That the arithmetic mean of a set of number is larger than the geometric mean is a foundational result in the theory of inequalities. This is a useful result for index numbers in particular as it gives that, for the same data, an index based on an arithmetic mean will always be larger than the corresponding index based on the geometric mean.

Comparing housing prices over time is difficult because of all the heterogeneity in housing that can both affect prices and vary over time. The typical solution to this problem is to use a hedonic price index that attempts to model the relationship between housing characteristics and price, and uses this to control the change in these characteristics over time.

What causes inflation in prices over time? Increases in the money supply, of course. Although there are many reasons for increasing prices over time, money supply is one of the few things that can do so and grow without bound. Case closed, right? Not quite. In practice we can’t perfectly observe how prices change over time and instead we must measure inflation with an index number.

A diversity index is a way to measure the prevalence species in an ecosystem. Although these arise naturally in ecology, diversity indexes show up elsewhere as well.