Statistics How To

Harmonic Mean: Definition, Formula, Examples

Statistics Definitions > Harmonic Mean

What is the Harmonic Mean?

The harmonic mean is a very specific type of average. It’s generally used when dealing with averages of units, like speed or other rates and ratios.

The formula is:
harmonic mean formula
If the formula above looks daunting, all you need to do to solve it is:

  • Add the reciprocals of the numbers in the set.
  • Divide the number of items in the set by your answer to Step 1.

Not that great with calculating reciprocals? Try this online calculator.

Example

What is the harmonic mean of 1,5,8,10?

  • Add the reciprocals of the numbers in the set: 1/1 + 1/5 + 1/8 + 1/10 = 1.425
  • Divide the number of items in the set by your answer to Step 1. There are 4 items in the set, so:
    4 / 1.425 = 2.80702

Tip: Check your calculations with this online calculator.

Difference Between the Harmonic Mean and Arithmetic Mean

The difference between the two means is a little bit of a mind-bender, so if you don’t quite get the concept at first, you aren’t alone. I can remember having to read the explanation several times and doing some examples on paper to assure myself it was correct.

Here is an example of a problem involving (the more common) arithmetic mean:
Joe drives a car at 20 mph for the first hour and 30 mph for the second. What’s his average speed?
The average speed is 20 + 30 / 2 = 25 mph.

And now, for an example with the harmonic mean:
Joe drives a car at 20 mph for the half of the journey and 30 mph for the second half. What’s his average speed?
For this problem note that we’re being told he went a certain speed for a journey segment. We need the harmonic mean:
= 2/(1/20 + 1/30)
= 2(0.05 + 0.033)
= 2 / 0.083
= 24.09624 mph.

The difference between these two is that the first problem is calculating an average speed based on time, while the second is based on distance. You’ll notice that the harmonic mean is slightly less than the arithmetic mean. This is always the case, that the harmonic mean will be the lowest average.

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