Harmonic Mean Calculator

Harmonic Mean Formula

The harmonic mean is calculated as the reciprocal of the arithmetic mean of the reciprocals of the numbers. It's particularly useful for calculating average rates, especially when dealing with rates of change.

Formula:

Harmonic Mean = n / (1/x₁ + 1/x₂ + ... + 1/xₙ)

Steps

How to Calculate Harmonic Mean

To calculate the harmonic mean, follow these steps:

1
Take the reciprocal of each number (1/x)
2
Find the arithmetic mean of these reciprocals
3
Take the reciprocal of the result

For example, to find the harmonic mean of 2, 4, 8:

Example Calculation:

Harmonic Mean = 3 / (1/2 + 1/4 + 1/8) = 3 / (0.5 + 0.25 + 0.125) = 3 / 0.875 = 3.43

Examples

Harmonic Mean - Practical Examples

Example 1 Average Speed

A car travels 60 km at 60 km/h and returns at 40 km/h. What is the average speed for the round trip?

Harmonic Mean = 2 / (1/60 + 1/40) = 2 / (0.0167 + 0.025) = 48 km/h

Example 2 Parallel Resistors

Two resistors of 4 ohms and 6 ohms are connected in parallel. What is the equivalent resistance?

Harmonic Mean = 2 / (1/4 + 1/6) = 2 / (0.25 + 0.167) = 2.4 ohms

Example 3 Work Rate

Three workers can complete a task in 2, 3, and 6 hours respectively. What is their average work rate?

Harmonic Mean = 3 / (1/2 + 1/3 + 1/6) = 3 / (0.5 + 0.333 + 0.167) = 3 hours

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