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

The harmonic mean, H, for n positive real numbers x_1, x_2, ..., x_n is defined as

H = n \div (1/x_1 + 1/x_2 + .. + 1/x_n) = n \div \sum_{i = 1}^{n}x_i^{-1}.

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:

  1. 60 km/hr for 120 km
  2. 40 km/hr for another 120 km

Then its average speed for this trip is

S_{avg} = 2 \div (1/60 + 1/40) = 48 \text{ km/hr}

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

S_{avg} = \Delta \text{distance} \div \Delta \text{time}

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

\Delta \text{ time} = 120 \text{ km} \times (1 \text{ hr}/60 \text{ km}) + 120 \text{ km} \times (1 \text{ hr}/40 \text{ km})

\Delta \text{time} = 5 \text{ hours}


S_{avg} = 240 \text{ km} \div 5 \text{ hours} = 48 \text { km/hr}

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!