long to start doing the job. Now its a test unit for me.