Analysis Overview & Setup

Purpose

This section visualizes how test scores are distributed across the student population, revealing whether performance follows normal patterns or exhibits skewness that might indicate floor/ceiling effects. Understanding score distributions is essential for identifying whether assessment difficulty is appropriately calibrated and whether demographic disparities (explored in earlier sections) reflect genuine performance gaps or measurement artifacts.

Key Findings

• Math Score Mean: 66.09 (SD: 15.16) – The primary target shows moderate central tendency with substantial variability, suggesting diverse performance levels
• Total Score Mean: 203.31 (SD: 42.77) – Composite performance across three subjects averages near two-thirds of the maximum possible range
• Distribution Shape: Positive skew (0.66) indicates a right-tailed distribution, with more students clustering below the mean and a tail extending toward higher scores
• Score Range: Values span from near-zero to 100+ (math) and up to 329 (total), confirming full utilization of the scoring scale without obvious ceiling effects

Interpretation

The moderate skew and spread suggest reasonably normal assessment performance without severe floor or ceiling constraints. The math score mean of 66.09 aligns with the overall pass rate of 89.7% (threshold: 50), indicating most students exceed minimum competency. However, the right skew combined with 28.

Purpose

This section quantifies the strength of relationships between the three test score domains. Understanding score correlations reveals whether academic performance is domain-specific or reflects a unified underlying ability. All correlations are statistically significant (p < 0.001), indicating these relationships are robust and not due to chance.

Key Findings

• Reading-Writing Correlation (r = 0.955): Exceptionally strong relationship; students' reading and writing performance are nearly interchangeable, suggesting these skills are highly interdependent or measure similar cognitive abilities.
• Math-Reading Correlation (r = 0.818): Strong positive relationship; students excelling in math tend to perform well in reading, though the relationship is notably weaker than reading-writing.
• Math-Writing Correlation (r = 0.803): Slightly weaker than math-reading; math performance shows the most independence from writing skills among the three pairs.

Interpretation

The correlation hierarchy reveals that reading and writing form a tightly integrated skill cluster, while math operates somewhat independently. This pattern suggests students may have distinct mathematical aptitude separate from language-based competencies. However, all correlations exceed 0.80, confirming that strong overall academic ability manifests across all three domains. The near-perfect reading-writing correlation (0.955) indicates these subjects could be treated as a single construct in predict

Purpose

This section establishes the baseline composition of the dataset by documenting how respondents distribute across five key demographic and program variables. Understanding these categorical distributions is essential for interpreting subsequent statistical tests and identifying whether certain groups are over- or under-represented, which affects the generalizability of findings across the analysis.

Key Findings

• Gender Distribution: Nearly balanced with 51.8% female (n=518) and 48.2% male (n=482), providing adequate representation for gender-based comparisons
• Lunch Program Participation: Heavily skewed toward standard lunch (64.5%, n=645) versus free/reduced (35.5%, n=355), indicating a majority-minority split
• Test Preparation Course: Only 35.8% (n=358) completed test prep, while 64.2% (n=642) did not, showing low adoption of preparation resources
• Parental Education: Distributed across six levels with associate's degree (22.2%) and high school (19.6%) as most common, reflecting diverse educational backgrounds
• Race/Ethnicity: Concentrated in group C (31.9%) with smaller representation in group A (8.9%), showing unequal demographic composition

Interpretation

The categorical landscape reveals a dataset with balanced gender representation but substantial disparities in socioeconomic indicators (lunch program

Purpose

This section evaluates whether categorical demographic and program variables are statistically independent of each other. Understanding these relationships is essential for identifying confounding factors and determining whether observed performance differences across groups stem from demographic characteristics or program participation patterns.

Key Findings

• Total Chi-Square Tests: 10 pairs tested; 0 showed statistical significance (p < 0.05)
• Strongest Association: Parental education × test preparation course (Cramer's V = 0.098) — still a weak association
• Independence Pattern: All categorical variables demonstrate statistical independence, with p-values ranging from 0.06 to 0.95

Interpretation

The absence of significant associations indicates that demographic characteristics (gender, race/ethnicity, parental education) and program participation (lunch type, test prep completion) operate independently within this population. This independence is analytically valuable because it suggests that performance differences observed across demographic groups are unlikely to be confounded by unequal distribution of test preparation or lunch program participation. The weak Cramer's V values (all ≤ 0.10) confirm minimal practical association strength.

Context

These findings assume adequate cell sizes and random sampling. The independence of categorical variables strengthens the validity of subsequent ANOVA analyses examining performance differences across demographic groups, as group membership is not systematically linked to program exposure.

