A barometer measueres momentum. A thermometer measures energy. If you want the exact explanation: A thermometer is a closed system which is in thermodynmic equilibrium with the system you want to measure. According to the zeroth law of thermodynamics then the temperature inside the thermometer and the other system are the same.

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now let's go to Maxwell-Boltzmann

first we need the kinetic temperature

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/kintem.html#c1you start out assuming the speeds in each direction are normal distributed. then we find out that the mean is zero. and the standard deviation is the same as the rms velocity. I'll write the rest with Mathematica:

http://stud4.tuwien.ac.at/~e0425252/mbdist/the first part is not mathematical mostly thinking a little bit. The second part goes easier with a little bit of advanced calculus but it's the same idea as in the radial probability function of electron orbitals for example.

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actually I'd like to add something to geodome. I'm kind of a part time mathematician ...

it's a general theorem of mathematics that with any set of numbers the root mean sqare is always larger or equal to the arithmetic average. (equal only if all the values are the same). same for arithmetic and geometric Sqrt(a*b). even smaller the harmonic average: 2/(1/a+1/b)