Generalized mean

Generalized_means_of_1,_x

MathematicalMeans

Generalized mean

The generalized mean, also known as the power mean or Hölder mean, is an abstraction of the more familiar arithmetic mean, geometric mean, and harmonic mean. It is a parameterized family of means that includes these and other means as special cases. The generalized mean is defined for a set of positive real numbers and a real parameter.

Definition

For a set of positive real numbers \( x_1, x_2, \ldots, x_n \) and a real number \( p \), the generalized mean \( M_p \) is defined as:

\[ M_p(x_1, x_2, \ldots, x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{\frac{1}{p}} \]

where \( p \) is a real number.

Special Cases

The generalized mean encompasses several well-known means as special cases:

• When \( p = 1 \), the generalized mean is the arithmetic mean:

\[ M_1(x_1, x_2, \ldots, x_n) = \frac{1}{n} \sum_{i=1}^n x_i \]

• When \( p = 0 \), the generalized mean is the geometric mean:

\[ M_0(x_1, x_2, \ldots, x_n) = \left( \prod_{i=1}^n x_i \right)^{\frac{1}{n}} \]

• When \( p = -1 \), the generalized mean is the harmonic mean:

\[ M_{-1}(x_1, x_2, \ldots, x_n) = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}} \]

Properties

• **Monotonicity**: If \( p < q \), then \( M_p(x_1, x_2, \ldots, x_n) \leq M_q(x_1, x_2, \ldots, x_n) \). This means that the generalized mean increases as the parameter \( p \) increases.
• **Continuity**: The generalized mean is continuous with respect to the parameter \( p \).
• **Homogeneity**: The generalized mean is homogeneous of degree 1, meaning that for any positive scalar \( \alpha \):

\[ M_p(\alpha x_1, \alpha x_2, \ldots, \alpha x_n) = \alpha M_p(x_1, x_2, \ldots, x_n) \]

Applications

The generalized mean is used in various fields such as statistics, economics, and engineering. It provides a flexible way to average data, allowing for different emphasis on the values depending on the choice of the parameter \( p \).

Related Pages

• Arithmetic mean
• Geometric mean
• Harmonic mean
• Mean
• Statistics

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