TFT

Harmonic Mean Calculator – Find Harmonic Average Online

Calculate the harmonic mean of any dataset with our free online harmonic mean calculator. Ideal for rates and ratios where harmonic averaging is more appropriate.

Numbers cannot be zero. Separate with commas, spaces, or newlines.

Examples:

Understanding Harmonic Mean

The harmonic mean is a type of average especially useful for rates and ratios. While the arithmetic mean adds values and divides by the count, the harmonic mean uses reciprocals – it's the reciprocal of the arithmetic mean of the reciprocals.

Use harmonic mean when averaging rates like speed, work rates, or prices per unit. For example, if you drive 60 mph one way and 40 mph back, your average speed isn't 50 mph – it's the harmonic mean: 48 mph. This is because you spend more time at the slower speed.

How to Calculate Harmonic Mean

  1. 1

    Find reciprocals

    Take 1 divided by each number. For 4, 8, 16: reciprocals are 1/4, 1/8, 1/16.

  2. 2

    Add the reciprocals

    Sum all the reciprocal values.

  3. 3

    Divide count by the sum

    HM = n / (sum of reciprocals). This gives the harmonic mean.

HM = n / (1/x₁ + 1/x₂ + ... + 1/xₙ)

Worked Examples

Example 1: Harmonic Mean of 4, 8, 16

Reciprocals: 1/4 = 0.25, 1/8 = 0.125, 1/16 = 0.0625
Sum of reciprocals: 0.25 + 0.125 + 0.0625 = 0.4375
n = 3
HM = 3 / 0.4375 = 6.857
Compare: AM = 9.33, GM = 8

Example 2: Average Speed Problem

Drive 120 miles at 60 mph, return 120 miles at 40 mph. What's the average speed?

Total distance: 240 miles
Time out: 120/60 = 2 hours
Time back: 120/40 = 3 hours
Total time: 5 hours
Average speed: 240/5 = 48 mph
Using harmonic mean: HM = 2/(1/60 + 1/40) = 2/(0.0167 + 0.025) = 48 mph ✓

NOT (60+40)/2 = 50! You spend more time at the slower speed.

Example 3: Two Numbers Formula

For two numbers a and b:
HM = 2ab / (a + b)
Example: a = 6, b = 12
HM = 2×6×12 / (6+12) = 144/18 = 8
Verify: 2/(1/6 + 1/12) = 2/(0.167 + 0.083) = 8 ✓

Example 4: Work Rate Problem

Worker A completes a job in 6 hours. Worker B completes it in 4 hours. Working together?

A's rate: 1/6 job per hour
B's rate: 1/4 job per hour
Combined rate: 1/6 + 1/4 = 5/12 job per hour
Time together: 12/5 = 2.4 hours

This uses the same reciprocal principle as harmonic mean.

Quick Fact

The harmonic mean was known to ancient Greek mathematicians and got its name from music theory. In a musical string, lengths in harmonic proportion (like 1, 2/3, 1/2) produce harmonious sounds. The Pythagoreans discovered these relationships around 500 BCE.

Frequently Asked Questions

When should I use harmonic mean instead of arithmetic mean?

Use harmonic mean for rates and ratios – speeds, work rates, prices per unit, fuel efficiency. Use arithmetic mean for quantities that add together directly. If you're averaging "per something" values, consider harmonic mean.

Why can't harmonic mean include zero?

The harmonic mean uses reciprocals (1/x). Division by zero is undefined. Also, if one rate is zero (like speed = 0), you'd never complete the journey, making the average meaningless.

What's the relationship between the three means?

For any positive numbers: Harmonic Mean ≤ Geometric Mean ≤ Arithmetic Mean. They're equal only when all values are identical. This is the HM-GM-AM inequality.

How does harmonic mean handle outliers?

Harmonic mean is pulled toward smaller values. A very large number has less effect than a very small number. This makes it useful when you want to penalize poor performance more than reward excellent performance.

Can harmonic mean be used with negative numbers?

Technically yes, but it's rarely meaningful. Rates and ratios are typically positive. If you have mixed positive and negative values, the harmonic mean might not give an interpretable result.

What's a real-world application?

The P/E ratio in finance often uses harmonic mean when averaging across companies. Fuel efficiency (mpg) should be averaged harmonically. In physics, parallel resistors combine like a harmonic mean.

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