Purpose

This analysis tests whether math scores differ significantly across five student ethnic groups using the Kruskal-Wallis non-parametric test. The test is appropriate for comparing medians across multiple groups when data may not meet normality assumptions. Understanding whether meaningful score differences exist between groups is critical for identifying potential equity gaps in educational outcomes.

Key Findings

• H-Statistic (57.08): Substantially exceeds the critical value, indicating strong evidence against the null hypothesis of equal group medians
• P-Value (≈0): Highly significant result, confirming that observed differences are not due to random chance
• Effect Size (ε² = 0.057): Small practical magnitude despite statistical significance, meaning group membership explains only ~5.7% of score variance
• Significant Pairwise Comparisons: 5 of 10 group pairs show significant differences; Group E (median 74.5) differs markedly from Groups A–D (medians 61–69)
• Group Distribution: Unequal sample sizes (89–319 observations), with Group C largest; median scores range 13.5 points across groups

Interpretation

The analysis confirms statistically significant differences in math scores across ethnic groups. However, the small effect size indicates that while differences are real and reproducible, group membership alone is a

Purpose

This section documents the data cleaning and preparation phase for the Kruskal-Wallis statistical analysis comparing five groups. Perfect retention indicates no missing values, duplicates, or outliers were removed during preprocessing, which is critical for maintaining statistical power in the hypothesis test and ensuring the final sample accurately represents the underlying population.

Key Findings

• Retention Rate: 100% (1,000 of 1,000 rows retained) - No data loss occurred during cleaning, preserving the full analytical dataset
• Rows Removed: 0 - No filtering, deduplication, or outlier removal was applied
• Data Completeness: All 1,000 observations remained eligible for analysis across all five groups
• Train/Test Split: Not applicable - This is a descriptive/inferential analysis rather than a predictive modeling task

Interpretation

The complete retention of all observations strengthens the Kruskal-Wallis test results (H=57.08, p<0.001) by maximizing sample size and statistical power. With no rows excluded, the group distributions reflect the raw data without artificial filtering. However, the absence of any data cleaning raises questions about whether missing values, outliers, or data quality issues were genuinely absent or simply not addressed during preprocessing.

Context

The lack of train/test split is appropriate for this statistical inference task,

EXECUTIVE SUMMARY

Purpose

This analysis tested whether five distinct groups exhibit statistically different distributions across a measured outcome variable using the Kruskal-Wallis non-parametric test. The results confirm significant group-level differences, establishing that group membership is a meaningful predictor of performance or outcome variation across the 1,000 observations analyzed.

Key Findings

• H Statistic (57.079): Highly significant omnibus test result (p < 0.001) rejecting the null hypothesis of equal group distributions
• Effect Size (ε² = 0.057): Group membership explains only 5.7% of outcome variance—a small but statistically meaningful effect
• Significant Pairwise Comparisons: 5 of 10 group pairs show significant differences after Bonferroni correction, with Group E consistently differing from Groups A, B, C, and D (median 74.5 vs. 61–69)
• Group Distribution: Group E demonstrates the highest median (74.5) and mean (73.82), while Group A shows the lowest median (61)

Interpretation

The analysis confirms genuine group-level differences in outcome distributions, though the small effect size indicates these differences, while statistically robust, account for modest variance. Group E substantially outperforms other groups, particularly Groups A

Purpose

This section tests whether meaningful differences exist in the distribution of values across five groups using a non-parametric approach. The Kruskal-Wallis H-test is ideal for this dataset because it evaluates rank-based distributions without assuming normality, making it robust for the observed data structure with 1,000 observations across unequal group sizes.

Key Findings

• H Statistic (57.079): Substantially exceeds the critical value, indicating strong evidence that at least one group's distribution differs from the others
• P-Value (< 0.001): Highly significant result with negligible probability of observing this test statistic by chance alone
• Effect Size (ε² = 0.057): Small practical magnitude—group membership explains only 5.7% of rank variance, despite statistical significance
• Group Medians: Range from 61 (Group A) to 74.5 (Group E), with Group E showing notably higher central tendency

Interpretation

The test confirms statistically significant differences in distributions across the five groups. However, the small effect size indicates these differences, while real, account for minimal variance in the outcome. Post-hoc Dunn comparisons reveal that Group E drives most significance, showing substantial separation from Groups A, B, C, and D, whereas early groups cluster more closely together.

Purpose

This section visualizes the central tendency and spread of each group's distribution using medians and interquartile ranges (IQR). By displaying where the middle 50% of data falls for each group, it provides a non-parametric view of group differences that complements the Kruskal-Wallis test result and enables visual identification of potential pairwise differences before post-hoc testing.

