The Chi-Square Test Statistic

The chi-square test statistic  is an overall measure of how close the observed frequencies are to the expected frequencies. It has the form

$$\chi^2= \mbox{SUM} \frac{(\mbox{observed frequency - expected frequency})^2} {\mbox{expected frequency}}.$$

The null hypothesis of independence is rejected if $\chi^2$ is large, because this means that observed frequencies and expected frequencies are far apart. The chi-square curve is used to judge whether the calculated test statistic is large enough. We reject H₀ if the test statistic is large enough so that the area beyond it (under the chi-square curve with (r-1)(c-1) degrees of freedom) is less than .05.

The P-value is the area greater than $\chi^2$ under the chi-square curve with (r-1)(c-1) degrees of freedom.