Overview & Data Preparation

Chi-Square Test Configuration and Data Quality

OV

Analysis Overview

Chi-Square Test Configuration

Analysis overview and configuration

Chi Square Test
Educational Research Institute
Analyze survey responses to identify factors affecting test performance
Module Configuration
significance_level 0.05
min_group_size 5
categorical_vars gender, race/ethnicity, parental level of education, lunch, test preparation course
target_vars math score, reading score, writing score
primary_target math score
Processing ID
test_1773382899
analysis type: Chi Square Test
company: Educational Research Institute
objective: Analyze survey responses to identify factors affecting test performance
IN

Key Insights

Analysis Overview

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.

IN

Key Insights

Analysis Overview

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.

PP

Data Preprocessing

Data Quality & Cleaning

1,000
Final Observations

Data preprocessing and column mapping

Data Pipeline
1,000
Initial Records
1,000
Clean Records
Column Mapping
student_gender
→
gender
race_ethnicity
→
race/ethnicity
parental_education
→
parental level of education
lunch_program
→
lunch
test_prep
→
test preparation course
math_score
→
math score
reading_score
→
reading score
writing_score
→
writing score
1,000 Records
MCP Analytics
1,000 Records
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Key Insights

Data Preprocessing

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

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Key Insights

Data Preprocessing

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

Executive Summary

Key Findings and Recommendations

TLDR

Executive Summary

Key Findings & Recommendations

0
Significant Pairs

Key Performance Indicators

N significant
0
Max cramers v
9.5%
Primary p value
6%

Key Findings

Key findings

finding value
Total variable pairs tested 10
Significant associations found 0 of 10 pairs
Strongest association gender x race/ethnicity
Cramers V (effect size) 0.095
Effect magnitude negligible
Chi-square statistic 9.027
Significance level used 0.05

Executive Summary

Bottom Line: Analyzed 10 variable pairs across 1000 observations. 0 significant associations found (Benjamini-Hochberg FDR correction applied).

Key Findings:
• Strongest association: gender x race/ethnicity is not statistically significant (p = 0.0604) with Cramers V = 0.095 (negligible effect)
• 0 of 10 pairs are significant at alpha = 0.05 after multiple comparison correction
• Maximum effect size observed: Cramers V = 0.098

Recommendation: No statistically significant associations detected between categorical variables. The variables appear to be approximately independent. Consider whether sample size is adequate to detect small effects.

Executive Summary
Statistical Test
Effect Size
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Key Insights

Executive Summary

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

IN

Key Insights

Executive Summary

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

Chi-Square Test Results

All Pairwise Tests with FDR Correction

CHI

Chi-Square Test Results

All Variable Pair Tests with FDR Correction

10
Pairs Tested

Chi-square test of independence results for all variable pairs

variable_pair chi_square df_val p_value p_adjusted cramers_v effect_size significant
student gender x race ethnicity 9.027 4.000 0.060 0.297 0.095 Negligible No
race ethnicity x parental education 29.459 20.000 0.079 0.297 0.086 Small No
parental education x test prep 9.544 5.000 0.089 0.297 0.098 Negligible No
race ethnicity x test prep 5.488 4.000 0.241 0.602 0.074 Negligible No
race ethnicity x lunch program 3.442 4.000 0.487 0.801 0.059 Negligible No
student gender x lunch program 0.457 1.000 0.499 0.801 0.021 Negligible No
lunch program x test prep 0.291 1.000 0.590 0.801 0.017 Negligible No
student gender x parental education 3.385 5.000 0.641 0.801 0.058 Negligible No
student gender x test prep 0.036 1.000 0.849 0.943 0.006 Negligible No
parental education x lunch program 1.111 5.000 0.953 0.953 0.033 Negligible No
Chi-Square
Pearson
FDR Correction
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Key Insights

Chi-Square Test Results

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

IN

Key Insights

Chi-Square Test Results

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

Effect Sizes

Cramér's V for All Variable Pairs

EF

Effect Sizes

Cramér's V for All Variable Pairs

Cramers V effect sizes for all tested variable pairs

Cramér's V
Effect Size
Association Strength
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Key Insights

Effect Sizes

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

IN

Key Insights

Effect Sizes

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

Contingency Table Heatmap

Observed Frequencies for Primary Variable Pair

CT

Contingency Table Heatmap

Observed Counts for Primary Variable Pair

Observed frequency distribution for the most significant variable pair

Contingency Table
Observed Counts
Frequency
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Key Insights

Contingency Table Heatmap

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

IN

Key Insights

Contingency Table Heatmap

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

Standardized Residuals

Which Cells Drive the Association

RES

Standardized Residuals

Cells Driving the Chi-Square Association

Standardized residuals showing which cells drive the chi-square association

Residuals
Association Drivers
Diagnostic
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Key Insights

Standardized Residuals

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

IN

Key Insights

Standardized Residuals

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

Group Distribution

Proportional Distribution Across Groups

GB

Group Distribution

Proportional Distribution Across Groups

Proportional distribution of one variable across levels of the other

Proportions
Grouped Bar
Distribution
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Key Insights

Group Distribution

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

IN

Key Insights

Group Distribution

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

Observed vs Expected

Cell-Level Chi-Square Breakdown

DET

Observed vs Expected

Cell-Level Chi-Square Breakdown

10
Cells Analyzed

Observed vs expected counts with standardized residuals for primary variable pair

row_category col_category observed expected std_residual
female group A 36.000 46.100 -2.245
female group B 104.000 98.400 0.900
female group C 180.000 165.200 2.004
female group D 129.000 135.700 -0.967
female group E 69.000 72.500 -0.642
male group A 53.000 42.900 2.245
male group B 86.000 91.600 -0.900
male group C 139.000 153.800 -2.004
male group D 133.000 126.300 0.967
male group E 71.000 67.500 0.642
Observed
Expected
Residuals
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Key Insights

Observed vs Expected

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

IN

Key Insights

Observed vs Expected

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

Job ID: test_analytics__statistical__groups__chi_square_test_20260312_232816
Secret Hash: 36f2d40ab80e