# (Social) inequality measure based on chi-square

Is there any social inequality measure for categorical or ordinal variables, based on a goodness of fit chi squared test (distance from an uniform distribution)?

Example: part of my data is unemployment rates by 4 age groups. I know that inequality is minimum if the rates are equal for the 4 age groups (uniform distribution), for example [0.04, 0.04, 0.04, 0.04].

N.B. The sum of these rates is $\neq$ 1.

• Can you explain your question more fully? What type of data do you have? What are you measuring? What is your research question? – Joel W. Aug 20 '18 at 23:53

## 1 Answer

For a possibly useful general discussion of 'Diversity indexes' you might want to look at Wikipedia, perhaps starting with the Simpson index $$\lambda = \sum_{i=1}^R p_i^2,$$ where $R$ is the number of different 'types' of individuals and $p_i$ are their respective proportions in the population.

For a large population, roughly speaking, $\lambda$ can be viewed as the probability that two individuals randomly chosen from the population are of the same 'type'. Thus, smaller $\lambda$ corresponds to greater diversity.

• Please see my edits. I am not sure if with the Simpson index $\sum_{i=1}^R{p_i}$ has to be $=1$. Thank you for the link to 'Diversity indexes'. – sbac Aug 21 '18 at 15:32
• In your unemployment example: conditionally on being unemployed, age groups are equally likely [.25,.25,.25,.25] and you could use the standard chi-squared test statistic for adherence to that proper probability distribution (with probabilities summing to 1). – BruceET Aug 21 '18 at 16:38
• And the Simpson index also works in this case? Even if the unemployment rates are for example [0.04, 0.08, 0.16, 0.32]? In this case should I force (compute) the sum of probabilities to be 1 as in [0.066, 0.133, 0.267, 0.533]? – sbac Aug 21 '18 at 17:20
• For the Simpson index, I think it would be $p_1 = p_2 =p_3=p_4 = 0.25,$ so that $\sum_{i=1}^4 p_i = 1.$ – BruceET Aug 21 '18 at 17:50
• You want the proportion of all unemployed subjects who happen to be members of each age group. 'Condition on unemployment` means $P(G_i|\text{Unemployed}).$ Not $P(\text{Unemployed}|G_i).$ Not $P(\text{Unemployed}\cap G_i).$ – BruceET Aug 21 '18 at 22:20