Overview
This lecture explains how to calculate and interpret expected counts in contingency tables, particularly in the context of a chi-square test for independence using an experiment on music and memory retention.
Experimental Design and Observed Counts
- Students attempted to memorize information under different music conditions: liked music, disliked music, or no music.
- Confounding variables (like music volume) were controlled, and students were randomly assigned to groups.
- Students were classified as having "high retention" or "low retention" based on their memory performance.
- The observed counts for each group were provided (e.g., 10, 14, 11, 15, 18, 7).
Calculating Expected Counts
- If the null hypothesis (music does not affect retention) is true, retention rates should be similar across all groups.
- Calculate overall high retention rate: 39/75 = 0.52 (52%); low retention: 36/75 = 0.48 (48%).
- Expected count for each cell = percentage (from the whole group) × group size (e.g., 0.52 × 24 = 12.48 for high retention, liked music group).
- Repeat this for each combination of group and retention type.
The Expected Count Formula
- Alternative formula: (row total × column total) ÷ grand total.
- This formula accounts for different group sizes and matches the previous method when computed.
- Used in most statistical software for computing expected counts in contingency tables.
Computing the Chi-Square Statistic
- For each cell: (observed count – expected count)² ÷ expected count.
- Sum these values for all cells to get the chi-square test statistic.
- Ensure observed and expected counts are matched correctly for each cell.
Table Structure and Degrees of Freedom
- The example uses a 2×3 contingency table (two retention levels, three music groups, not counting totals).
- Degrees of freedom = (number of rows – 1) × (number of columns – 1) = 2–1 × 3–1 = 2.
Key Terms & Definitions
- Observed Count — The actual number of cases counted in each category of the contingency table.
- Expected Count — The number of cases expected in each category if the null hypothesis is true.
- Null Hypothesis — The assumption that variables (e.g., music and retention) are independent.
- Chi-Square Statistic — A sum of squared differences between observed and expected counts, divided by expected count.
- Degrees of Freedom — (Rows – 1) × (Columns – 1), used in significance testing.
Action Items / Next Steps
- Practice calculating expected counts and chi-square statistics using sample tables.
- Review how to determine degrees of freedom for contingency tables.
- Prepare for problems applying these concepts to different experimental designs.