1

Suppose we distributed $100$ coins to $10$ persons and the $i$-th person got ${x}_{i}$ coins, how to judge the distribution $X=\{{x}_{1}, {x}_{2}, ..., {x}_{n}\}$ (e.g., $X=\{5, 20, 15, 5, 10, 10, 10, 15, 5, 5\}$) is (almost) balanced or not? Is there a mathematical definition or empirical criterion of the unbalancedness?

Lijie Xu
  • 123

2 Answers2

0

There are many different ways you could measure unbalancedness/inequality. The first one to mind is the entropy, but you could alternatively use any of the many ways economic inequality is measured (like the Gini coefficient).

Andy Jones
  • 2,236
0

There were also two measures developed by Ahrens/Pincus (1981) for the unbalancedness of (panel) data sets, which would also work in your case.

If a data set is balanced, the measures equal one and the more unbalanced the data, the lower the measures (but >0).

However, use of that measure seems not very widespread. Most often, it is found in econometric context (if at all). An implementation is e. g. in the current development version latest CRAN versions of the package plm for panel data for the R environment (punbalancedness is the function).

Reference is:

Ahrens/Pincus, On Two Measures of Unbalancedness in a One-Way Model and Their Relation to Efficiency, Biometrical Journal, Volume 23, Issue 3, pages 227–235, 1981.

Helix123
  • 1,467