Geometric mean interval

The geometric mean interval is a multiplicative measure that is sensitive to interval ratios. This property makes it a cornerstone in measuring Order.

Mathematical Definition

The geometric mean interval can be calculated

from Intervals Chain

Let \(IC\) is Intervals Chain described as n-tuple

\[IC = <\Delta_1, \Delta_2, ..., \Delta_n> | \forall j \in \{1,...,n\} \exists \Delta_j \in \{1,...,n\}\]

\[\Delta_g=\sqrt[n]{\prod_{i=1}^{n} \Delta_{i}}\]

from Intervals Distribution

Let \(ID\) is Intervals Distribution described as function

\[ID : \{1,...,n\} \longrightarrow \{1,...,n\}\]

\[\Delta_g(ID)=\sqrt[l]{\prod_{i=1}^{n} i^{ID(i)}}, l = \sum_{i=1}^{n} ID(i)\]

Properties

Geometric mean is less or equals arithmetic mean

\[\Delta_g \le \Delta_a\]