The ambiguity of the meaning of the term “average”

Question

Suppose $\{x_1, x_2, \ldots , x_n\}$ is a set of data of n weights. The average weight is then (the sum of these weights divided by $n$), right?

Now, suppose $\{x_1, x_2, \ldots , x_n\}$ is a set of data of n speeds. The average speed, under the same logic, is then (the sum of these speeds divided by n), right?

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The main problem that I want to point out is:-

Suppose that a person travels from A to C via a straight road ABC by 3 stages.

In stage 1 (going from A to B), dts = [distance traveled, time taken, speed used] = $[D_1, T_1, S_1]$.

In stage 2 (resting), dts = $[0, T_2, 0]$.

In stage 3 (going from B to C), dts = $[D_3, T_3, S_3]$.

Now, which of the following correctly describes his “average speed”?

(1) [Using the method described previously], average speed = $\frac {S_1 + 0 + S_3}{3}$

Or

(2) average speed = $\dfrac {\text{total distance traveled}}{\text{total time taken}} = \dfrac {D_1 + 0 + D_3}{T_1 + T_2 + T_3}$

Answer 1

I think the problem here is that you're mixing up your ideas of what exactly is meant by average speed.

For the question you pose about a person traveling between 3 points, the average speed given by method 1 is a discrete measure. The "average" in this case is the average of the person's speeds in a discrete sense, i.e. it is the average of the speeds he travels at at any single point in time. However, if you multiply this "average" speed by the total time it takes for him to get from point A to point C then clearly the answer doesn't come out right.

The correct method is method 2. In this case you're determining the average speed the person travels at over the entire time interval.

The difference between these two methods is subtle but important. Basically, it all boils down to what you're averaging over: the number of different speeds you travel at or the time interval during which you are traveling.

Answer 2

The problem lies in the definition of average. What does $$ \frac{1}{3} \left(S_{1}+S_{2}+S_{3} \right), $$ really mean? It is hard to attribute an actual physical interpretation to this quantity. It clearly does not coincide with the definition in (2) that feels right when we think of average speed.

But there is a concept of average that does agrees with (2): the weighted average (or simply average, depending on your definition of average!). The above is just a special case of the weighted average giving equal weight to all three measurements/samples.

When we say average speed, we think of average with respect to time. Hence, the weights for each sample/speed correspond to the fraction of time for which we maintained that speed. Then, $$ S_{\text{avg}} = \frac{S_{1}\cdot T_{1}+S_{2}\cdot T_{2}+S_{3}\cdot T_{3}}{T_{1}+T_{2}+T_{2}} = \frac{ D_{1}+D_{2}+D_{3}}{T_{1}+T_{2}+T_{2}}, $$ which coincides exactly with the definition in $(2)$.