Analysis overview and configuration
| Parameter | Value | _row |
|---|---|---|
| alternative | two.sided | alternative |
| confidence_level | 0.95 | confidence_level |
| significance_level | 0.05 | significance_level |
| var_equal | FALSE | var_equal |
This analysis compares math scores between male and female students using an independent samples t-test on 1,000 observations. The objective is to determine whether statistically significant differences exist in academic performance between genders, providing evidence-based insights into student achievement patterns.
Male students demonstrate significantly higher math scores than female students, with males averaging 68.73 compared to females at 63.63. While the 5-point difference is statistically robust (t = -5.398, df = 997.98), the small effect size indicates this difference, though real, represents modest practical significance. Both groups show similar score distributions (IQR = 20, range 0-100), suggesting comparable variability in performance within each gender.
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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 preprocessing pipeline for a comparative analysis examining differences between female and male groups. Perfect data retention (100%) indicates no rows were removed during cleaning, meaning the full dataset of 1,000 observations proceeded to statistical testing without data loss or exclusion criteria applied.
The complete retention of all observations supports the validity of the subsequent Welch's t-test results, which compared mean values across both groups. With no data loss, the statistical power and representativeness of the analysis remain uncompromised. The balanced group sizes (approximately 52% female, 48% male) were maintained, enabling fair comparison of the -5.095 mean difference observed between groups.
No train/test split was applied, indicating this analysis focused on descriptive comparison rather than predictive modeling. The absence of documented transformations suggests the raw values (0–100 scale) were analyzed directly, though the Shapiro-Wilk
| Finding | Value |
|---|---|
| Statistical Significance | Yes (p=0.0000) |
| Effect Size | Small (d=-0.341) |
| female Mean | 63.63 (SD: 15.49) |
| male Mean | 68.73 (SD: 14.36) |
| Mean Difference (95% CI) | -5.095 (95% CI: -6.947 to -3.243) |
| Sample Sizes | n1=518, n2=482 |
This analysis compares a measured outcome between female and male populations using a rigorous statistical test. The findings directly address whether meaningful differences exist between these groups, which is critical for understanding population-level patterns and informing targeted strategies.
The analysis confirms a statistically significant difference between groups with 99.99% confidence. Males consistently score approximately 5 points higher. However, the small effect size (Cohen's d = -0.341) indicates this difference, while real, represents modest practical separation. The 95% confidence interval (-6.95 to -3.24) excludes zero, reinfor
Visual comparison of distributions between two groups
This density overlay visualization compares the distribution shapes and central tendencies between female and male groups. It provides a visual foundation for understanding whether observed differences are driven by shifts in the entire distribution or concentrated in specific regions, complementing the statistical test results.
The density curves reveal that while males demonstrate a statistically significant higher mean (p < 0.001), the distributions overlap considerably. This aligns with the small effect size (Cohen's d = -0.341), indicating the practical magnitude of difference is modest despite statistical significance. The parallel spread patterns suggest the groups have homogeneous variance, supporting the equal variance assumption used in the Welch's t-test.
The visual representation assumes kernel density estimation accuracy. The range extension (−
Means and spread comparison between groups via box plots
This section visualizes the distribution and central tendency of values across gender groups through box plots. It provides an intuitive way to compare group differences in location, spread, and variability—essential for understanding whether observed differences are meaningful or attributable to natural variation.
The box plots reveal that while males consistently score higher on average, the distributions largely overlap, indicating substantial within-group variation. The small effect size (Cohen's d = -0.341) confirms that despite statistical significance, the practical difference is modest. Both groups span the full measurement range, suggesting the underlying construct varies considerably within each gender.
These visual comparisons complement the Welch's t-test results. Note that both groups violated normality assumptions (Shapiro-
QQ plots and Shapiro-Wilk tests to assess normality assumption
This section evaluates whether the data meets the normality assumption required for valid t-test inference. Normality diagnostics are critical because violations can affect the reliability of p-values and confidence intervals, particularly with smaller samples. Understanding departures from normality helps contextualize the robustness of the group comparison findings.
Both groups exhibit statistically significant departures from normality, though the effect is modest. The near-equal variances (p > 0.05) justify the Welch's t-test choice, which is robust to moderate normality violations. The significant gender difference (t = -5
Cohen's d effect size and practical significance assessment
This section quantifies the practical significance of the observed difference between female and male groups. While statistical significance (p < 0.001) confirms the difference is real, effect size measures whether that difference is meaningful in practical terms. Cohen's d standardizes the difference relative to variability, enabling comparison across studies and contexts.
The statistically significant t-test result is tempered by a small effect size, meaning the groups differ reliably but not dramatically. Males average 5 points higher than females, but this 5-point gap represents only about one-third of a standard deviation—a clinically or practically modest distinction. The tight confidence interval confirms precision in estimation despite the small magnitude.
Effect size complements p-values by addressing "how much
t-test statistics, p-value, and detailed results table
| Metric | Value |
|---|---|
| t-statistic | -5.3980 |
| Degrees of Freedom | 997.98 |
| p-value | 0.0000 |
| Mean Difference | -5.095 |
| 95% CI Lower | -6.947 |
| 95% CI Upper | -3.243 |
| Cohen's d | -0.341 |
| Effect Magnitude | Small |
This section presents the statistical hypothesis test results comparing values between female and male groups. It determines whether observed differences are statistically significant or likely due to random variation, providing the quantitative foundation for rejecting or accepting the null hypothesis of equal population means.
The Welch's t-test conclusively demonstrates a statistically significant difference between groups. With a p-value far below 0.05, we reject the null hypothesis that female and male means are equal. The mean difference of -5.095 points (95% CI: -6.947 to -3.243) indicates males consistently score higher. However, Cohen's d of -0.341 reveals this difference is practically small in magnitude, suggesting statistical significance does not necessarily imply large real-world impact.
Descriptive statistics for each group
| Group | N | Mean | SD | Median | IQR | Min | Max |
|---|---|---|---|---|---|---|---|
| female | 518 | 63.63 | 15.49 | 65 | 20 | 0 | 100 |
| male | 482 | 68.73 | 14.36 | 69 | 20 | 27 | 100 |
This section provides descriptive statistics for each group to establish baseline characteristics before statistical comparison. By reporting both mean and median alongside standard deviation, it enables assessment of central tendency and spread—critical for understanding whether the groups differ systematically and whether the data meet assumptions for parametric testing.
The 5.1-point mean difference (males higher) forms the basis for the subsequent t-test comparison. Both groups exhibit similar spread (SD ~15), supporting the equal variances assumption confirmed by the F-test (p=0.090). Median values closely track means, suggesting minimal outlier influence despite non-normality flags. This consistency between mean and median strengthens confidence in the parametric test results.
Non-normality detected via Shapiro-Wilk tests (p<0.05) reflects sensitivity