Harmonic Mean — Computation, Merits and Limitations

Harmonic mean (HM) is defined as the reciprocal of the arithmetic mean of the reciprocals of the observations. It is the correct average to use for quantities expressed as a rate per unit — most classically, average speed when equal distances are covered at different speeds.

1. Individual series:

2. Discrete series:

3. Continuous series — use the class mid-point $m$ in place of $x$:

Classic application: if a vehicle travels equal distances at speeds $x_1, x_2, \ldots, x_n$, the AVERAGE speed for the whole journey is the harmonic mean of the individual speeds — NOT their arithmetic mean, which would overstate the true average because more TIME is actually spent travelling at the slower speed.

Merits of harmonic mean:

• Based on all observations.
• Rigidly defined, capable of further algebraic treatment.
• The most suitable and mathematically correct average for rates and ratios (speed, price per unit when equal amounts of money are spent, etc.).
• Gives greater weight to smaller values, which is exactly appropriate for rate-type data.

Limitations of harmonic mean: …