The formula
Geometric = ⁿ√(x₁ · x₂ · … · xₙ) · Harmonic = n / Σ(1/xᵢ)
What it means
The arithmetic mean adds and divides, and that is right when what you are averaging accumulates by adding. But some quantities accumulate by multiplying, such as chained returns: gaining 50 % and then losing 50 % does not leave you where you started, it leaves you 25 % down, and the arithmetic mean would say zero. There the geometric one is right. And when you are averaging ratios with the same numerator — speeds over the same distance, prices per unit — the harmonic one is.
How to work it out by hand
- For the arithmetic mean, add the values and divide by how many there are
- For the geometric mean, multiply them all and take the nth root
- For the harmonic mean, add the reciprocals and divide n by that total
- Check the ordering: harmonic ≤ geometric ≤ arithmetic, always
What is worth knowing
The inequality between the three always holds and they are only equal when every value is, so it works as a check on the arithmetic. The classic harmonic case is the round trip: drive out at 60 km/h and back at 30 along the same road and your average speed is not 45 but 40, because the return leg takes twice as long and those hours count for more. The arithmetic mean says 45 and it is simply wrong.