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Mathematics

Harmonic Mean Calculator

n / Σ(1/xᵢ). Free online Harmonic Mean Calculator. Calculate harmonic mean online — fast, accurate, mobile-friendly, no signup needed.

HM
3.428571

Derivation

  1. ├── 01Givena = 2, b = 4, c = 8
  2. ├── 02Formula3 / (1 / e.a+1 / e.b+1 / e.c)
  3. ├── 03Substitute3 / (1 / e.2+1 / e.4+1 / e.8)
  4. └── 04Compute HM3.428571
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§01What is

Understanding the Harmonic Mean Calculator

The Harmonic Mean Calculator computes HM from 3 inputs: a, b, c. n / Σ(1/xᵢ).

Mathematics shows up in every corner of daily life — budgeting, cooking, construction, engineering, even reading a bus schedule. A calculator like this lets you skip the scratch-paper step and move straight to the answer, without the arithmetic mistakes that creep in when the numbers get messy. The Harmonic Mean Calculator sits in that toolkit — it n / Σ(1/xᵢ). Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

3 / (1 / e.a+1 / e.b+1 / e.c)

Where

a
a
b
b
c
c
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: a = 2, b = 4, c = 8.

  1. 01Start by noting the input — a: 2.
  2. 02Start by noting the input — b: 4.
  3. 03Start by noting the input — c: 8.
  4. 04Substitute these values into the formula: 3 / (1 / e.a+1 / e.b+1 / e.c)
  5. 05Compute HM: the calculator returns 3.42857.
  6. 06Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Harmonic Mean Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

a halved

a = 1 (from 2)

Keep every other input at its default and halve the a. See how hm responds.

  1. 01New a: 1
  2. 02Baseline HM: 3.42857
  3. 03New HM: 2.18182
  4. 04HM decreases by 36.4% → use this sensitivity to plan for real-world variation.
02 · PATTERN

a doubled

a = 4 (from 2)

Keep every other input at its default and double the a. See how hm responds.

  1. 01New a: 4
  2. 02Baseline HM: 3.42857
  3. 03New HM: 4.8
  4. 04HM increases by 40% → use this sensitivity to plan for real-world variation.
03 · PATTERN

b halved

b = 2 (from 4)

Keep every other input at its default and halve the b. See how hm responds.

  1. 01New b: 2
  2. 02Baseline HM: 3.42857
  3. 03New HM: 2.66667
  4. 04HM decreases by 22.2% → use this sensitivity to plan for real-world variation.
04 · PATTERN

b doubled

b = 8 (from 4)

Keep every other input at its default and double the b. See how hm responds.

  1. 01New b: 8
  2. 02Baseline HM: 3.42857
  3. 03New HM: 4
  4. 04HM increases by 16.7% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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