harmonic mean
Definition
- Noun:
- A specific type of average: The harmonic mean is a mathematical measure of central tendency. It is calculated as the reciprocal of the arithmetic mean of the reciprocals of a given set of numbers. It is particularly useful for averaging rates or ratios.
Usage
The harmonic mean is used primarily in mathematics, statistics, and various applied fields like finance and physics. It is the appropriate average to use when dealing with rates, such as speeds, or other quantities defined by a reciprocal relationship. - Formula: For a dataset of n positive numbers (x₁, x₂, ..., xₙ), the harmonic mean H is: H = n / (1/x₁ + 1/x₂ + ... + 1/xₙ)
Examples
- Noun:
- To find the average speed for a round trip, you must calculate the harmonic mean of the two speeds.
- The harmonic mean of the numbers 2 and 8 is 3.2.
- Financial analysts sometimes use the harmonic mean to average price-to-earnings ratios.
Advanced Usage
- "Weighted harmonic mean": A variation where each number has an associated weight. The formula is modified to account for these weights, making it useful for averaging ratios where frequencies differ.
Variants and Related Words
- Arithmetic mean (n): The most common type of average, calculated by summing numbers and dividing by the count.
- Geometric mean (n): An average calculated by multiplying numbers and taking the nth root, often used for growth rates.
- Pythagorean means (n): The collective term for the arithmetic, geometric, and harmonic means.
Synonyms
- Subcontrary mean (n): A less common historical synonym for the harmonic mean.
Related Concepts
- Reciprocal (n): The multiplicative inverse of a number (e.g., the reciprocal of is ). The harmonic mean is fundamentally based on the reciprocals of the data.
- Rate (n): A measure of one quantity per unit of another (e.g., speed). The harmonic mean is the correct average for certain rate problems.
Noun
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the mean of n numbers expressed as the reciprocal of the arithmetic mean of the reciprocals of the numbers
