Geometric mean extension for data sets with zeros

TL;DR

This paper reviews existing methods for computing the geometric mean with zero values, identifies their limitations, and proposes a new modified geometric mean that handles zeros while satisfying recovery and monotonicity conditions.

Contribution

The paper introduces a novel modified geometric mean that effectively incorporates zeros and adheres to key mathematical properties, improving data summarization.

Findings

Existing solutions fail to meet recovery and monotonicity conditions.

The proposed modified geometric mean satisfies both conditions.

The new method provides a more accurate summary for datasets with zeros.

Abstract

There are numerous examples in different research fields where the use of the geometric mean is more appropriate than the arithmetic mean. However, the geometric mean has a serious limitation in comparison with the arithmetic mean. Means are used to summarize the information in a large set of values in a single number; yet, the geometric mean of a data set with at least one zero is always zero. As a result, the geometric mean does not capture any information about the non-zero values. The purpose of this short contribution is to review solutions proposed in the literature that enable the computation of the geometric mean of data sets containing zeros and to show that they do not fulfil the `recovery' or `monotonicity' conditions that we define. The standard geometric mean should be recovered from the modified geometric mean if the data set does not contain any zeros (recovery condition). Also, if the values of an ordered data set are greater one by one than the values of another data set then the modified geometric mean of the first data set must be greater than the modified geometric mean of the second data set (monotonicity condition). We then formulate a modified version of the geometric mean that can handle zeros while satisfying both desired conditions.