Quasi-arithmetic mean

Quasi-arithmetic mean, also known as the generalized mean or power mean, is a mathematical concept that generalizes the Pythagorean means including the arithmetic mean, geometric mean, and harmonic mean. It is a powerful tool in both pure and applied mathematics, particularly useful in the fields of statistics, economics, and various branches of engineering. The quasi-arithmetic mean is defined for a set of numbers and a continuous and invertible function, providing a flexible framework for aggregating numbers.

Definition

Given a set of numbers \(x_1, x_2, ..., x_n\) and a continuous, invertible function \(f\), the quasi-arithmetic mean \(M_f\) of these numbers is defined as:

\[M_f(x_1, x_2, ..., x_n) = f^{-1}\left(\frac{f(x_1) + f(x_2) + ... + f(x_n)}{n}\right)\]

where \(f^{-1}\) is the inverse function of \(f\).

Special Cases

The quasi-arithmetic mean encompasses several well-known means as special cases, which occur for specific choices of the function \(f\):

• For \(f(x) = x\), \(M_f\) is the Arithmetic mean.
• For \(f(x) = \ln(x)\), \(M_f\) is the Geometric mean.
• For \(f(x) = \frac{1}{x}\), \(M_f\) is the Harmonic mean.

Properties

The quasi-arithmetic mean has several important properties:

• It is symmetric, meaning the order of the numbers does not affect the mean.
• It is homogeneous, meaning that scaling all numbers by a positive factor scales the mean by the same factor.
• Under certain conditions on the function \(f\), it satisfies the intermediate value theorem, implying that the mean of any set of numbers lies between the smallest and largest numbers in the set.

Applications

Quasi-arithmetic means find applications in various fields:

• In Statistics, they are used to define generalized measures of central tendency.
• In Economics, they can model preferences and utility functions.
• In Engineering, they are used in signal processing and control theory to aggregate data points or signals.

See Also

• Mean
• Central tendency
• Inequality of arithmetic and geometric means