Expected Frequencies

Statistics · Axiom Academy

Understanding the theoretical foundation of chi-square tests 1. What Expected Frequencies Represent Expected frequencies are the theoretical counts we would anticipate in each category if the null hypothesis (H₀) were true . For example, if we're testing whether a die is fair, the expected frequency for each face would be n/6 where n is the total number of rolls. For contingency tables (cross-tabulations), we calculate expected frequencies using row and column totals: This formula ensures that the expected frequencies preserve both the row and column marginal totals. 3. Why We Compare Observed to Expected The chi-square test statistic measures how far our observed frequencies deviate from the expected frequencies: Large deviations between observed (O) and expected (E) values produce a large χ² statistic, providing evidence against the null hypothesis. Small χ²: Observed data closely matches expectations under H₀ Large χ²: Observed data differs substantially from expectations under H₀ 4. When Expected Frequencies Are Too Small The chi-square distribution is only a good approximation when expected frequencies are sufficiently large. Small expected frequencies make the chi-square approximation unreliable The test may give inaccurate p-values Type I error rates may be inflated Combine categories (if theoretically justified) Use Fisher's exact test (for 2×2 tables) Collect more data to increase sample size

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