Power Mean — Definition, Formula & Examples

Power mean is a family of averages that includes the arithmetic mean, geometric mean, and harmonic mean as special cases. By changing a single parameter

p

, you smoothly transition between different types of means.

For positive real numbers $x_1, x_2, \ldots, x_n$ and a nonzero real number $p$ , the power mean of order $p$ (also called the generalized mean) is $M_p = \left(\frac{1}{n}\sum_{i=1}^{n} x_i^{\,p}\right)^{1/p}$ . The case $p = 0$ is defined as the limit $M_0 = \left(\prod_{i=1}^{n} x_i\right)^{1/n}$ , which equals the geometric mean.

Key Formula

M_p = \left(\frac{1}{n}\sum_{i=1}^{n} x_i^{\,p}\right)^{1/p}

Where:

$M_p$ = Power mean of order p
$p$ = Real-valued parameter that selects the type of mean (e.g., 1 = arithmetic, −1 = harmonic)
$x_i$ = The ith positive data value
$n$ = Number of data values

How It Works

Choose a value of

p

to select which type of average you want. Setting

p = 1

gives the ordinary arithmetic mean. Setting

p = -1

gives the harmonic mean. The limit as

p \to 0

yields the geometric mean, while

p \to \infty

approaches the maximum value and

p \to -\infty

approaches the minimum. A key property is that

M_p

is non-decreasing in

p

: if

p < q

, then

M_p \leq M_q

. This single inequality encapsulates the classical AM–GM–HM inequality.

Worked Example

Problem: Compute the power mean of order 2 (the quadratic mean) for the values 1, 2, and 3.

Raise each value to the power p = 2: Square each data value.

1^2 = 1,\quad 2^2 = 4,\quad 3^2 = 9

Take the arithmetic mean of the powers: Sum the squared values and divide by n = 3.

\frac{1 + 4 + 9}{3} = \frac{14}{3}

Take the (1/p)-th root: Raise the result to the power 1/2 (i.e., take the square root).

M_2 = \sqrt{\frac{14}{3}} \approx 2.160

Answer: The power mean of order 2 is

\sqrt{14/3} \approx 2.160

, which is slightly larger than the arithmetic mean of 2.

Why It Matters

The power mean appears in signal processing (root-mean-square voltage uses

p = 2

), inequality proofs in analysis, and optimization. In statistics, understanding that different means respond differently to extreme values helps you choose the right summary measure for skewed data.

Common Mistakes

Mistake: Using the power mean formula with p = 0 by plugging in zero directly, which causes division by zero in the exponent.

Correction: The case p = 0 is defined as the geometric mean via a limit:

M_0 = (x_1 \cdot x_2 \cdots x_n)^{1/n}

. You must use this separate formula, not substitute p = 0 into the general expression.