# Readability Formulas for Elementary School Texts in Mexican Spanish
## 4 Experimental Study
### Experimental Results
The \(\mu\) index displays a more gradual  
trend, although with considerable overlap between distributions. The Crawford and SOL formulas show a modest upward trend. While Crawford formula indicates a range of age between 0 and 5 for most of the texts, the SOL formula provides a pretty realistic range between 3 and 11. Among the cross-linguistic formulas, the Gulpease formula displays a discernible downward trend but with greater score dispersion, while the Osman and Flesch formulas exhibit wide overlaps and irregular patterns, indicating lower alignment with the Spanish-language educational context. Figure 3: Familywise error rate by using [PERSON]’s HSD test accross elementary grade levels. Color intensity reflects the magnitude of the error rate. (0.179), closely followed by Linear-MX, indicating their superior alignment with grade-level expectations. Although these formulas do not yield the lowest MSE or RMSE values, their higher \(R^{2}\) suggests they better capture the overall trend and variability in the data, offering more meaningful predictions of readability across grade levels. By contrast, Flesch and Osman show extremely poor performance, with very high RMSE values (over 70 and 45, respectively) and large negative \(R^{2}\), suggesting that these models explain far less variance than a constant baseline. Traditional formulas such as Fernandez-Huerta and Szigriszt-Pazos yield the lowest RMSE and MSE values, but exhibit negative \(R^{2}\), indicating poor fit. Interestingly, Fernandez-Huerta has the second-lowest RMSE but a notably low \(R^{2}\), hinting at low variance in predictions rather than true alignment with target grades. Notice that the Crawford and SOL formulas were excluded from this comparison, since they compute the number of years of education necessary to understand a text instead of a readability level in the range from 0 to 100, so the MSE and RMSE would be biased. Figure 5 presents the values of the metric \(\Delta_{p}\), as defined in Equation (13), which measures the separability of readability levels based on the statistical significance of [PERSON]'s HSD post hoc comparisons across educational grades. The results offer several key insights into the performance of both traditional and proposed readability formulas. The Linear-MX model achieved the highest score (\(\Delta_{p}=-6.53\)), indicating superior performance in distinguishing between texts intended for different grade levels. This suggests the model is more effective than the rest of the formulas at capturing linguistic features aligned with educational progression in the corpus. The GP-MX model followed closely with an \(\Delta_{p}\) value of \(-8.12\), demonstrating that even a compact, symbolic representation derived through GP can yield strong differentiation across grade levels. Among the traditional Spanish-language formulas, the Fernandez-Huerta index performed the worst (\(\Delta_{p}=-49.92\)), followed by Szigriszt-Pazos (\(\Delta_{p}=-15.81\)) and Osman (\(\Delta_{p}=-15.92\)). These results indicate that such formulas may not align well with the specific characteristics of Mexican elementary-level texts. The Gutierrez-Polini (\(\Delta_{p}=-14.55\)) and SOL (\(\Delta_{p}=-10.57\)) formulas showed moderately better results but still fell short compared to the proposed models. Indices originally developed for other languages, such as Flesch, Gulpease, and Osman, also underperformed in this context, underscoring the limitations of applying traditional formulas across different linguistic and educational settings.