# Is RMS velocity of gas particles always larger than the average velocity?

Is the root-mean-square velocity of gas particles always faster than the average velocity? I've done a couple of calculations with both and that seems to be the case; however can you make a formal mathematical proof of it? (i.e. maybe using Cauchy-Schwarz)

• Do you mean speed or velocity? – Farcher Sep 25 '18 at 8:08

EDIT (09/25/2018): So the OP confirmed that in his comment. So you can consider the difference between the squares of the rms velocity and average velocity $$(\sum_i V_i^2)/n-(\sum_i V_i)^2/n^2$$, which is a quadratic form, and find its minimum requiring that all derivatives with respect to $$V_i$$ vanish.