Purpose

This chi-square test analysis examines associations between categorical variables (gender, race/ethnicity, parental education, lunch program, test preparation) and test performance outcomes. The analysis tests whether these demographic and socioeconomic factors show statistically significant relationships with student performance, directly supporting the institute's objective to identify performance drivers.

Key Findings

• Number of Variable Pairs Tested: 10 pairs evaluated for independence
• Significant Associations Found: 0 out of 10 pairs (0% significance rate)
• Primary P-Value: 0.06 (marginally above 0.05 threshold)
• Maximum Cramér's V: 0.098 (negligible effect size across all pairs)
• Data Quality: 100% retention with no low expected cell frequencies; all assumptions met

Interpretation

Despite testing 10 variable combinations, no statistically significant associations emerged between demographic factors and test performance at the conventional 0.05 significance level. The closest relationship (student gender × race/ethnicity, p=0.06) remains non-significant. Effect sizes are uniformly negligible (Cramér's V ≤ 0.10), indicating that even where p-values approach significance, practical associations are minimal. This suggests demographic characteristics alone do not meaningfully predict test performance variation in this sample.

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Purpose

This section documents the data cleaning and preparation phase for the chi-square independence test analyzing factors affecting test performance. Perfect retention indicates no rows were excluded during preprocessing, meaning the full dataset of 1,000 survey responses remained available for statistical analysis. This is critical for maintaining statistical power and ensuring the test results reflect the complete sample.

Key Findings

• Retention Rate: 100% (1,000 of 1,000 rows retained) - No observations were removed during data cleaning, preserving the full analytical sample
• Rows Removed: 0 - No missing values or data quality issues necessitated exclusion from the categorical variables analyzed
• Data Integrity: Complete dataset available for all 10 variable pairs tested in the chi-square analysis

Interpretation

The perfect retention rate indicates robust data quality in the survey responses. Since the chi-square test requires complete cases for contingency table construction, maintaining all 1,000 observations strengthens the reliability of the statistical findings. The absence of missing data in categorical columns (gender, race/ethnicity, parental education, lunch program, test prep) means no information loss occurred that could bias association estimates or reduce statistical power.

Context

The analysis note mentions "Missing values in categorical columns" as a removal reason, yet zero rows were actually removed—suggesting either no missing values existed or they were handled through imputation rather than deletion

Purpose

This chi-square analysis examined 10 variable pairs from 1,000 survey responses to identify factors affecting test performance. The objective was to detect statistically significant associations between categorical variables (gender, race/ethnicity, parental education, lunch program, and test preparation) that could inform educational interventions.

Key Findings

• Primary P-Value: 0.0604 – Falls just above the 0.05 significance threshold; the strongest association (gender × race/ethnicity) narrowly misses conventional statistical significance
• Maximum Cramér's V: 0.095 – Indicates negligible effect size even for the strongest relationship detected
• Significant Associations Found: 0 of 10 pairs – After multiple comparison correction (Benjamini-Hochberg FDR), no associations remain statistically significant
• Data Quality: 100% retention with no low expected cell frequencies; all assumptions met

Interpretation

The analysis reveals that the surveyed categorical variables are approximately independent of one another. Despite testing 10 variable pairs, no statistically significant associations emerged that would suggest demographic or program factors meaningfully predict test performance groupings. The near-significant gender × race/ethnicity relationship (p = 0.0604) carries negligible practical effect, suggesting minimal real-world differentiation.

Context

This null finding does not confirm

Purpose

This section evaluates whether demographic and academic factors show statistically significant associations with test performance outcomes. By testing 10 variable pairs, the analysis identifies which survey-measured characteristics (gender, race/ethnicity, parental education, lunch program, test prep) are meaningfully related to student performance, directly addressing the research objective to understand factors affecting test outcomes.

Key Findings

• Primary p-value: 0.06 (gender × race/ethnicity) — marginally above the 0.05 significance threshold, indicating no statistically significant association
• Cramér's V (max): 0.098 — negligible effect size across all tested pairs, suggesting weak practical relationships even where associations exist
• Significant pairs after correction: 0 of 10 — multiple testing adjustment (Benjamini-Hochberg FDR) eliminated all associations, confirming no robust relationships survive conservative statistical control
• Pattern observed: Uniform non-significance across all variable combinations, with effect sizes consistently negligible

Interpretation

The analysis found no statistically significant associations between the measured demographic/academic variables and test performance. The strongest candidate (gender × race/ethnicity, p = 0.06) falls just outside conventional significance thresholds and exhibits negligible effect size. This suggests that within this sample, these categorical factors do not independently predict test performance variation in a

Purpose

This section quantifies the strength of association between each tested variable pair using Cramér's V, a standardized effect size metric. It directly addresses whether factors affecting test performance show meaningful relationships—moving beyond statistical significance to practical magnitude. Understanding effect sizes is critical for identifying which demographic and educational factors have the strongest real-world associations with student outcomes.

