Assessing Skew Normality in Marks Distribution, a Comparative Analysis of Shapiro Wilk Tests

TL;DR

This work addresses the problem that student marks distributions often deviate from normality due to skewness, making standard normality tests potentially misleading. It introduces a skew-normal–aware modification of the Shapiro–Wilk test that uses a data transformation $Y_i = U_i \cdot |X_i - \hat{\xi}|$, with parameters $\xi,\omega,\lambda$ estimated by maximum penalized likelihood, to assess skew-normality without bootstrapping. Empirical results on two course datasets show that while the classical Shapiro–Wilk test rejects normality, the modified test indicates a skew-normal distribution, a conclusion supported by visualizations of transformed data. The study demonstrates the practical value of skew-normal modeling in educational data analysis and provides a framework for more robust statistical assessment of student performance metrics, with broad applicability to other skewed domains.

Abstract

This paper investigates the distribution of marks obtained by students across multiple courses to explore whether the data conforms to a skew-normal distribution. Traditional methods for assessing normality, such as the Shapiro Wilk test, often reject normality in datasets with evident skewness. To address this, we apply a modified Shapiro Wilk test tailored for skew-normal distributions, as described in the literature, to evaluate the suitability of skew-normal models for these datasets. The analysis includes both classical and modified tests, complemented by visualizations such as histograms and Q-Q plots of transformed data. Our findings highlight the relevance of using specialized statistical methods for skew normality, offering valuable insights into the characteristics of academic performance data. This study provides a framework for robust statistical analysis in educational research, emphasizing the need to account for distributional properties when analyzing student performance metrics.

Figures (4)

• Figure 1: Histogram of Total Marks
• Figure 2: Histogram of Modified Total Marks
• Figure 3: Histogram of Midterm Marks
• Figure 4: Histogram of Modified Midterm Marks