harmonic mean
C2Academic / Technical
Definition
Meaning
A specific type of average, calculated as the reciprocal of the arithmetic mean of the reciprocals of a set of numbers.
In statistics and mathematics, it is particularly useful for averaging rates or ratios (e.g., speed). In finance, it is sometimes used in portfolio analysis.
Linguistics
Semantic Notes
The term is mathematically precise; its meaning outside of technical contexts is zero. It is often contrasted with the arithmetic mean and the geometric mean.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or semantic differences. The spelling 'harmonick' is obsolete.
Connotations
Identically technical in both varieties.
Frequency
Used with identical frequency in academic and technical texts in both regions.
Vocabulary
Collocations
Grammar
Valency Patterns
The harmonic mean of X and Y is...To calculate the harmonic mean, take...X is the harmonic mean of the set.Vocabulary
Synonyms
Neutral
Weak
Usage
Context Usage
Business
Rare; might appear in specialised quantitative finance or economics papers discussing average rates of return.
Academic
Primary domain. Common in mathematics, statistics, physics, and engineering textbooks and papers.
Everyday
Virtually never used.
Technical
Core term in fields dealing with rates, ratios, or data smoothing (e.g., signal processing).
Examples
By Part of Speech
adjective
British English
- The harmonic mean calculation is required.
- They applied a harmonic mean estimator.
American English
- The harmonic mean calculation is required.
- They used a harmonic mean estimator.
Examples
By CEFR Level
- The harmonic mean is a type of average used in maths.
- If you travel at different speeds, your average speed is the harmonic mean of those speeds.
- To average ratios like speed, the harmonic mean is more appropriate than the arithmetic mean.
- The researcher calculated the harmonic mean of the precision and recall scores to get the F1 score.
Learning
Memory Aids
Mnemonic
Think of HARMONY in music: notes work together. The harmonic mean works with RECIPROCALS (1/x) together to find an average for things like speed.
Conceptual Metaphor
BALANCING RECIPROCALS / THE FAIR SHARE FOR RATES
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid translating 'harmonic' as 'гармоничный' (peaceful). The correct term is 'среднее гармоническое'.
- Do not confuse with 'арифметическое среднее' (arithmetic mean) or 'геометрическое среднее' (geometric mean).
Common Mistakes
- Using it for data sets containing zero (undefined).
- Confusing it with the arithmetic mean when averaging rates.
- Incorrectly stating the formula: n / (sum of numbers) instead of n / (sum of reciprocals).
Practice
Quiz
When is the harmonic mean most appropriately used?
FAQ
Frequently Asked Questions
It is 2ab/(a+b).
The harmonic mean gives less weight to high outliers and is designed for averaging rates (e.g., if you travel a fixed distance at different speeds, your overall average speed is the harmonic mean of the speeds).
No, because it involves taking the reciprocal (1/x) of each number, and division by zero is undefined.
For a set of positive numbers (where not all are equal), the harmonic mean is always less than or equal to the geometric mean, which is less than or equal to the arithmetic mean.