(weighted) mean of standard deviations

#1
Dear all,

I have the following problem. Suppose I have three results of temperature measurements:

10 +/- 2
2 +/- 6
9 +/- 3

Here the +/- are the standard deviations, or uncertainties on the measurements. The average here is 7. I do not want a weighted average because lower temperatures have a larger error bar, and a weighted average would obviously skew the mean towards higher temperatures.

Now, does anyone know a reasonable way to calculate the uncertainty on this average? I figure it is not simply the mean of the standard deviations, but unsure.

Any help very much appreciated!