Harmonic Mean Calculator

Calculate the harmonic mean for a set of numbers

Enter a list of non-zero numbers, separated by commas, to find their harmonic mean.

Other Titles
Understanding the Harmonic Mean: A Comprehensive Guide
Learn what the harmonic mean is, how it differs from other averages like the arithmetic mean, and its specific use cases, especially for averaging rates.

Understanding the Harmonic Mean Calculator: A Comprehensive Guide

  • The harmonic mean is a type of numerical average.
  • It is calculated as the number of values divided by the sum of their reciprocals.
  • It is the most appropriate average for rates and ratios.
The harmonic mean is one of the three Pythagorean means, alongside the arithmetic mean and the geometric mean. While the arithmetic mean is the most common type of 'average', the harmonic mean is specifically designed for situations involving the average of rates.
The formula for the harmonic mean (H) of a set of n numbers (x₁, x₂, ..., xₙ) is: H = n / (1/x₁ + 1/x₂ + ... + 1/xₙ). It tends to give less weight to larger values and more weight to smaller values compared to the arithmetic mean. It cannot be used if any of the values are zero.

Basic Examples

  • Harmonic Mean of 1, 2, and 4 is 3 / (1/1 + 1/2 + 1/4) = 3 / (1.75) ≈ 1.714.
  • Arithmetic Mean of 1, 2, and 4 is (1+2+4)/3 = 7/3 ≈ 2.333.
  • Note that the Harmonic Mean is always the smallest of the three Pythagorean means.

Step-by-Step Guide to Using the Harmonic Mean Calculator

  • Enter your set of numbers, separated by commas.
  • All numbers must be non-zero.
  • Click the button to instantly get the harmonic mean.
Our calculator simplifies the formula for you.
Input Guidelines:
  • Numbers: Enter a list of numbers separated by commas. These can be integers or decimals, but they cannot be zero.
  • Precision: You can set the number of decimal places for the final answer.
The Calculation Process:
The tool first counts how many numbers you've entered (n). Then, it calculates the reciprocal of each number (1/x) and sums these reciprocals. Finally, it divides n by this sum.

Usage Examples

  • To find the harmonic mean of 10, 20, and 30: Enter '10, 20, 30'. Result: 3 / (1/10 + 1/20 + 1/30) = 3 / (0.1833...) ≈ 16.364.
  • For the set {2, 4, 8}: Enter '2, 4, 8'. Result: 3 / (0.5 + 0.25 + 0.125) = 3 / 0.875 ≈ 3.429.

Real-World Applications of Harmonic Mean Calculations

  • Physics: Calculating average speed over a journey with varying speeds.
  • Finance: Averaging price-to-earnings (P/E) ratios.
  • Computer Science: Evaluating performance of parallel processors.
The harmonic mean is crucial in fields where the average of rates is needed.
Average Speed:
This is the classic example. If you travel a certain distance at a speed 'x' and return the same distance at a speed 'y', your average speed is NOT the arithmetic mean (x+y)/2. It is the harmonic mean of x and y. For example, if you drive to a city 100 miles away at 50 mph and return at 100 mph, your average speed for the entire trip is the harmonic mean of 50 and 100, which is 66.67 mph.
Finance:
In finance, the harmonic mean is used to average multiples like the P/E ratio. If you invest the same amount of money in several stocks with different P/E ratios, the harmonic mean gives a more accurate picture of the overall valuation than the arithmetic mean.

Real-World Scenarios

  • A car travels at 60 km/h for the first 10 km and 40 km/h for the next 10 km. The average speed is the harmonic mean of 60 and 40, which is 48 km/h.
  • You invest $1000 in a stock with a P/E of 20 and $1000 in another with a P/E of 30. The average P/E of your portfolio is the harmonic mean of 20 and 30, which is 24.

Common Misconceptions and Correct Methods

  • Using arithmetic mean for rates.
  • Attempting to use zero values.
  • Understanding its sensitivity to small numbers.
The biggest misconception is using the wrong type of average for a given problem.
Arithmetic vs. Harmonic Mean
  • Wrong: To find the average speed for a trip to a store at 30 mph and back at 60 mph, calculating (30+60)/2 = 45 mph is incorrect.
  • Correct: The distance is the same, so we average the speeds using the harmonic mean. H = 2 / (1/30 + 1/60) = 2 / (0.05) = 40 mph. The true average speed is 40 mph.
Sensitivity to Low Values
The harmonic mean is heavily influenced by the smallest values in a dataset. A single very small number can pull the harmonic mean down significantly, much more than it would affect the arithmetic mean. This is because small numbers have large reciprocals, which dominate the sum in the denominator of the formula.

Correction Examples

  • Dataset {1, 10, 100}: Arithmetic Mean = 37. Harmonic Mean ≈ 2.7.
  • Dataset {10, 100, 1000}: Arithmetic Mean = 370. Harmonic Mean ≈ 27.3.

Mathematical Derivation and Properties

  • Relationship to other Pythagorean means.
  • The formula for weighted harmonic mean.
  • Its origin in music and harmony.
The harmonic mean is part of a trio of classical means with interesting relationships.
Pythagorean Means Inequality
For any set of positive numbers, the following inequality holds: Harmonic Mean (H) ≤ Geometric Mean (G) ≤ Arithmetic Mean (A). The three means are only equal if all the numbers in the set are identical.
Weighted Harmonic Mean
A weighted version exists for when certain values are more important. The formula is: Weighted H = (w₁ + w₂ + ... + wₙ) / (w₁/x₁ + w₂/x₂ + ... + wₙ/xₙ), where 'w' are the weights corresponding to each value 'x'.
Name Origin
The term 'harmonic' is derived from its use in music theory by ancient Greeks. The harmonic mean of two numbers represents the length of a string that produces a tone which is the average of the tones produced by two other strings. For example, the harmonic mean of 6 and 12 is 8, which corresponds to the musical interval of a fourth relative to the octave.

Mathematical Properties

  • For the set {2, 8}: A = 5, G = 4, H = 3.2. This demonstrates A > G > H.
  • For the set {5, 5}: A = 5, G = 5, H = 5. All three means are equal.