Expected Count vs Chi-Square Test

Expected counts are the tallies a chi-square model predicts for each cell when the null hypothesis holds, and the test measures how far the observed counts drift from them. For a two-way table, each cell's expected count is $E = \frac{(\text{row total}) \times (\text{column total})}{\text{grand total}}$. For example, in a 200-person table a cell whose row totals 80 and column totals 50 has $E = \frac{80 \times 50}{200} = 20$. Every expected count should be at least 5 for the chi-square approximation to be trustworthy.

A chi-square test works with counts in categories rather than means, checking goodness of fit, independence, or homogeneity. It adds up the squared gaps between observed and expected counts, each scaled by the expected count, into one statistic. The formula is $\chi^2 = \sum \frac{(O - E)^2}{E}$, where $O$ is an observed count and $E$ an expected count. For example, rolling a die 60 times and comparing each face's tally to the expected 10 tests whether the die is fair, and a large $\chi^2$ gives a small p-value.