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.
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!