Overview
This lecture covers how to calculate and interpret a chi-square statistic for testing associations between categorical variables using observed and expected counts.
Calculating the Chi-Square Statistic
- Subtract the expected count from the observed count for each group.
- Square the result, then divide by the expected count: (Observed - Expected)² / Expected.
- Example for "liked music, low retention": (14 - 11.5)² / 11.5.
- Example for "disliked music, low retention": (15 - 12.48)² / 12.48.
- Example for "no music group": (7 - 12)² / 12.
- Add all calculated values to get the total chi-square test statistic.
- In this example, the chi-square statistic is 6.012.
Interpreting the Test Statistic
- Use degrees of freedom (df) to determine the distribution to use for comparing the statistic.
- Use a theoretical distribution calculator or software (like StatKey or Statcato) to find the critical value.
- Compare your chi-square statistic to the critical value to decide if it falls in the rejection region (tail) based on your significance level.
Key Terms & Definitions
- Observed Count — the actual number recorded in each category.
- Expected Count — the number that would be expected in each category based on the null hypothesis.
- Chi-Square Statistic — sum of (Observed - Expected)² / Expected across all categories; measures the difference between observed and expected values.
- Degrees of Freedom (df) — parameter needed to determine the appropriate chi-square distribution.
- Significance Level — cutoff probability for rejecting the null hypothesis.
Action Items / Next Steps
- Practice calculating chi-square statistics by matching observed and expected counts.
- Use a statistical calculator (StatKey, Statcato) to find critical values and make decisions about hypothesis tests.