Formula & Calculator
Harmonic Mean
Calculates the central tendency of a dataset by averaging the reciprocals, most appropriate for rates like speed over fixed distances.
Interpretation
HM = n / Σ(1/xᵢ). Used for rates and averages of ratios (e.g., speed, efficiency). Favours small values. Used in physics and finance.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| HM | Harmonic mean | |
| n | Number of values | |
| xᵢ | Each data value |
What it means
The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the data. It is used when averaging rates (e.g., speed, price‑earnings ratios). It is always less than or equal to the geometric mean. It is particularly useful for averaging ratios with different denominators. In finance, it is used for P/E ratios; in physics, for parallel resistances; in geometry, for calculating average speed over equal distances. Understanding the harmonic mean is important for proper averaging in contexts where arithmetic mean would be misleading.
Worked example
Harmonic Mean – Two Examples
Real‑World| Parameter | Value |
|---|---|
| Data | 60, 40 |
| n | 2 |
| Parameter | Value |
|---|---|
| Data | 10, 20, 30 |
| n | 3 |
Common mistakes
- Harmonic mean: Used for rates and averages of ratios.
- All data non‑zero: The harmonic mean requires all values to be non‑zero.
- Positive values: Typically used with positive values.
- Reciprocals: Sum the reciprocals of the data, then divide n by that sum.
- Comparison: For the same data, HM ≤ GM ≤ AM (arithmetic mean).
Applications
The harmonic mean is the reciprocal of the arithmetic mean of reciprocals. It is used when averaging rates, ratios, or quantities that are inversely proportional to the variable of interest. Engineers use it for average speeds (when distances are equal), for hydraulic conductivity, and for parallel resistances in electronics. In finance, it helps in averaging price‑to‑earnings ratios. By using the harmonic mean, professionals can avoid over‑ or under‑estimation that would occur with arithmetic means in such contexts. It is particularly useful when the data involve rates or frequencies. Understanding the harmonic mean ensures that averages are computed appropriately for the problem at hand, improving the accuracy of engineering and financial analyses.
- Average speed when distances are equal
- Parallel circuit equivalent resistance
- Hydraulic conductivity averaging (vertical flow)
- Portfolio valuation – averaging price‑earnings ratios
- Average rates of production or throughput
Frequently Asked Questions
HM = n / Σ(1/xᵢ). It is the reciprocal of the arithmetic mean of the reciprocals. It is used for rates and ratios where the average of the rates is desired.
Use it for average rates such as speed (distance/time), or when averaging per‑unit quantities like miles per gallon, or in finance for average price of multiple purchases.
If you travel the same distance at speeds v₁ and v₂, the average speed is the harmonic mean: 2 / (1/v₁ + 1/v₂). This is less than the arithmetic mean.
For positive data, the harmonic mean ≤ geometric mean ≤ arithmetic mean. The harmonic mean is the smallest of the three.
No, if any value is zero, the reciprocal is infinite, making the harmonic mean undefined. All values must be positive.
It is used to compute the average price when buying a fixed dollar amount of a stock at different prices (cost averaging). The harmonic mean gives the correct average cost per share.
HM = 2 / (1/4 + 1/6) = 2 / (0.25 + 0.1667) = 2 / 0.4167 = 4.8.
The F1‑score is the harmonic mean of precision and recall: F1 = 2·(precision·recall)/(precision+recall). It balances both metrics.
It is very sensitive to small values (since reciprocals are used). A single small value can drastically reduce the harmonic mean. It should only be used when the context calls for it.
HM_w = (Σ wᵢ) / Σ (wᵢ / xᵢ). This generalises the harmonic mean to weighted data.