Giorgi Kvizhinadze and Haizhen Wu

Victoria University of Wellington

Diversity analysis in multiple choice questionnaires

Consider a multiple choice questionnaire with q questions, and n individuals are asked to fill out this questionnaire so we obtain n "opinions". Our questions are: how many different opinions will we observe? What is the proportion of unique opinions? What is the proportion of opinions we will see twice or any k times?

One would reasonably assume that the answers to our questions depend on various properties of the questionnaire structure, such as the number of questions, the number of possible answers in each question, the probabilities of the answers, the number of interviewees and so on. However, we discovered that the asymptotic behaviour of the quantities we are interested in follows the Karlin-Rouault law, which depends on only one parameter that incorporates all information about the properties of the questionnaire structure.