I think I had an intuitive sense of the issue, but after thinking about it, I can formulate the exact problem from my point of view.

Suppose that the thermometer is 21. That means in reality it's somewhere between 21.5 and 20.5 assuming perfect measurement and rounding.

And if it's 20, then it must be between 19.5 and 20.5.

So looking at the temperature moving by one degree, it could really be anywhere from zero to two degrees. But whether or not it moved one degree (C) is what I want to know. So the precision being equal to the meaningful amount is inadequate.

I don't consider this fancy science stuff, but something that bothered me at a gut level before figuring out why.

> Suppose that the thermometer is 21. That means in reality it's somewhere between 21.5 and 20.5.

Isn't that completely dependent on the thermostat? The tolerance could be lower, say between 20.9 and 21.1, or higher, such as between 20 and 22, or even something terrible like 18 and 24 (come to New York City some time and experience the wonder of our ancient heating systems).

Sorry, I wasn't talking about the precision that the temperature is maintained at, but just the precision that the temperature is measured with. Nothing to do with my thermostat.