Contraharmonic mean

An average that gives more weight to larger values

The contraharmonic mean is a type of average that is particularly useful in certain mathematical and statistical contexts. It is defined for a set of non-negative numbers and is calculated as the sum of the squares of the numbers divided by the sum of the numbers themselves. This mean gives more weight to larger values in the dataset.

Definition

For a set of non-negative real numbers \( x_1, x_2, \ldots, x_n \), the contraharmonic mean \( C \) is given by:

\[ C = \frac{\sum_{i=1}^{n} x_i^2}{\sum_{i=1}^{n} x_i} \]

This formula implies that the contraharmonic mean is always greater than or equal to the arithmetic mean, and it is particularly sensitive to larger values in the dataset.

Properties

• The contraharmonic mean is undefined if all the values in the dataset are zero, as this would result in division by zero.
• It is always greater than or equal to the arithmetic mean, and less than or equal to the maximum value in the dataset.
• The contraharmonic mean is a special case of the power mean with exponent 2.

Applications

The contraharmonic mean is used in various fields such as image processing, where it can be used to enhance images by emphasizing larger pixel values. It is also used in the calculation of certain types of averages in statistics and probability.

Example

Consider the set of numbers \( \{1, 2, 3\} \). The contraharmonic mean is calculated as follows:

\[ C = \frac{1^2 + 2^2 + 3^2}{1 + 2 + 3} = \frac{1 + 4 + 9}{6} = \frac{14}{6} = \frac{7}{3} \approx 2.33 \]

Also see

• Arithmetic mean
• Harmonic mean
• Geometric mean
• Power mean
• Weighted mean