Peter Newman points me to this claim by Chiefio that it is impossible to obtain 0.01°C precision by taking the average of measurements that are rounded to the nearest 1°C. Actually, he talks about °F, but it's the same argument. Chiefio appears to be overlooking two points. First, a thermometer that rounds to the nearest 1°C correctly must be able to distinguish between 1.49999°C and 1.50001°C. The perfectly-rounding thermometer is a perfectly-accurate thermometer. Second, every measurement has noise, and noise has some marvelous properties when it comes to taking averages.
Suppose we have an insulated box whose temperature is exactly constant at 20.23°C. We measure its temperature with a thermometer that is perfectly accurate but rounds to the nearest 1°C. So, we get a measurement of 20°C for the box, and we're wrong by 0.23°C. No matter how many perfectly-accurate thermometers we place in the box, we will always get a measurement of 20°C, and we will always be off by 0.23°C.
But thermometers are not perfect. Let's suppose we have a factory that produces thermometers that are on average perfectly accurate, but they each have an offset that is evenly-distributed in the range ±1°C. So far as we are concerned, the thermometer detects the temperature, adds the offset, and rounds to the nearest °C. To obtain the correct temperature, we subtract the offset from the thermometer reading. Half the thermometers have offset >0°C, a quarter have offset >0.5°C, and none have an offset >1°C.
Now we put 100 of these thermometers in our box and take the average of their measurements. Any thermometer with offset <0.27°C and >−0.73°C will give us a measurement of 20°C. Any thermometer with offset >0.27°C will give us a measurement of 21°C. And those with offset <−0.73°C will say 19°C. Applying our even distribution of offsets, we see that we have 13.5% saying 19°C, 50% saying 20°C, and 36.5% saying 21°C. So the average temperature of a large number of thermometers will be 0.135×19 + 0.5×20 + 0.365×21 = 20.23 °C, which is exactly correct.
In addition to permanent offsets, thermometers are subject to random errors from one measurement to the next. By taking many measurements with the same thermometer, we can obtain more precision. The underlying physical quantity tends to vary too, with short-term random fluctuations. These too we can overcome with many measurements, in many places, and at many times. The rounding error of a thermometer ends up being one of many sources of error.
In my field, we deal with rounding error all the time. We call it quantization noise. If we round to the nearest 1°C, the quantization noise is ±0.5°C, is evenly-distributed, and has standard deviation 1/√12 = 0.29 °C.
Our many evenly-distributed errors end up being added together. A consequence of the Central Limit Theorem is that the sum of many evenly-distributed errors will look like a gaussian distribution. So we tend to think of each measurement as being the correct physical value plus a random error with gaussian distribution. The arguments we have presented above will still work when applied to gaussian errors, because the gaussian distribution is symmetric.
So, despite Cheifio's skepticism, we can indeed obtain an exact measurement from a large number of thermometers that each round to the nearest °C.