1) Definition
- The harmonic mean of $n$ positive numbers $x_1, x_2, …, x_n$ is:
$H = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_n}}$
It is the reciprocal of the arithmetic mean of reciprocals.
2) Comparison with Other Means
For two numbers $a$ and $b$:
- Arithmetic mean (AM): $\frac{a+b}{2}$
- Geometric mean (GM): $\sqrt{ab}$
- Harmonic mean (HM): $\frac{2}{\frac{1}{a} + \frac{1}{b}} = \frac{2ab}{a+b}$
Relationship: $\text{HM} \leq \text{GM} \leq \text{AM}$
3) Why it is useful
- Balances ratios: Harmonic mean is appropriate when numbers are rates or ratios (like speed, precision/recall).
- It gives more weight to smaller values.
- Example: If one number is very small, HM will also be small.
- Prevents one large number from dominating the average.
4) Example: Average Speed
Suppose you drive:
- 60 km at 60 km/h, then 60 km at 120 km/h.
- Arithmetic mean speed = (60 + 120) / 2 = 90 km/h (wrong).
- True average speed:
- Time1 = 60/60 = 1h
- Time2 = 60/120 = 0.5h
- Total distance = 120 km
- Total time = 1.5h
- Avg speed = 120/1.5 = 80 km/h
Now check with Harmonic mean: $H = \frac{2}{\frac{1}{60} + \frac{1}{120}} = 80$
Correct.
5) Example in Machine Learning: F1-score
- F1 = harmonic mean of precision and recall:
$F1 = \frac{2}{\frac{1}{\text{Precision}} + \frac{1}{\text{Recall}}}$
If either precision or recall is very low, F1 becomes low (balanced measure).
6) Summary
- Harmonic mean = reciprocal of arithmetic mean of reciprocals.
- Best for rates or when smaller values should dominate.
- Ensures balance (one very low value pulls mean down strongly).