Harmonic Averages

This is a short note to talk about Harmonic Averages. Most people are familiar with Weighted Averages, as they are a valuable tool for aggregation. For instance, with the data below, the average profitability (~1735) can be easily calculated using weighted averages.

Avg_Profit = (1000*1200 + 200*300 + 500*2500 + 10*600 + 100*300) / (1000 + 200 + 500 + 10 + 100)

or

Avg_Profit = SUMPRODUCT(UnitsSold,ProfitPerUnit)/SUM(UnitsSold)

I’m using Excel notation, and assuming it is clear from the context that UnitsSold is a range that covers the second column, for all models, etc.

A less known way of averaging are Harmonic Averages. It is relevant when the data to aggregate is actually a ratio whose denominator is proportional to the weighting factor. A typical case is miles per gallon (MPG) for a bunch of vehicles. Gas consumption is directly proportional to the number of units.

Let’s add some MPG data to the table above.

Using Weighted Averages for an inverse ratio like MPG is plain wrong (24.3 MPG is NOT the average fuel economy)

The right thing is to use Harmonic Average:

Harm_Avg_MPG = (1000 + 200 + 500 + 10 + 100) / (1000/22.5 + 200/15.0 + 500/32.0 + 10/12.0 + 100/24.0)

As Excel doesn’t have a similar function to SUMPRODUCT for adding 1000/22.5, 200/15.0, etc. I will not use Excel notation, but plain math notation:

UPDATED formula

If you have to deal with Harmonic Averages, you may find interesting this note on how to do PivotTable Multidimensional Analysis with Harmonic Averages. There’s a similar one for Weighted Averages as well.

Let me know what you think.