Key Findings

• Maximum Cramér's V: 0.10 (parental education × test prep and student gender × race/ethnicity)—both classified as Negligible despite approaching the small effect threshold
• Mean Effect Size: 0.05 across all 10 variable pairs, indicating uniformly weak associations
• Effect Size Distribution: 90% of pairs show negligible associations; only 1 pair (race/ethnicity × parental education, V=0.09) approaches small effect classification
• No Statistical Significance: Zero of 10 pairs achieved statistical significance at conventional thresholds, despite p-values ranging from 0.06 to 0.95

Interpretation

The analysis reveals that demographic and educational factors tested show remarkably weak associations with test performance groupings. Even the strongest relationships (Cramér's V ≈ 0.10) fall below the small effect threshold, suggesting these variables explain minimal variance in the outcome. This pattern indicates that test performance variation is not

Purpose

This contingency table heatmap visualizes the observed frequency distribution across gender and five categorical groups, revealing how respondents are distributed across these demographic combinations. It serves as the foundation for the chi-square test of independence, allowing visual identification of patterns that may indicate association between gender and group membership in the context of analyzing factors affecting test performance.

Key Findings

• Highest Concentration: Female respondents in Group C (180 observations, 18% of total) represent the largest single cell, suggesting uneven distribution across categories
• Gender Distribution: Females comprise 518 observations (51.8%) and males 482 (48.2%), indicating near-parity overall
• Group C Dominance: Both genders show elevated representation in Group C (180 females, 139 males), accounting for 31.9% of the sample
• Lowest Frequency: Female respondents in Group A (36 observations, 3.6%) represent the sparsest cell

Interpretation

The chi-square statistic of 9.027 with p = 0.0604 indicates the observed distribution is marginally close to statistical significance but does not meet the conventional 0.05 threshold. The standardized residuals (ranging from -2.24 to 2.24) show modest deviations from expected frequencies, particularly in Groups A and C

Purpose

Standardized residuals pinpoint which specific category combinations deviate most from statistical independence. This section identifies the cells driving the chi-square statistic, revealing where observed frequencies differ meaningfully from expected values. Understanding these deviations is critical for interpreting whether the overall test's marginal significance (p=0.06) stems from concentrated patterns or dispersed differences.

Key Findings

• Maximum Absolute Residual: 2.24 (female × group A and male × group A) - These cells exceed the |2| threshold, indicating statistically significant deviations from independence
• Symmetric Pattern: Residuals show perfect symmetry across gender categories (e.g., female group A = -2.24, male group A = +2.24), suggesting gender-driven disparities in group distribution
• Concentration: Four of ten cells exceed |2|, with the strongest deviations in groups A and C, indicating non-random gender representation in these categories

Interpretation

The residuals reveal that females are significantly underrepresented in group A (36 observed vs. 46.1 expected) while overrepresented in group C (180 observed vs. 165.2 expected). Males show the inverse pattern. This concentrated deviation in specific groups explains why the overall chi-square test approaches significance despite negligible effect size (Cramér's

Purpose

This grouped bar chart visualizes how five demographic or performance categories (Groups A–E) are distributed differently across gender (female vs. male). It serves as a visual complement to the chi-square test, allowing you to see whether the proportional composition differs between groups. If the variables were truly independent, bar heights would be identical across genders; visible differences suggest potential association.

Key Findings

• Group C Concentration: Females show the highest proportion in Group C (34.7%), while males are more evenly distributed (28.8%), indicating a 5.9 percentage-point difference
• Group A Underrepresentation: Females comprise only 6.9% in Group A versus 11% for males—the largest relative disparity
• Overall Distribution Pattern: Female respondents skew toward Groups C and D (59.6% combined), while males show more balanced spread across all five categories
• Sample Size Consistency: Counts range from 36–180 observations per cell, providing adequate statistical power

Interpretation

These proportional differences align with the chi-square test result (p=0.06, Cramér's V=0.10), which approached but did not reach statistical significance at α=0.05. The visual pattern suggests gender and category membership are weakly associated rather than independent, though the relationship is marginal. The concentration of females

Purpose

This section examines cell-level deviations between observed and expected frequencies in the gender × race/ethnicity contingency table. Standardized residuals identify which category combinations occur significantly more or less often than independence would predict, revealing patterns of association that drive the overall chi-square test result.

Key Findings

• Standardized Residuals Range: -2.24 to +2.24 - Four cells exceed the ±2 threshold, indicating statistically significant departures from independence at p < 0.05
• Female × Group A: Observed=36, Expected=46.1, Residual=-2.24 - Females underrepresented in this category
• Male × Group A: Observed=53, Expected=42.9, Residual=+2.24 - Males overrepresented (mirror pattern)
• Female × Group C: Observed=180, Expected=165.2, Residual=+2.0 - Females overrepresented
• Male × Group C: Observed=139, Expected=153.8, Residual=-2.0 - Males underrepresented (inverse pattern)

Interpretation

Despite the overall chi-square test yielding p=0.06 (non-significant at α=0.05), cell-level analysis reveals