Harmonic Mean Calculator
Compute the harmonic mean for a list of positive numbers. Supports weighted harmonic mean, shows calculation steps, and compares with arithmetic mean.
How it works
Unweighted harmonic mean for n positive values xᵢ: H = n / Σ(1 / xᵢ).
Weighted harmonic mean with positive weights wᵢ: H = (Σ wᵢ) / Σ( wᵢ / xᵢ ).
The harmonic mean is appropriate for rates (e.g. speeds over equal distances) and gives more weight to smaller values. It is defined only for strictly positive values.
📌 What Is a Harmonic Mean Calculator?
A harmonic mean calculator is an online tool that helps you compute the harmonic mean of a set of positive numbers quickly and accurately. Unlike the arithmetic mean, the harmonic mean is especially useful when working with rates, ratios, speeds, and financial averages, where smaller values have a greater impact on the result.
This calculator supports:
- Standard harmonic mean calculation
- Weighted harmonic mean
- Step-by-step formula breakdown
- Accurate results with adjustable decimal precision
📘 Harmonic Mean Formula
The harmonic mean is calculated using the following formula:
H = n / Σ(1 / xᵢ)
Where:
- n = total number of values
- xᵢ = each individual value
Weighted Harmonic Mean Formula
When weights are applied, the formula becomes:
H = Σwᵢ / Σ(wᵢ / xᵢ)
This is useful in real-world scenarios such as finance, economics, and performance metrics.
🧮 Harmonic Mean Calculation Example
Suppose you have the values:
4, 5, 6
Step 1: Calculate reciprocals
1/4 + 1/5 + 1/6 = 0.6167
Step 2: Divide total count by the sum
H = 3 / 0.6167 ≈ 4.86
This harmonic mean calculator performs all these steps automatically and displays the result instantly.
📊 Harmonic Mean vs Arithmetic Mean
| Type | Best Used For |
|---|---|
| Arithmetic Mean | Simple averages |
| Harmonic Mean | Rates, ratios, speeds |
| Geometric Mean | Growth rates |
The harmonic mean is always less than or equal to the arithmetic mean when values vary.
🎯 When Should You Use Harmonic Mean?
Use a harmonic mean when:
- Calculating average speed
- Finding average price per unit
- Working with financial ratios
- Analyzing rates over equal intervals
This makes a harmonic mean calculator an essential tool for students, analysts, and professionals.
⚡ Why Use This Harmonic Mean Calculator?
- ✅ Free & online
- ✅ Works offline once loaded
- ✅ No ads or tracking
- ✅ Step-by-step explanation
- ✅ Supports weighted data
- ✅ Accurate & fast calculations
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❓ FREQUENTLY ASKED QUESTIONS
⚠️ Do NOT change wording unless you also update schema
❓ What is a harmonic mean calculator?
A harmonic mean calculator computes the harmonic average of positive numbers using the standard or weighted formula.
❓ How do you calculate harmonic mean?
Divide the number of values by the sum of their reciprocals using the formula:
H = n / Σ(1/x).
❓ When should harmonic mean be used?
Harmonic mean is best for rates, ratios, and averages like speed or price per unit.
❓ Can harmonic mean be calculated with weights?
Yes, this calculator supports weighted harmonic mean where each value contributes according to its assigned weight.
❓ Is harmonic mean always smaller than arithmetic mean?
Yes, when values are different, the harmonic mean is always less than the arithmetic mean.