Three cars goes from A to B in 2mph, 3mph and 4 mph.

In this case Harmonic mean of speed is210/13 mph and while the arithmetic mean is 3 mph.

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

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- Jan 16th 2013, 03:47 PMhisajeshharmonic 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 PMrichard1234Re: 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 PMHallsofIvyRe: harmonic mean vs arithmetic mean.
- Jan 16th 2013, 07:04 PMhisajeshRe: 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 AMrichard1234Re: harmonic mean vs arithmetic mean.