Averages
There are three candidates for what is called the "average" value of the speed of the Maxwell speed distribution.
Firstly, by finding the maximum of the MSD (by differentiating, setting the derivative equal to zero and solving for the speed), we can determine the most probable speed. Calling this v_{mp}, we find that:
Second, we can find the mean value of v from the MSD. Calling this :
Third and finally, we can find the root mean square of the speed by finding the expected value of v2. (Alternatively, and much simpler, we can solve it by using the equipartition theorem.) Calling this v_{rms}:
Notice that
These are three different ways of defining the average velocity, and they are not numerically the same. It is important to be clear about which quantity is being used.

Read more about this topic: Maxwell Speed Distribution
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