Pythagorean Means — Definition, Formula & Examples

The Pythagorean means are three classical types of average — the arithmetic mean, geometric mean, and harmonic mean — studied together because of a elegant inequality that always relates them.

For positive real numbers $a_1, a_2, \ldots, a_n$ , the Pythagorean means are the arithmetic mean $A$ , geometric mean $G$ , and harmonic mean $H$ , which satisfy the inequality $A \geq G \geq H$ , with equality holding if and only if all values are equal.

Key Formula

A = \frac{a_1 + a_2 + \cdots + a_n}{n}, \quad G = \sqrt[n]{a_1 \cdot a_2 \cdots a_n}, \quad H = \frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}}

Where:

$a_1, a_2, \ldots, a_n$ = The positive real numbers being averaged
$n$ = The count of values
$A$ = Arithmetic mean
$G$ = Geometric mean
$H$ = Harmonic mean

How It Works

Each Pythagorean mean summarizes a set of positive numbers in a different way. The arithmetic mean (AM) adds the values and divides by the count. The geometric mean (GM) multiplies the values and takes the

n

th root. The harmonic mean (HM) takes the reciprocal of the arithmetic mean of the reciprocals. For any set of positive numbers that are not all identical, the AM is always the largest and the HM is always the smallest, with the GM falling in between.

Worked Example

Problem: Find the arithmetic, geometric, and harmonic means of 4 and 16, and verify the AM–GM–HM inequality.

Arithmetic Mean: Add the values and divide by 2.

A = \frac{4 + 16}{2} = 10

Geometric Mean: Multiply the values and take the square root.

G = \sqrt{4 \times 16} = \sqrt{64} = 8

Harmonic Mean: Take the reciprocal of the average of the reciprocals.

H = \frac{2}{\frac{1}{4} + \frac{1}{16}} = \frac{2}{\frac{5}{16}} = \frac{32}{5} = 6.4

Answer:

A = 10

G = 8

H = 6.4

, confirming

10 \geq 8 \geq 6.4

Visualization

Why It Matters

The AM–GM inequality is a fundamental tool in competition mathematics and optimization proofs. The harmonic mean appears in physics (e.g., average speed for equal-distance trips) and in computing (e.g., the F1 score in machine learning). Understanding all three means together gives you flexibility in choosing the right average for a given context.

Common Mistakes

Mistake: Assuming the three means can appear in any order for positive numbers.

Correction: The inequality

A \geq G \geq H

always holds for positive values. The arithmetic mean is never smaller than the geometric mean, which is never smaller than the harmonic mean.