Statistical math does allow for calculating an average and a tolerance, but he denies statistical math.
Among it's requirements are unbiased raw data (you can't cook it!). Since time is significant and location grouping is significant, the only way to get that is to use uniformly spaced thermometers (they aren't uniformly spaced!) and to read them at the same time by the same authority.
You are quite properly bringing up the requirement that you must declare and justify your variance. A variance of 20 deg/mile is a conservative estimate, since temperature can vary that much across storm fronts, from grass to asphalt, changes with altitude or by moving in or out of a warm or cold current, etc.
NOAA uses the larger number of thermometers. Given the thermometers they use, and given the variance I just declared, that means one thermometer will be measuring an area the size of Virginia (assuming they are uniformly spaced, which they aren't). With this variance, the margin of error is greater than the highest and lowest temperatures ever measured on Earth, so that becomes the fence.
Mathematically speaking, any claim of the temperature of the Earth is just guessing.
It's even worse trying to measure the global atmospheric content of CO2. There are very few stations, all on the surface capable of measuring CO2 at their sensors at all, and CO2 is not uniformly distributed in the atmosphere.
Due to similar problems, you cannot measure the temperature of the Earth, the global atmospheric content of CO2, the pH of the oceans, the total snow and ice on Earth, the range of every animal on Earth, the location of every plant on Earth, the total precipitation on Earth, or the emissivity of Earth.
The Church of Global Warming routinely denies and discards statistical mathematics, probability mathematics, random number mathematics, and even algebra.
Further, they routinely deny and discard the 0th, 1st, and 2nd laws of thermodynamics, and the Stefan-Boltzmann law.
Climate cannot change. It has no values that can 'change'. Climate is a subjective description, and is devoid of any objective values.