Harmonic Mean Calculator

What Harmon isic Mean?

The harmonic mean is one of the three Pythagorean means. It is particularly useful when dealing with rates or ratios, such as speed calculations or in financial mathematics.

When to Use Harmonic Mean

Use the harmonic mean when:

• Dealing with rates (like speed, where distance/time is important)
• Working with reciprocals of values
• Calculating average rates when the total quantity is fixed
• Financial calculations like average returns when investment amounts differ

Example Use Cases

Example 1: Average Speed

If you drive 60 km/h for 2 hours and 40 km/h for 3 hours, what's your average speed?

Harmonic mean is used here: (2 + 3) / (1/60 + 1/40) = 2.55 km/h

Example 2: Average Rate of Work

If a worker completes 3 units of work in 2 hours and 4 units in 3 hours, what's their average rate?

Harmonic mean: (2 + 3) / (1/1.5 + 1/1.333) = 1.727 units per hour

Formula Reference

For a set of n numbers x₁, x₂, ..., xₙ:

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

Note: The harmonic mean is always less than or equal to the arithmetic mean, and it's undefined when any of the values is zero.