Mean Calculator

In statistics and mathematics, the "mean" encompasses the Pythagorean means: the Arithmetic Mean (standard additive average), the Geometric Mean (mult

Mathematics

Formula & Methodology

Arithmetic Mean (AM) = (Σx) ÷ n Geometric Mean (GM) = (x₁ · x₂ · ... · xₙ)^(1/n) = exp((Σ ln(x)) ÷ n) Harmonic Mean (HM) = n ÷ Σ(1/x)

The three Pythagorean means satisfy the fundamental inequality HM ≤ GM ≤ AM for all positive datasets: 1. Arithmetic Mean: Sum all numbers and divide by count n. Used for additive quantities. 2. Geometric Mean: Take the n-th root of the product of all positive numbers. Used for investment returns and growth scaling factors. 3. Harmonic Mean: Divide the count n by the sum of reciprocals. Used for average speeds, rates, and densities.

Arithmetic Mean (Σx / n)

Overview

In statistics and mathematics, the "mean" encompasses the Pythagorean means: the Arithmetic Mean (standard additive average), the Geometric Mean (multiplicative rate average), and the Harmonic Mean (rate/ratio average). This free Mean Calculator computes all three classical means simultaneously for any dataset.

Understanding all three means is essential: financial compound growth rates require geometric means, velocity and fuel efficiency calculations demand harmonic means, and standard distributions rely on arithmetic means.

Worked Examples

Example 1: Pythagorean Means for [14, 22, 28, 36, 40]

Dataset: 14, 22, 28, 36, 40 (n = 5)

Arithmetic Mean: (14 + 22 + 28 + 36 + 40) ÷ 5 = 140 ÷ 5 = 28.00.
Geometric Mean: (14 × 22 × 28 × 36 × 40)^(1/5) = 26.06.
Harmonic Mean: 5 ÷ (1/14 + 1/22 + 1/28 + 1/36 + 1/40) = 24.16.

Final Answer: AM: 28.00, GM: 26.06, HM: 24.16

Example 2: Average Speed over a Round Trip (60 mph and 40 mph)

Dataset: 60, 40 (Harmonic Mean applied to rates)

Harmonic Mean = 2 ÷ (1/60 + 1/40) = 2 ÷ (2/120 + 3/120) = 2 ÷ (5/120) = 48 mph.

Final Answer: 48.00 mph (Harmonic Mean)

Example 3: Compound Investment Growth Multipliers (1.10, 1.25, 1.05)

Dataset: 1.10, 1.25, 1.05 (Geometric Mean applied to growth)

Product: 1.10 × 1.25 × 1.05 = 1.44375.
Cube root: 1.44375^(1/3) = 1.1302 (13.02% annualized return).

Final Answer: 1.1302 (Geometric Mean)

How to Use This Calculator

Enter numbers separated by spaces or commas into the dataset field.

Select which mean (Arithmetic, Geometric, or Harmonic) you want as the primary display.

Click "Calculate" to evaluate all three means.

Compare the results in the secondary metrics and visual bar chart.

Click "Reset" to analyze a different dataset.

Dataset	Arithmetic Mean	Geometric Mean	Harmonic Mean
2, 8	5.00	4.00	3.20
4, 9	6.50	6.00	5.54
10, 20, 30	20.00	18.17	16.36
14, 22, 28, 36, 40	28.00	26.06	24.16
50, 100	75.00	70.71	66.67

Dataset

Arithmetic Mean

Geometric Mean

Harmonic Mean

2, 8

5.00

4.00

3.20

4, 9

6.50

6.00

5.54

10, 20, 30

20.00

18.17

16.36

14, 22, 28, 36, 40

28.00

26.06

24.16

50, 100

75.00

70.71

66.67

Frequently Asked Questions

What is the Pythagorean Inequality of Means?

For any set of positive numbers, the Harmonic Mean is always less than or equal to the Geometric Mean, which is always less than or equal to the Arithmetic Mean (HM ≤ GM ≤ AM). Equality holds only if all numbers are identical.

Why can the Geometric Mean not accept negative numbers?

Taking fractional roots of negative products yields complex (imaginary) numbers or mathematical undefined errors in real arithmetic.

When should I use the Harmonic Mean instead of the Arithmetic Mean?

Use the harmonic mean whenever you are averaging rates, ratios, or speeds over fixed distances (e.g. miles per hour, price-to-earnings ratios).

Why do finance analysts use the Geometric Mean for stock returns?

Because investment returns compound multiplicatively over time. The geometric mean correctly reflects the true compound annual growth rate (CAGR).

What happens if a dataset contains zero?

The geometric mean becomes 0, and the harmonic mean becomes undefined due to division by zero (1/0). The arithmetic mean continues to work normally.