# Mathematics and Applied Statistics Lesson of the Day – The Harmonic Mean

June 12, 2014 Leave a comment

The **harmonic mean, H,** for positive real numbers is defined as

.

This type of mean is useful for measuring the average of **rates**. For example, consider a car travelling for 240 kilometres at 2 different speeds:

- 60 km/hr for 120 km
- 40 km/hr for another 120 km

Then its average speed for this trip is

Notice that the speed for the 2 trips have equal weight in the calculation of the harmonic mean – this is valid because of the equal distance travelled at the 2 speeds. If the distances were not equal, then use a **weighted harmonic mean** instead – I will cover this in a later lesson.

To confirm the formulaic calculation above, let’s use the definition of average speed from physics. The **average speed** is defined as

We already have the elapsed distance – it’s 240 km. Let’s find the time elapsed for this trip.

Thus,

**Notice that this explicit calculation of the average speed by the definition from kinematics is the same as the average speed that we calculated from the harmonic mean**!

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