Analysis overview and configuration
| Parameter | Value | _row |
|---|---|---|
| significance_level | 0.05 | significance_level |
| min_group_size | 5 | min_group_size |
| categorical_vars | gender, race/ethnicity, parental level of education, lunch, test preparation course | categorical_vars |
| target_vars | math score, reading score, writing score | target_vars |
| primary_target | math score | primary_target |
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.
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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Data preprocessing and column mapping
| Metric | Value |
|---|---|
| Initial Rows | 1,000 |
| Final Rows | 1,000 |
| Rows Removed | 0 |
| Retention Rate | 100% |
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.
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.
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
| 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 |
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.
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.
This null finding does not confirm
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 | 0.0604 | 0.297 | 0.095 | Negligible | No |
| race ethnicity x parental education | 29.46 | 20 | 0.0791 | 0.297 | 0.0858 | Small | No |
| parental education x test prep | 9.544 | 5 | 0.0892 | 0.297 | 0.0977 | Negligible | No |
| race ethnicity x test prep | 5.488 | 4 | 0.241 | 0.602 | 0.0741 | Negligible | No |
| race ethnicity x lunch program | 3.442 | 4 | 0.487 | 0.801 | 0.0587 | Negligible | No |
| student gender x lunch program | 0.457 | 1 | 0.499 | 0.801 | 0.0214 | Negligible | No |
| lunch program x test prep | 0.291 | 1 | 0.59 | 0.801 | 0.017 | Negligible | No |
| student gender x parental education | 3.385 | 5 | 0.641 | 0.801 | 0.0582 | Negligible | No |
| student gender x test prep | 0.036 | 1 | 0.849 | 0.943 | 0.006 | Negligible | No |
| parental education x lunch program | 1.111 | 5 | 0.953 | 0.953 | 0.0333 | Negligible | No |
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.
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
Cramers V effect sizes for all tested variable pairs
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.
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
Observed frequency distribution for the most significant variable pair
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.
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 showing which cells drive the chi-square association
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.
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
Proportional distribution of one variable across levels of the other
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.
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 counts with standardized residuals for primary variable pair
| row_category | col_category | observed | expected | std_residual |
|---|---|---|---|---|
| female | group A | 36 | 46.1 | -2.245 |
| female | group B | 104 | 98.4 | 0.9 |
| female | group C | 180 | 165.2 | 2.004 |
| female | group D | 129 | 135.7 | -0.967 |
| female | group E | 69 | 72.5 | -0.642 |
| male | group A | 53 | 42.9 | 2.245 |
| male | group B | 86 | 91.6 | -0.9 |
| male | group C | 139 | 153.8 | -2.004 |
| male | group D | 133 | 126.3 | 0.967 |
| male | group E | 71 | 67.5 | 0.642 |
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.
Despite the overall chi-square test yielding p=0.06 (non-significant at α=0.05), cell-level analysis reveals