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I have a sample problem: Bob travels up a hill at $X$ miles an hour and down the hill at $Y$ miles an hour (You can assume that the length of the trail going up is the length of the trail going down in terms of distance).

If a question were to ask:

What is Bob's average speed in miles per hour?

How would I know whether the question is talking about the average miles per hour from a time perspective (where the answer would be the harmonic mean of $X$ and $Y$)?

$$\frac{2XY}{X+Y}$$

Or if the question is asking about the average miles per hour from a distance perspective (where the arithmetic mean would be the answer)

$$\frac{X+Y}{2}$$

To further clarify time perspective:

If $X = 3$, $Y = 12$ and the length of the hill is 3 miles:

The time it would take to go up the hill is one hour while the time it takes to go down the hill is 15 minutes. From a time perspective $X$ is four times more weighted than $Y$, not equally as weighted (they are equally weighted in the distance perspective since the distances up and down the hill are the same).

How would I go about a problem like the one "What is Bob's average speed in miles per hour?" My reason for this interest is due to my observation of a question that had the answer developed from a time perspective. I have also seen questions with the answers derived from a distance perspective.

Thank you.

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  • $\begingroup$ I am using means by its second definition @LiveForever, "For a data set, refers to a central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values." (arithmetic mean). I did not find a tag for harmonic mean sadly. $\endgroup$ – Mario Ishac Feb 2 '16 at 5:32
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    $\begingroup$ The whole point of speaking of "average speed" for a trip with varying speeds is that it answers the question: at what constant speed would the person have needed to travel in order to travel the total distance in the same amount of time? For this question, the arithmetic mean is pointless, and the harmonic mean is only useful when you've got two speeds over equal distances. The general answer is: total distance divided by total time. $\endgroup$ – symplectomorphic Feb 2 '16 at 6:43
  • $\begingroup$ (Compare this to the average value of a continuous function over an interval, which tells you what constant function -- i.e. what rectangle -- gives the same area as the area under your nonconstant function.) $\endgroup$ – symplectomorphic Feb 2 '16 at 6:46
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The generally accepted meaning is:

$$ \text{average speed} = \frac{\text{total distance}}{\text{total time}} $$

Besides, the arithmetic mean of the two speeds has no particular meaning of interest in this scenario. For example, if Bob ran up the hill at X mph, then circled the top of the hill at Y mph for 30 seconds, the arithmetic mean of the two speeds would still be the same.


[EDIT] P.S. To clear up any confusion about harmonic mean possibly having anything to do with average rates in general, that's just an artifact of the given scenario - where Bob runs equal distances up and down the hill.

Suppose instead that Bob ran at $X$ mph for a time $T_1$, then at $Y$ mph for $T_2$. The total distance covered would be $X T_1 + Y T_2$, and the average speed is $(X T_1 + Y T_2) / (T_1 + T_2)$.

If - and only if - the distances are equal $X T_1 = Y T_2 = L$ then it so happens that the average speed calculates to $(L + L) / (L / X + L / Y) = 2 X Y / (X + Y)$ the harmonic mean.

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