# harmonic mean vs arithmetic mean.

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• Jan 16th 2013, 03:47 PM
hisajesh
harmonic mean vs arithmetic mean.
Three cars goes from A to B in 2mph, 3mph and 4 mph.

In this case Harmonic mean of speed is 2 10/13 mph and while the arithmetic mean is 3 mph.

How do you logically prove that harmonic mean is more exact in this situation.
• Jan 16th 2013, 03:56 PM
richard1234
Re: harmonic mean vs arithmetic mean.
Instead of three cars, let's make the problem more intuitive by using one car going from A to B at 2 mph, then B to A at 3 mph, then A to B at 4 mph. We wish to find the average speed.

WLOG, suppose the distance from A to B is 1 mile. Then

$\displaystyle \text{avg. speed} = \frac{\text{tot. distance}}{\text{tot. time}}$

$\displaystyle = \frac{3}{\frac{1}{2} + \frac{1}{3} + \frac{1}{4}} \text{ mph}$

$\displaystyle = 2 \frac{10}{13} \text{ mph}$, which is the harmonic mean of 2, 3, and 4.
• Jan 16th 2013, 05:12 PM
HallsofIvy
Re: harmonic mean vs arithmetic mean.
Quote:

Originally Posted by hisajesh
Three cars goes from A to B in 2mph, 3mph and 4 mph.

In this case Harmonic mean of speed is 2 10/13 mph and while the arithmetic mean is 3 mph.

How do you logically prove that harmonic mean is more exact in this situation.

That question really doesn't make sense. What do you mean by "more exact"?
• Jan 16th 2013, 07:04 PM
hisajesh
Re: harmonic mean vs arithmetic mean.
@ all: Books suggest to use harmonic mean instead of arithmetic mean in such scenarios, could you explain why do we use harmonic mean and why not arithmetic mean.
• Jan 17th 2013, 08:29 AM
richard1234
Re: harmonic mean vs arithmetic mean.
Quote:

Originally Posted by hisajesh
@ all: Books suggest to use harmonic mean instead of arithmetic mean in such scenarios, could you explain why do we use harmonic mean and why not arithmetic mean.

Because the car travels at 2 mph for a longer time than it does at 3 mph, etc, similar to a weighted average (as opposed to an average or arithmetic mean, which weighs 2,3,4 equally). See above solution.