How does the error in the mean, of the mean, of the mean, etc. propagate? This is my first question here in the Mathematics section, so please forgive my transgressions.  I'm working with a large body of time series temperature measurements from weather stations.  The data is summarized for the weather station by the month, then the year.  Finally, all of the stations' annual means are themselves averaged to get an overall value for the year for all stations.
So here is my question/problem:  starting with $\pm$ 0.05 C as the uncertainty in the measurement of the daily high (TMAX) and low (TMIN), I get the uncertainty in the daily average (TAVG) as 
$$0.07C = \sqrt{0.05C^2 + 0.05C^2}$$
and in the monthly mean as
$$\Delta \overline {TAVG} _{est} = \frac{\Delta TAVG}{\sqrt(31)}$$
(e.g., 31 being the number of days in January) But it's been a long time since my University statistics courses, and the reading I've done online has not delved so far into propagating  the propagation.
Now I've got a month's worth of daily TAVG calculations, each with their daily uncertainty attached.  Here's the money question:  when I calculate the estimated uncertainty in the monthly mean, what do I do with the daily uncertainties?  All my online examples didn't cover having an uncertainty in the initial population.
And I have to go on from there, too.  After the monthly average is the yearly average, and then the hundreds of other stations' averages averaged together.  I need to keep track of this uncertainty all the way through.  But am I just overthinking this?
Thanks for your help!
 A: I think I may have figured it out, but I'm going on general principles, so I could be way off, too.  I'll start with a basic series of five daily high/low (TMAX/TMIN) temperature measurements with a thermometer that has a $\pm 0.05$  °C uncertainty.
20.5 °C $\pm$0.05°     11.4 °C $\pm$0.05°C
21.5 °C $\pm$0.05°     10.6 °C $\pm$0.05°C
23.6 °C $\pm$0.05°     9.9 °C $\pm$0.05°C
28.0 °C $\pm$0.05°     8.4 °C $\pm$0.05°C
22.3 °C $\pm$0.05°     12.1 °C $\pm$0.05°C  
To get the uncertainty in the daily average, I believe you first get the square root of the sum of the squares:$\sqrt{0.5^2 + 0.5^2 }$ = 0.7. The sum of the first two readings is 31.9°C. Since we're dividing by a constant, the relative uncertainty is used:  0.7/31.9 = 0.02.
Here is where I'm a bit unsure, but I think that once you have the relative uncertainty, it doesn't change with the multiplication or division.  31.9/2 = 16.0, and that result is multiplied by the relative uncertainty to get the absolute uncertainty:  16.0 * 0.02 = 0.32°C.  The final answer for the daily average would be 16.0 $\pm$0.3°C.
If that's correct, then you would do the same for each pair (I'm NOT going to do that here, we will just imagine it so.)
Now we have a full month of averaged pairs just like the example.  To average those and propagate the uncertainty, I believe the same process would be followed with the uncertainties for each day:  get the square root of the sum of the squares, divide that by the sum of each day's average to get the relative uncertainty.  Since we have a larger series of 30 or so daily averages, the formula for getting the estimated error in the mean, dividing the standard error of the series by the square root of the number of measurements.  After calculating the average in the usual way, you have the average  and the estimated error in the mean.  The only thing left to do is multiply the relative uncertainty by the monthly average to get the absolute uncertainty... and then add the two  uncertainties together?
This is where I'm out on a limb, but it seems right.  There's the estimated error in the mean from its formula, there's the straight-up average of the values, but you have to something with the other error as well, don't you?
I'm really interested in being able to propagate the uncertainty all the way as far as is necessary.  Thanks to anyone who reads this and offers some input regarding my method.  
