Executive Summary
Overall significance, effect size, and key pairwise findings
Kruskal-Wallis test on outcome across 6 groups (H = 26.506, df = 5, p = 1e-04): statistically significant. Effect size ε² = 0.0265 (small effect). 1000 observations retained after removing groups smaller than 5. 4 out of 15 pairwise comparisons are significant after bonferroni correction.
Group Descriptive Statistics
Median, IQR, and sample size per group for each outcome variable
| Group | Outcome | N | IQR | Outcome 2 Median | Outcome 3 Median |
|---|---|---|---|---|---|
| master's degree | 73 | 59 | 25.5 | 76 | 75 |
| bachelor's degree | 68 | 118 | 18 | 73 | 74 |
| some college | 67.5 | 226 | 17 | 70.5 | 70 |
| associate's degree | 67 | 222 | 23 | 72.5 | 70.5 |
| some high school | 65 | 179 | 21 | 67 | 66 |
| high school | 63 | 196 | 18.2 | 66 | 64 |
The group 'master's degree' has the highest median outcome at 73. Table shows median (middle value), IQR (spread), and n (sample size) per group. Higher medians indicate better typical performance for that group.
Score Distributions by Group
Box plots of outcome by group — shows spread, median, and outliers
The box plot reveals how outcome is distributed within each group. Group 'master's degree' has the highest median while 'high school' has the lowest, a range of 10 points. Overlapping boxes suggest groups are similar; non-overlapping boxes suggest real differences.
Kruskal-Wallis Test Results
H statistic, degrees of freedom, p-value, and epsilon-squared for each outcome
| Outcome VAR | Outcome | H Stat | Df | P Value | Eps Sq | N | Significant |
|---|---|---|---|---|---|---|---|
| outcome | 0.0265 | 26.51 | 5 | 0.0001 | 0.0265 | 1000 | Yes |
| outcome_2 | 0.0387 | 38.66 | 5 | 0 | 0.0387 | 1000 | Yes |
| outcome_3 | 0.0624 | 62.33 | 5 | 0 | 0.0624 | 1000 | Yes |
3 of 3 outcome(s) show significant group differences: outcome, outcome_2, outcome_3. H is the test statistic — larger values mean greater rank dispersion across groups. Epsilon-squared (ε²) measures practical effect size: 0.01 = small, 0.06 = medium, 0.14+ = large.
Median Scores by Education Level
Horizontal bar chart of median outcome per group, sorted low to high
Group 'master's degree' has the highest median outcome at 73 while 'high school' has the lowest at 63. The bar lengths make it easy to see the rank ordering of groups and estimate how large the between-group gap is in practical terms.
Pairwise Group Comparisons (Dunn Test)
Adjusted p-values heatmap — cells below 0.05 (bonferroni corrected) indicate significant pairs
4 of 15 group pairs are significantly different after bonferroni correction (α = 0.05). Darker cells indicate smaller (more significant) adjusted p-values. Pairs with p_adj < 0.05 suggest real distributional differences.
Significant Pairwise Differences
All 15 group pairs ranked by adjusted p-value
| Group | Outcome | Z Stat | P Raw | P Adj | Significant |
|---|---|---|---|---|---|
| bachelor's degree vs high school | 0.0014 | 3.909 | 0.0001 | 0.0014 | Yes |
| associate's degree vs high school | 0.0073 | 3.487 | 0.0005 | 0.0073 | Yes |
| high school vs master's degree | 0.0155 | -3.282 | 0.001 | 0.0155 | Yes |
| high school vs some college | 0.0209 | -3.196 | 0.0014 | 0.0209 | Yes |
| bachelor's degree vs some high school | 0.063 | 2.863 | 0.0042 | 0.063 | No |
| master's degree vs some high school | 0.2007 | 2.474 | 0.0134 | 0.2007 | No |
| associate's degree vs some high school | 0.3686 | 2.248 | 0.0246 | 0.3686 | No |
| some college vs some high school | 0.7522 | 1.959 | 0.0501 | 0.7522 | No |
| associate's degree vs bachelor's degree | 1 | -0.998 | 0.3185 | 1 | No |
| associate's degree vs master's degree | 1 | -0.993 | 0.3205 | 1 | No |
| associate's degree vs some college | 1 | 0.316 | 0.7521 | 1 | No |
| bachelor's degree vs master's degree | 1 | -0.2 | 0.8416 | 1 | No |
| bachelor's degree vs some college | 1 | 1.263 | 0.2065 | 1 | No |
| high school vs some high school | 1 | -1.122 | 0.262 | 1 | No |
| master's degree vs some college | 1 | 1.199 | 0.2304 | 1 | No |
4 of 15 pairwise comparisons are significant at α = 0.05 after bonferroni correction. The most significant pair is 'bachelor's degree vs high school' with p_adj = 0.0014. Pairs marked 'Yes' have distributions that differ beyond what chance alone would produce.