Yes I am one of those people who constantly ask those stupid questions and frequently I get stupid answers in return. When you get a stupid answer it's normally because you've asked how do we know something that everyone just automatically assumes we know and in reality we don't know. So what is the daily temperature?
If I were tasked with producing an average temperature to represent a given day for official records, I would take either a continuous strip record of the temperatures (best technique) or take numerous discrete measurements such as every 15 minutes and average them together. I think if you asked a 7th grade science classes to come up with a number to represent a whole day's temperature, they would quickly come to a consensus to use a very similar method; and most reasonable people assume the this is how professional meteorologists do it. This professional meteorological data then is used by the professional Climatologists.
When you ask most people this they go to a table or a website and give you a number and
then report for example, BLACKSBURG VA, on Dec 21, 2008 the average temperature was 29 degrees F, but was it?
Definitions for the data Preliminary Climate Data(Form F-6) define
Maximum temperature. This is the highest temperature (F) recorded for the calendar day.(42 in this case),
Minimum temperature. This is the lowest temperature (F) recorded for the calendar day. (16 in this case) and ,
Average temperature. The sum of the previous two columns, divided by 2, and rounded, gives the value for this column.
This means while we are assuming that the dataset was rich and the reported average consisted of an average of 24-96 data-points per day, they are in fact using two! The main problem is while every day has a maximum and a minimum, they are not equally spaced. It may make sense if the minimum and maximum temperatures were 12 hours apart and the transition from one to the other was gradual, however in my neck of the woods temperatures can and often do change quite abruptly when storm fronts move into and out of the area. Because the "average" they use in the US is not representative of the reality on the ground, any subsequent use of the data taints that product as well.