The most important is to remember that Aref is dirrectly connect to the internal reference.
On Aref you can mesure the exact value of the "2.56V internal reference".
And if Aref is noisy Mesurment is also noisy.
ATMega328 Datasheet (for ATMega2560 it is the same):
ADC Voltage Reference
The reference voltage for the ADC (VREF) indicates the conversion range for the ADC. Single
ended channels that exceed VREF will result in codes close to 0x3FF. V REF can be selected as
either AVCC, internal 1.1V reference, or external AREF pin.
AVCC is connected to the ADC through a passive switch. The internal 1.1V reference is gener-
ated from the internal bandgap reference (VBG) through an internal amplifier. In either case, the
external AREF pin is directly connected to the ADC, and the reference voltage can be made
more immune to noise by connecting a capacitor between the AREF pin and ground.
VREF can also be measured at the AREF pin with a high impedance voltmeter. Note that VREF is a high
impedance source, and only a capacitive load should be connected in a system.
If the user has a fixed voltage source connected to the AREF pin, the user may not use the other
reference voltage options in the application, as they will be shorted to the external voltage. If no
external voltage is applied to the AREF pin, the user may switch between AVCC and 1.1V as ref-
erence selection.The first ADC conversion result after switching reference voltage source may
be inaccurate, and the user is advised to discard this result.
Edit : doing an average without remove the origin of the default is like "to hide the dust under the carpet"