Purpose

This section compares performance across three academic subjects (math, reading, writing) within each demographic group to identify whether achievement gaps are consistent or subject-specific. Understanding these patterns reveals whether certain groups face universal barriers or experience advantages/disadvantages in particular subjects, informing targeted intervention strategies.

Key Findings

• Gender Gap Reversal: Males score 5.1 points higher in math (68.73 vs 63.63), but females score 7.14 points higher in reading and 9.16 points higher in writing, indicating fundamentally different performance profiles by subject
• Test Preparation Impact: Completion shows consistent gains across all three subjects, with the largest effect on writing (9.91-point gap), suggesting preparation benefits compound across domains
• Mean Score Range: Across all demographic groups and subjects, scores range from 58.92 to 75.68 (mean=67.97), with relatively low variance (SD=4.09), indicating moderate consistency in performance patterns
• Parental Education Influence: This variable shows the most granular breakdown (18 observations), suggesting education level creates differentiated subject performance profiles

Interpretation

The data reveals that demographic disparities are not uniform across subjects. Gender shows the most dramatic subject-specific variation, with males excelling in quantitative reasoning but females demonstrating stronger literacy skills. Test preparation's consistent positive

Purpose

This section identifies which demographic and program groups achieve higher or lower math scores, revealing performance disparities across the student population. Understanding these group differences is essential for identifying where targeted interventions may be needed and which factors most strongly influence academic outcomes.

Key Findings

• Lunch Program Effect: Standard lunch students score 70.03 vs. 58.92 for free/reduced (11.11-point gap)—the largest observed difference with non-overlapping confidence intervals
• Test Preparation Impact: Completed test prep yields 69.7 vs. 64.08 for no prep (5.62-point gap), indicating meaningful preparation benefits
• Gender Gap: Males average 68.73 vs. females at 63.63 (5.10-point difference), statistically significant
• Race/Ethnicity Range: Scores span 61.63–64.46 across groups, with narrower variation than lunch or test prep factors
• Overall Spread: Mean scores range from 58.92 to 73.82 across all groups, representing substantial performance variation

Interpretation

The data reveals that socioeconomic status (lunch program) and test preparation are the strongest differentiators of math performance, with effect sizes substantially larger than demographic factors. The 11-point lunch program gap suggests resource disparities significantly impact achievement.

Purpose

This section identifies where categorical variables show the strongest associations in the dataset. The mosaic plot visualizes the relationship between gender and race/ethnicity, with residuals indicating which demographic combinations occur more or less frequently than statistical independence would predict. Understanding these patterns helps identify whether demographic groups are distributed evenly across the population or show meaningful clustering.

Key Findings

• Strongest Association: Parental education × test preparation course (Cramér's V = 0.098) - This represents the weakest association among all categorical pairs tested, indicating minimal dependency between these variables
• Gender × Race/Ethnicity Residuals: Range from -1.49 to +1.54, all below the ±2 threshold for notable deviation - No cell combinations show statistically meaningful departures from expected frequencies
• Pattern Observed: All 10 chi-square tests yielded non-significant results (p > 0.05), suggesting categorical variables operate largely independently of one another

Interpretation

Despite testing 10 categorical variable pairs, none demonstrate meaningful statistical associations. The residuals clustering near zero indicate that observed frequencies closely match expected values under independence. This suggests demographic characteristics (gender, race/ethnicity, parental education, lunch program status, test preparation) are distributed relatively uniformly across the sample without strong interdependencies.

Context

These weak associations contrast sharply with the strong

Purpose

This section evaluates student performance distribution across letter grades (A–F) and identifies which demographic groups concentrate in top versus bottom performance tiers. Understanding grade distribution reveals whether performance gaps observed in earlier analyses translate into meaningful disparities in final outcomes, directly addressing how socioeconomic and demographic factors influence academic achievement.

Key Findings

• Pass Rate (≥50): 89.7% - The vast majority of students meet the minimum threshold, indicating broad baseline competency across the cohort
• Grade A Concentration: Only 52 students (5.2%) achieve top-tier performance, revealing a narrow elite group despite high pass rates
• Grade F Prevalence: 285 students (28.5%) receive failing grades, the single largest grade category, indicating substantial performance inequality
• Grade Distribution Skew: The distribution is heavily right-skewed with 54.2% of students earning C or below, while only 19.8% earn B or above

Interpretation

The high pass rate masks a deeply stratified performance landscape. While 89.7% of students technically pass, the concentration of F grades (28.5%) and scarcity of A grades (5.2%) demonstrate that most passing students cluster in the C–D range. This pattern aligns with earlier ANOVA findings showing significant group effects by lunch program (η²=0.123