The random error in a single thermometer reading is about 0.2◦ C (1 sigma) [folland et al., 2001]; the monthly average will be based on at least two readings a day throughout the month, giving 60 or more values contributing to the mean. So the error in the monthly average will be at most 0.2/ sqrt(60) = 0.03◦ C and this will be uncorrelated with the value for any other station or the value for any other month.