I use the analog reference value which I connect to the AREF pin at 4.9V. The problem is that this value is never exactly 4.9V, causing the conversion to be inaccurate and resulting in different values in each conversion.
You cannot get exactly 4.9 V if you didn't apply very clean, stable and precise AREF from precise reference voltage source and the same clean and the same clean stable and precise voltage on ADC input. And even if you do that, any ADC has it's own noise figure and non linearity, so result still can be varied.
Any wires also add interference for measurements, so you're needs to use proper PCB layout to minimize such interference. If you're using wires/breadboard to connect AREF or AIN, it is almost impossible to protect it from significant RF interference. As result you can get high noise on measurement results.
Note that usual voltage regulator has not enough stability for using with AREF. You're needs to use precise reference voltage source chip.
Also you're needs to understand that power rails don't have stable and clean 5V, it's actual voltage depends on current consumption which depends on MCU program. Also power rail for digital circuit consists a lot of digital noise. So using power rail as reference voltage is a bad idea. Even you're don't needs stable measurement, you're needs to add at least low pass filter with ferrite beads between power rail and AREF to reduce it's noise.