In the context of a chi-square test, determining the values one anticipates under the assumption of no association between categorical variables is a crucial step. These anticipated frequencies, known as expected values, are derived from the marginal totals of the contingency table. For each cell within the table, the expected value is calculated by multiplying the row total by the column total, and then dividing the result by the grand total of all observations. For instance, if analyzing the relationship between gender and political affiliation, and the row total for females is 200, the column total for Democrats is 150, and the grand total is 500, the expected value for female Democrats would be (200 * 150) / 500 = 60.
The calculation of these values is fundamental to the chi-square test because it provides a baseline against which the observed frequencies are compared. This comparison quantifies the extent to which the observed data deviates from what would be expected if the variables were independent. Significant deviations suggest an association, prompting further investigation into the nature of that relationship. The concept of comparing observed and expected frequencies has been integral to statistical hypothesis testing since the development of the chi-square test by Karl Pearson in the early 20th century, providing a valuable tool across various fields including social sciences, healthcare, and market research.
