The Theil index measures an entropic "distance" the population is away from the egalitarian state of everyone having the same income. The numerical result is in terms of negative entropy so that a higher number indicates more order that is further away from the complete equality. Formulating the index to represent negative entropy instead of entropy allows it to be a measure of inequality rather than equality. The Theil index can be transformed into an Atkinson index, which has a range between 0 and 1 (0% and 100%), where 0 indicates perfect equality and 1 (100%) indicates maximum inequality. (See Generalized entropy index for the transformation.) The Theil index is derived from Shannon's measure of information entropy formula_14, where entropy is a measure of randomness in a given set of information. In information theory, physics, and the Theil index, the general form of entropy is  When looking at the distribution of income in a population, formula_17 is equal to the ratio of a particular individual's income to the total income of the entire population. This gives the observed entropy formula_27 of a population to be:  The Theil index formula_36 measures how far the observed entropy (formula_27, which represents how randomly income is distributed) is from the highest possible entropy (formula_38, which represents income being maximally distributed amongst individuals in the population– a distribution analogous to the [most likely] outcome of an infinite number of random coin tosses: an equal distribution of heads and tails). Therefore, the Theil index is the difference between the theoretical maximum entropy (which would be reached if the incomes of every individual were equal) minus the observed entropy:  When formula_34 is in units of population/species, formula_27 is a measure of biodiversity and is called the Shannon index. If the Theil index is used with x=population/species, it is a measure of inequality of population among a set of species, or "bio-isolation" as opposed to "wealth isolation". The Theil index measures what is called redundancy in information theory. It is the left over "information space" that was not utilized to convey information, which reduces the effectiveness of the price signal. The Theil index is a measure of the redundancy of income (or other measure of wealth) in some individuals. Redundancy in some individuals implies scarcity in others. A high Theil index indicates the total income is not distributed evenly among individuals in the same way an uncompressed text file does not have a similar number of byte locations assigned to the available unique byte characters.