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Harmonic & Geometric Mean Calculator

The three classical Pythagorean means — Arithmetic (AM), Geometric (GM), and Harmonic (HM) — capture central tendencies across different contexts like growth rates and speeds.

Enter positive numbers separated by commas or spaces (e.g. 2, 4, 8, 16).

Calculated Result
HM: 4.267 | GM: 5.657

Pythagorean Means

Harmonic Mean (HM)

4.267

Geometric Mean (GM)

5.657

Arithmetic Mean (AM)

7.500

Pythagorean Inequality

AM ≥ GM ≥ HM holds true

Calculation Breakdown

  1. Arithmetic Mean (AM)AM = (∑x) ÷ n = 7.500.
  2. Geometric Mean (GM)GM = (∏x)^(1/n) = exp((∑ ln x) ÷ n) = 5.657.
  3. Harmonic Mean (HM)HM = n ÷ ∑(1/x) = 4.267.

Pythagorean Means Comparison (Inputs: 2, 4, 8, 16)

Interactive visualization based on your current inputs

Value
0.01.93.85.67.5Harmonic Mean (HM)Geometric Mean (GM)Arithmetic Mean (AM)Mean TypeValue

What Is the Harmonic & Geometric Mean Calculator?

A harmonic and geometric mean calculator computes classical Greek mathematical central tendencies.

How Does the Harmonic & Geometric Mean Calculator Work?

Evaluates arithmetic, exponential logarithmic products, and reciprocal summation.

Harmonic & Geometric Mean Calculator Formula & Variables

The core mathematical equation utilized by this calculator is expressed as:

AM = ∑x ÷ n · GM = (∏x)^(1/n) · HM = n ÷ ∑(1/x)

Computes sum average, n-th root product (geometric), and reciprocal average reciprocal (harmonic).

How to Use the Harmonic & Geometric Mean Calculator

  1. Input a list of positive numbers separated by spaces or commas.

Step-by-Step Example Calculation

Series: 2, 4, 8, 16

Input Values:

numberList:2, 4, 8, 16

Understanding Your Result

Shows Harmonic, Geometric, and Arithmetic means and confirms ordering.

Factors That Affect the Result

  • Outliers pull the arithmetic mean upward; harmonic mean gives higher weight to smaller values.

When Should You Use This Calculator?

  • Investment portfolio CAGR, physics velocity averaging, computer benchmark scoring, and economics.

Assumptions & Limitations

  • All inputs must be strictly positive non-zero real numbers.

Frequently Asked Questions

Calculation Accuracy & Reference Note

Standard mathematical algebra evaluated via log-sums to prevent numerical overflow.

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