Analysis overview and configuration

Configuration

Analysis TypeKruskal Wallis

CompanyEducational Research Institute

ObjectiveTest whether math scores differ significantly across student ethnic groups using Kruskal-Wallis non-parametric test

Analysis Date2026-03-15

Processing Idtest_1773614312

Total Observations1000

Module Parameters

Parameter	Value	_row
significance_level	0.05	significance_level
min_group_size	5	min_group_size

Kruskal Wallis analysis for Educational Research Institute

Interpretation

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

Data preprocessing and column mapping

Data Quality

Initial Rows1000

Final Rows1000

Rows Removed0

Retention Rate100

Data Quality

Metric	Value
Initial Rows	1,000
Final Rows	1,000
Rows Removed	0
Retention Rate	100%

Processed 1,000 observations, retained 1,000 (100.0%) after cleaning

Interpretation

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,

Key Metrics

Overall Result: Significant Group Differences Found
H Statistic: H(4) = 57.079
P-Value: p < 0.001
Effect Size (ε²): 0.0571 — Small
Sig. Pairs: 5 of 10 pairs significant (Dunn+Bonferroni)

Key Findings

Finding	Value
K-W Test Result	Significant Group Differences Found (p < 0.001)
Effect Size	eps2=0.0571 (Small effect)
Post-Hoc Pairs	5/10 significant (Dunn, Bonferroni)
Groups Analyzed	5
Total N	1000

Summary

Bottom Line: Group rank distributions differ significantly — H(4) = 57.079, p < 0.001.

Effect Size: Epsilon-squared (ε²) = 0.0571 (Small effect — group membership explains 5.7% of rank variance)

Post-Hoc (Dunn + Bonferroni): 5 of 10 pairwise comparisons were significant.

Recommendation: Review the Dunn post-hoc table to identify which specific group pairs differ, and focus interventions on groups with the lowest median outcomes.

Interpretation

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

Kruskal-Wallis H-test results with H statistic, degrees of freedom, p-value, and effect size

statistic	degrees_freedom	p_value	effect_size	effect_magnitude	significant	n_total	n_groups
57.08	4	0	0.0571	Small	True	1000	5

Interpretation

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.

Group medians with interquartile range (IQR) error bars for visual comparison

Interpretation

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,

Overlaid distributions per group with density curves to visualize shape differences

Interpretation

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.

Dunn post-hoc pairwise comparisons with Bonferroni-adjusted p-values and significance indicators

comparison	group1	group2	z_stat	p_value	p_adj	significant	sig_label
group A - group B	group A	group B	-1.299	0.194	1	False	ns
group A - group C	group A	group C	-1.829	0.0674	0.6741	False	ns
group B - group C	group B	group C	-0.5719	0.5674	1	False	ns
group A - group D	group A	group D	-3.473	5.00e-04	0.0051	True	**
group B - group D	group B	group D	-2.721	0.0065	0.065	False	ns
group C - group D	group C	group D	-2.482	0.0131	0.1308	False	ns
group A - group E	group A	group E	-6.174	0	0	True	***
group B - group E	group B	group E	-6.017	0	0	True	***
group C - group E	group C	group E	-6.094	0	0	True	***
group D - group E	group D	group E	-3.925	1.00e-04	9.00e-04	True	***

Interpretation

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

Rank distributions per group — shows the actual ranks used in the Kruskal-Wallis test

Interpretation

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

Descriptive statistics per group — median, IQR, mean, SD, min, max

group	n	median_val	q1	q3	iqr	mean_val	sd_val	min_val	max_val
group A	89	61	51	71	20	61.63	14.52	28	100
group B	190	63	54	74	20	63.45	15.47	8	97
group C	319	65	55	74	19	64.46	14.85	0	98
group D	262	69	59	77	18	67.36	13.77	26	100
group E	140	74.5	64.75	85	20.25	73.82	15.53	30	100

Interpretation

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

Analysis Overview

Configuration

Module Parameters

Interpretation

Purpose

Key Findings

Interpretation

Data Preprocessing

Data Quality

Data Quality

Interpretation

Purpose

Key Findings

Interpretation

Context