High ADC accuracy and repeatability takes dedication at the PCB layout stage. There must be analog ground right beside the analog input and overall copper pour ground planes. Keeping all analog circuitry within its own fenced are of copper usually helps quite a bit.

Another good technique (thanks Melanie) is to take 16 readings, sort them, throw out the top 4 and the bottom 4 then average the eight values that are left. That technique eliminates the outliers and tames the last one or two counts.

I have also found several ADC ticks of noise/variability can be eliminated by locking the sampling point to the mains. I bring low voltage AC to an optocoupler and use that to trigger the ADC conversion. Varying the delay after the mains zero crossing will usually show a sweet spot where the fluorescent lights and other electrical loads are fairly quiet. I NEVER attempt 16 bit ADC without mains locking.

Sloppy techniques that are invisible with 8 bit conversion start showing up with 10 bits and can make the transition to 12 or 16 bit ADC a worthless expedition.

HTH
BrianT