Average of (harmonic) average speeds?

Question

Suppose I have travel speeds $x_i$ at times $z_i$ ($i = 1, \ldots, n$) from point $A$ to point $B$.

Furthermore, assume that I have travel speeds $y_i$ at times $z_i$ ($i = 1, \ldots, n$) from point $B$ to point $A$.

Now, I know that the average speed at time $z_i$ ($i = 1, \ldots, n$) is given by the harmonic mean $$\frac{2{x_i}{y_i}}{x_i + y_i}.$$

Here is my question:

What formula do we use to compute for the average of the (harmonic) average speeds, say, per quarter?

I am confused as to whether we still need to use the harmonic average formula (since we are still dealing with equal travel distances [from $A$ to $B$ or from $B$ to $A$]), or if we could already use the (ordinary) arithmetic average formula.

Answer 1

Calling this the “average speed” is ambiguous (and this ambiguity may have contributed to your confusion). If (as in this case) two speeds are measured over the same distance $s$, in the sense that the distance was covered in times $t_1$ and $t_2$ and the corresponding speeds $v_1=\frac s{t_1}$ and $v_2=\frac s{t_2}$ are calculated, then the harmonic mean is the speed that would have been calculated if the two measurements had been combined into one:

$$ \frac{2s}{t_1+t_2}=\frac{2s}{\frac s{v_1}+\frac s{v_2}}=\frac{2v_1v_2}{v_1+v_2}\;. $$

Now the question is what you want to calculate for the quarter. You could just take the (arithmetic) average of the speeds if that's what you happen to be interested in. However, if you want to answer the same question again for the entire quarter that was answered by the harmonic mean for two measurements, namely, what speed would have been measured if all the measurements in the quarter had formed a single combined measurement, then you need to take the generalized harmonic mean of all the speeds measured:

$$ \frac{ns}{\sum_{i=1}^nt_i}=\frac{ns}{\sum_{i=1}^n\frac s{v_i}}=\left(\frac1n\sum_{i=1}^nv_i^{-1}\right)^{-1}\;. $$

You'll get the same result whether you first combine the speeds pairwise for each time $z_i$ and then combine the resulting speeds, or whether you combine all the $x_i$ and $y_i$ together in one go.

Basically, what you're doing is just averaging inverse speeds instead of speeds, so as long as you take the reciprocal before and after each averaging operation, you can treat this like a normal average.

(Note the terminological connection to the harmonic numbers and the harmonic series, which also involve reciprocals.)

Answer 2

I'll just add to the nice answer from @joriki.

Every "average" has a certain use. You don't just use an average without having a reason behind it (not just statistics but physics). For example, if you wanted to find the velocity that you can use in the equation for kinetic energy instead of summing over all objects (for example, calculating the "average" velocity of molecules at some temperature), you would calculate the quadratic average (root mean square) because you would get the definition of average from $(\sum m_i)v_{avg}^2=\sum m_iv_i^2$. Another example is the center of mass. In linear gravity (on the surface of the earth, not too far), the center of mass is the linear average of position over all objects. But when you are dealing with Newtonian gravity (big picture, planets and stuff), it's the harmonic average that determines the point around which the objects orbit.

So - there is no one "right" average - each has a different meaning, and is used differently.

That being said, back to your question. If these velocities are fundamental individual results (measurements with a radar or something) and you are doing statistics on these measurements... then use the arithmetic average, because that's what you do in statistical analysis when you are looking for the most likely value of some quantity, when a Gaussian distribution is assumed (symmetric to deviations up or down). But I don't think this applies to your situation.

Averaging inverse velocities is just like averaging times... which is usual in sports. And in sports, even doing statistics, you will do statistics with times, not velocities (and thus compute average times). So... as far as I understand, if talking about velocities is simply a convenient conversion, but the time is the true thing that was measured, then I'd say that just use the harmonic average for everything, because talking about velocities is just a "mask" over the true quantities in the system and velocities themselves were not really measured or detected directly. If I was doing this... I'd use harmonic average, or just stop pretending that we are talking about velocities (using times instead).