Key Findings

• Median Range: 61 to 74.5 across groups, with Group E showing the highest median (74.5) and Group A the lowest (61)—a 13.5-point spread
• IQR Consistency: All groups maintain similar IQR widths (18–20.25), indicating comparable within-group variability despite different sample sizes
• Non-Overlapping Ranges: Group E's IQR (64.75–85) shows minimal overlap with Groups A–C, suggesting strong pairwise differences
• Sample Size Variation: Group C (n=319) and Group D (n=262) dominate the sample, while Group A (n=89) is smallest

Interpretation

The data reveals a clear upward trend in central tendency from Group A through Group E, with Group E substantially elevated. The Kruskal-Wallis test (H=57.08,

Purpose

This section visualizes the complete distributional characteristics across five groups to identify whether differences stem from location shifts (medians), spread variations, or shape asymmetries. The Kruskal-Wallis test detects all these distributional differences simultaneously, so examining overlaid distributions helps pinpoint the specific nature of group disparities beyond central tendency alone.

Key Findings

• Overall Distribution Shape: Nearly symmetric (skew=0.02) with mean=66.09 and median=66, indicating balanced central tendency across the full dataset
• Value Range: Spans 0–100 with standard deviation of 15.16, showing moderate variability
• Group Composition: Group C dominates (31.9%, n=319), while Group A is smallest (8.9%, n=89), affecting statistical power per group
• Significant K-W Result: H-statistic=57.08 (p<0.001) confirms meaningful distributional differences exist among the five groups despite overall symmetry

Interpretation

The significant Kruskal-Wallis test (p≈0) indicates that at least one group's distribution differs meaningfully from others. Given the overall dataset symmetry, these differences likely manifest as location shifts (higher/lower central values) rather than shape distortions. The median summary data shows Group E (median=74.

Purpose

Dunn's post-hoc test identifies which specific group pairs differ significantly, following the omnibus Kruskal-Wallis result (H=57.08, p<0.001). The Bonferroni correction controls false positives across all 10 pairwise comparisons by adjusting p-values, ensuring the 5% significance threshold applies to the entire test family rather than individual comparisons.

Key Findings

• Significant Pairs: 5 of 10 (50%) - Half of all pairwise comparisons show statistically significant differences after correction
• Adjustment Method: Bonferroni - Each raw p-value multiplied by 10, making significance harder to achieve but more trustworthy
• Pattern Observed: Group E shows the strongest differentiation (4 significant comparisons with groups A, B, C, D at p<0.001), while early groups (A, B, C) show minimal pairwise differences

Interpretation

The Kruskal-Wallis test confirmed overall group differences; Dunn's test reveals Group E is distinctly elevated compared to all others, while groups A–D form a relatively homogeneous cluster. The conservative Bonferroni correction means only robust differences survive, reducing false positives but potentially masking subtle effects. This pattern

Purpose

This section displays the rank distributions that form the foundation of the Kruskal-Wallis test. By converting raw values to ranks (1–1000), the test becomes robust to outliers and non-normal distributions. Groups with systematically higher ranks indicate higher underlying values, while similar rank distributions suggest no meaningful group differences.

Key Findings

• Rank Range: 1–997 across all 1,000 observations, with mean rank of 500.5 (perfectly centered, as expected)
• Group C Dominance: Largest sample (319 observations, 31.9%) provides stable rank representation
• Rank Spread: Standard deviation of 288.75 reflects full utilization of the rank scale across groups
• Symmetry: Skewness near 0 (0.05) indicates balanced rank distribution with no systematic bias toward high or low ranks

Interpretation

The rank data confirms that the Kruskal-Wallis test (H = 57.08, p < 0.001) compared how each group's observations cluster within the overall ranking. Group E shows systematically higher ranks (median 74.5 on original scale), while Groups A–C occupy lower rank positions. This rank separation directly drives the significant test result, validating that group differences are not due to chance.

Context

Ranks

Purpose

This section presents non-parametric descriptive statistics for each of the five groups, with emphasis on medians and interquartile ranges (IQR) rather than means and standard deviations. Since the Kruskal-Wallis test is rank-based, these robust measures are the appropriate summary statistics to interpret alongside the K-W results. Comparing medians across groups reveals the direction and magnitude of differences detected by the statistical test.

Key Findings

• Median Range: 61 to 74.5 across groups—Group E shows the highest median (74.5), while Group A shows the lowest (61), a 13.5-point spread
• IQR Consistency: Interquartile ranges are remarkably stable (18–20.25), indicating similar within-group variability despite median differences
• Sample Size Imbalance: Group C dominates with 319 observations (31.9%), while Group A has only 89 (8.9%), affecting statistical power
• Monotonic Trend: Medians increase progressively from Group A through Group E, suggesting an ordered relationship

Interpretation

The Kruskal-Wallis test (H=57.08, p<0.001) confirms statistically significant differences among groups. The median progression from Group A (61) to Group