Power Mean Calculator

Input a set of numbers and a power value to quickly compute the Power Mean.

Calculate the Power Mean

Result

How to Calculate the Power Mean

The Power Mean is a type of average that incorporates a power value to weight data points. The formula is: \( M_p = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{\frac{1}{p}} \) Where:

\( M_p \) is the Power Mean with power \( p \).
\( x_i \) represents each value in the dataset.
\( n \) is the total number of values in the dataset.
\( p \) is the specified power.

Special cases of the Power Mean include:

When \( p = 1 \), it becomes the Arithmetic Mean.
When \( p = 0 \), it becomes the Geometric Mean (all values must be positive).
When \( p = -1 \), it becomes the Harmonic Mean.

Steps to Compute the Power Mean

Raise Each Value to the Power \( p \): Compute \( x_i^p \) for each value \( x_i \).
Sum and Divide by the Total Count: Add all the powered values and divide by the total number of values \( n \).
Take the \( \frac{1}{p} \) Root: Raise the result to the power \( \frac{1}{p} \) to get the Power Mean \( M_p \).

Examples

Example 1: Calculate the Power Mean of [2, 4, 6, 8] with p = 2

Solution:

1. Raise each value to \( p = 2 \):

\( 2^2 = 4 \)

\( 4^2 = 16 \)

\( 6^2 = 36 \)

\( 8^2 = 64 \)

2. Sum and divide by the total count:

\( \frac{4 + 16 + 36 + 64}{4} = \frac{120}{4} = 30 \)

3. Compute the square root (\( \frac{1}{2} \)-th power):

\( M_2 = 30^{\frac{1}{2}} = \sqrt{30} \approx 5.48 \)

Result: The Power Mean is approximately 5.48.

Example 2: Calculate the Power Mean of [3, 5, 7] with p = -1

Solution:

1. Raise each value to \( p = -1 \) (find the reciprocal):

\( 3^{-1} = \frac{1}{3} \)

\( 5^{-1} = \frac{1}{5} \)

\( 7^{-1} = \frac{1}{7} \)

2. Sum and divide by the total count:

\( \frac{\frac{1}{3} + \frac{1}{5} + \frac{1}{7}}{3} = \frac{0.67619}{3} \approx 0.2254 \)

3. Compute the reciprocal (\( -1 \)-th power):

\( M_{-1} = 0.2254^{-1} \approx 4.4366 \)

Result: The Power Mean is approximately 4.4366.