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

In a previous Statistics Lesson of the Day on the harmonic mean, I used an example of a car travelling at 2 different speeds – 60 km/hr and 40 km/hr.  In that example, the car travelled 120 km at both speeds, so the 2 speeds had equal weight in calculating the harmonic mean of the speeds.

What if the cars travelled different distances at those speeds?  In that case, we can modify the calculation to allow the weight of each datum to be different.  This results in the weighted harmonic mean, which has the formula

H = \sum_{i = 1}^{n} w_i \ \ \div \ \ \sum_{i = 1}^{n}(w_i \ \div \ x_i).

 

For example, consider a car travelling for 240 kilometres at 2 different speeds and for 2 different distances:

  1. 60 km/hr for 100 km
  2. 40 km/hr for another 140 km

Then the weighted harmonic mean of the speeds (i.e. the average speed of the whole trip) is

(100 \text{ km} \ + \ 140 \text{ km}) \ \div \ [(100 \text{ km} \ \div \ 60 \text{ km/hr}) \ + \ (140 \text{ km} \ \div \ 40 \text{ km/hr})]

= 46.45 \text{ km/hr}

 

Notice that this is exactly the same calculation that we would use if we wanted to calculate the average speed of the whole trip by the formula from kinematics:

\text{Average Speed} = \Delta \text{Distance} \div \Delta \text{Time}

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}

Thus,

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!