## Tuesday, May 14, 2019

### How to calculate the average of city and highway fuel efficiencies.

This post discusses why harmonic averages -- not weighted or geometric averages -- are used to evaluate fuel efficiency and fuel use by cars.

Question:  The Honda Pilot (the best ranking eight passenger SUV) gets fuel efficiency of 17 mpg in the city and 24 mpg in the highway.    A person travels 10,000 city miles and 5,000 highway miles per year in this vehicle.   This information is summarized below.

 Honda Pilot Fuel Efficiency and Use City HWAY EPA MPG 17 24 Miles Driven 10,000 5,000

What is the total amount of gasoline consumed per year by this individual?

What is the weighted (city/highway) average fuel efficiency for this car?

What is the geometric average fuel efficiency for this car?

What is the harmonic average fuel efficiency for this car?

What average fuel efficiency statistic – weighted, geometric or harmonic – should be used to calculate gas consumed for this driver of the Honda Pilot?

Answer:    The total fuel consumed by this Honda Pilot owner is the sum of gasoline consumed in the city and gasoline consumed on the highway.

Gas = (10000/17) + (5000/24) = 796.6 gallons

The weighted average fuel efficiency is

WA = (10000 * 17 + 5000 * 24)/ (10000+5000)

=19.3333

Note that 15000/19.3333 = 775.8 gallons, which is less than the actual fuel consumed.   Hence the actual overall fuel efficiency for the Honda Pilot for this individual is less than 19.33 gallons.

The geometric average fuel efficiency is given by

GA= EXP ((10000 x ln(17) + 5000 * ln(24))/(10000+5000))

= 19.07

Note that 15000/19.07 = 786.5.   This is less than actual fuel consumed.  The geometric average fuel efficiency statistic also overestimates actual fuel efficiency.

The Harmonic Average fuel efficiency statistic is:

HA=(10000 + 5000)/ ((10000/17) + (5000/24))

=18.83.

Note 15000/18.83 = 796.6

The harmonic average provides us with an overall fuel efficiency statistic that is consistent with overall fuel consumed by this person.

Resources on different types of averages:

Wikipedia provides some very nice information on the different types of averages.

Harmonic Average:

Weighted Mean:

Weighted Geometric Mean: