Definition
Pearson's chi-squared test is used to assess two types of comparison: tests of goodness of fit and tests of independence.
- A test of goodness of fit establishes whether or not an observed frequency distribution differs from a theoretical distribution.
- A test of independence assesses whether paired observations on two variables, expressed in a contingency table, are independent of each other (e.g. polling responses from people of different nationalities to see if one's nationality affects the response).
The first step is to calculate the chi-squared test statistic, X2, which resembles a normalized sum of squared deviations between observed and theoretical frequencies (see below). The second step is to determine the degrees of freedom, of that statistic, which is essentially the number of frequencies reduced by the number of parameters of the fitted distribution. In the third step, X2 is compared to the critical value of no significance from the distribution, which in many cases gives a good approximation of the distribution of X2. A test that does not rely on this approximation is Fisher's exact test; it is substantially more accurate in obtaining a significance level, especially with few observations.
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