Average over two variables: Why do standard error of mean and error propagation differ and what does that mean? I'm doing an experiment with a cryostat to determine the critical temperature for lead. To avoid asymmetries, I determine the critical temperature both through heating (going from 2 K to 10 K) and cooling (10 K -> 2 K).
Now I have two values, that differ slighty and I average them. So a measurement of (6.942 $\pm$ 0.020) K and (6.959 $\pm$ 0.019) K gives me an average of 6.951 K.
Now the question is: what is the error of that average?
One way to do it would be to calculate the variance of this sample (containing two points), take the square root and divide by $\sqrt{2}$. This gives me an SEM of 0.0085 K, which is too low for my opinion (where does this precision come from?)
The other way is to say the the mean is a function of two variables, $\bar{T} = \frac{T_1 + T_2}{2}$, therefore by error propagation the error is $\Delta T = \frac12\sqrt{(\Delta T_1)^2+(\Delta T_2)^2}$, and that gives me a much more rational value of 0.014.
I see how those values differ in terms of numbers, but which one is correct when talking about the correct estimate for the standard deviation?
 A: If you take the standard deviation of your two values (first way you suggested), this does not include the error within each measurement, only the error between the two methods (heating vs. cooling). That is why the first way shows surprisingly low standard deviation- it's only incorporating one of the sources of error.
Therefore, the correct way is to propagate error (the second way you suggested). This incorporates both the measurement error of each value and the error between your two methods.

(Also a note based on the question comments: the manufacturer error listed is the +/- range guaranteed by the company based on all the devices they make. So yes, it does save you time if you use it as the equipment's error, but if you take many measurements of the same thing yourself you can calculate the error of your specific equipment and that can be more precise than the range guaranteed by the manufacturer. I don't know about thermometers, but this is what analytical chemists do with their glassware.)
