Your theory is completely misconceived.

The A/D sampling on the arduino chip can detect 0-5 volts in 1023 steps. In round numbers 5 mV per step.

If you feed 5 mV to the chip, you will get the a/d answer 1

If you feed 10 mV to the chip, you will get the a/d answer 2 and so on.

If you feed 7.5 mV to the chip, you will MAYBE get sometimes 1, and sometimes 2. And if you do this a bunch of times, and get 1 and 2 a bunch of times, then you can do your averaging and get an average outcome of 1.5, and then maybe you can conclude that the voltage is somewhere between 5 mV and 10 mV, so let's call it 7.5 mV.

So, maybe, you can get 1 extra bit of implied precision.

But here is where your theory falls down in a twisted wreck.

If your input signal is 6.25 millivolts, your theory would depend on getting the a/d result 1, 75% of the time, and the a/d result 2, 25% of the time. You could then, supposedly, calculate an "average" a/d outcome of 1.25, and then interpolate the result of 6.25 mV in between the 5 mV and 10 mV.

The problem is, the a/d conversion does not work that way. The response of the device to the input voltage, looks like a staircase. For some input voltage range, between 3 and 7 mV, you will get the answer 1. For some input voltage range between 8 and 12 mV, you will get the answer 2. Between 13 and 17 mV, you will get 3. And so on.

These are the flat parts of the staircase. In some small region - which might be between 7 and 8 mV, or it might not be, you will get an uncertain answer, either 1 or 2. That's where your concept might work, for about 1 extra bit. but that is the only place it will work.

For any constant input in the flat step regions, you will always get the same reason. If you are not close to the vertical part of the step ( and you don't know exactly where that is ), you won't get different measurement outcomes. If you take a bunch of measurements with a 5.1 mV input, you'll always get the outcome 1. If you repeat this measurement with 5.2 mV, you'll also get the outcome 1. And at 5.3 mV. And at 6.3 mV. And anywhere else, up to the transition region of unknown location and extent, where the device outcome will jump to 2.

You cannot distinguish a/d readings up to 21 bits of precision, as you ludicrously claim.