Chi-square

Abstract

The chi-square (χ2) is a non-parametric procedure used to test hypotheses about the independence of frequency counts in various categories. For example, the data may be the proportions of smokers preferring each one of three brands of low-tar cigarettes or the proportion of people favouring a repeal of the abortion laws. Categories of responses are set up, such as Brands A, B, and C of cigarettes or Approve—Disapprove of a certain issue, and the number of individuals or events falling into each category is recorded. In such a situation one can obtain nothing more than the frequency, or number of times, that a particular category is chosen, which constitutes nominal data. With such data the only analysis possible is to determine whether the frequencies observed in the sample differ significantly from hypothesized frequencies. There are many social science variables which involve nominal data for which chi-square is a simple and appropriate means of analysis, e.g. social class levels, illness categories, age groups, sex, voting preferences, pass — fail dichotomies etc. The symbol χ is the Greek letter chi which is pronounced to rhyme with ‘sky’. The distribution of chi-square is based on chance as with ‘t’ and the various standard errors we have dealt with. Imagine you have a container holding a lot of tombola tickets of which half are even numbered, and half are odd numbered. After mixing them up well a given number of tickets are drawn out by a blindfolded person. You would expect half of the tickets to be even numbered and half to be odd numbered. This exact expectation would rarely be met; the obtained frequencies of odd and even numbers would not match the expected frequencies. Over a large number of similar draws this congruence might be achieved. This activity and principle should remind you of earlier activities with standard error and sampling (Chapter 7). So the χ2 test provides us with the probability that any obtained frequency arose by chance (null hypothesis).