Hi Jim. I kinda get this, but surely that narrow range isn't a case of critical measures such that a degree, or a fraction thereof, makes a vital difference. Of course I get that there are situations where one degree more hits fail line. But here's a point - when an official temperature measuring station records, say 40 degrees C the actual temperatures experienced by people might be as high as 50+ degrees C - depending on the location. In Australia official temperature measures are taken in a covered box with slatted sides. A person standing beside the box in full sun will experience a much higher temperature. So when we were told that Penrith had a temperature of 43 degrees C last summer all that meant was that it was 43 degrees C at the point of controlled and standardised measuring. In winter I record temperatures 5 or more degrees cooler than the official recorder for my district. I have had snow in my back garden while in the front all the snow melted days ago - and on days when the official temperature is around 4 degrees.
It is obvious that measurements of daily temperature can only be good to a finite accuracy. While Jim is right that you can use a thermometer to make measurements that are slightly more accurate than the gradations on the side, most of the measurements taken from weather stations, were taken for the purposes of weather, and I don't suppose those who took the measurements bothered to achieve the ultimate accuracy. Thus when it is claimed that the earth is 0.8 C hotter than it was in 1880, even that number may be wrong.
The practice of averaging a large number of such measurements and then looking for 0.01 C variations from year to year, seems to me to be just voodoo science.
In chemistry, melting points are important as a measure of purity, but even so, back in the 1970's, our thermometers were marked in tenths of a degree, and I notice that even now melting point data is normally quoted to this accuracy. In order to obtain that level of accuracy, within a laboratory, the sample is held in a small glass tube in a well stirred bath of liquid, which is slowly heated to ensure that the temperature is actually homogeneous to that degree of precision. Nowadays you probably put the sample into a machine! Measurements like that took time, and we had to do them as part of our practical exams - kind of stressful!