I definitely considered (and I am using) software smoothing, but the "noise" doesn't seem to be with mean = 0.
This what I'd expect, with
interference 60 Hz from electrical grid noise is not follow Gaussian math. But oversampling / averaging still would help to decrease variation when sampling frequency isn't a multiple of interference. Best of all, probably, take a samples with random delay, let say 100 millisec - 200 millisec, or implement software notch filter.
Regarding "off-set" you have in two measurements, it's symptom of great non-linearity presents:
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ADC input is non-linear. Solution: hardware filter, before signal get to input.
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Averaging is done after non-linear equation (with thermistor cubic polynomial?) Solution: only raw samples must be averaged. I'm guessing here, could you confirm that you've made this mistake, and average output results?