> But you can't really assume that the estimate of the mean represents the real value. For example, if the sensor is equally likely to show 80 or 81 BPM when the real heartrate is 80.7, the mean estimator will be biased.

Bias is different from precision. If both conditions have the same bias, their difference is still unbiased.

> Also, wearables aren't taking 100 measurements of the BPM at a given point in time. I think the highest frequency they usually have is 1 second measurement interval. So they don't really have a lot of measurements for each point in time.

I did not mean taking multiple measurements in succession. Those are likely to have correlated noise, meaning the assumptions do not hold. But between participants measurement noise is very unlikely to be correlated.

> That as a rule of thumb, you should not assume that repeating measurements will give you more precision than what the tool can offer. E.g., trying to measure down to milimeters with a ruler that has only 1cm marks will not really work well.

If you quantize so much that you have no variance in the measurements, then sure. But watches typically have 1 bpm quantization, which is fine at the scale of variation in HR.

If you have independent error in measurements and quantization that gives you variance in measurement, you very much can assume repeating measurements will give you more precision than the tool can offer. This is how e.g. particle physics (and many many other fields of science) is done.

> Bias is different from precision. If both conditions have the same bias, their difference is still unbiased.

Not really, and I wouldn’t assume that when the condition under study can also affect how the measurement is taken.

> If you quantize so much that you have no variance in the measurements, then sure. But watches typically have 1 bpm quantization, which is fine at the scale of variation in HR.

At the scale of variation in HR in general yes, but not if the difference you’re trying to measure is in the range of 3 BPM.

> If you have independent error in measurements and quantization that gives you variance in measurement

That’s a big if. I am not confident we can claim that the errors are independent in this case, using sauna or not can affect how the sensor measures BPM.