## Violin Plot: Elementary Math ### Overview The image is a violin plot comparing the accuracy of different models on elementary math problems. The models include a baseline with no Large Language Model (LLM), an LLM alone, and LLMs with confidence measures using random selection (Rand), query-based selection (Query), and a context-tracking method (CT). The plot shows the distribution of accuracy scores for each model. A red dashed line is present at y=0.3. ### Components/Axes * **Title:** Elementary Math * **Y-axis:** Accuracy, ranging from 0.2 to 1.0 in increments of 0.2. * **X-axis:** Categorical labels representing different models: * No LLM (Blue) * LLM (Orange) * LLM + Conf (Rand) (Green) * LLM + Conf (Query) (Red) * LLM + Conf (CT) (Purple) * **Horizontal dashed lines:** Present within each violin plot at approximately y=0.75, y=0.6, and y=0.3. * **Horizontal dashed red line:** Present at y=0.3. ### Detailed Analysis * **No LLM (Blue):** The violin plot is broad, indicating a wide range of accuracy scores. The distribution appears to be skewed towards higher accuracy, with the bulk of the data above 0.6. The minimum accuracy is approximately 0.15, and the maximum is approximately 1.0. * **LLM (Orange):** The violin plot is narrower than "No LLM," suggesting a smaller range of accuracy scores. The distribution is centered around 0.7, with a minimum accuracy of approximately 0.27 and a maximum of approximately 0.95. * **LLM + Conf (Rand) (Green):** The violin plot is similar in width to "No LLM" but appears to have a more concentrated distribution around 0.7. The minimum accuracy is approximately 0.08, and the maximum is approximately 1.0. * **LLM + Conf (Query) (Red):** This violin plot is the narrowest, indicating the most consistent accuracy scores. The distribution is centered around 0.7, with a minimum accuracy of approximately 0.05 and a maximum of approximately 1.0. * **LLM + Conf (CT) (Purple):** The violin plot is wider than "LLM + Conf (Query)" but narrower than "No LLM." The distribution is centered around 0.7, with a minimum accuracy of approximately 0.25 and a maximum of approximately 1.0. ### Key Observations * All models using an LLM, with or without confidence measures, generally perform better than the baseline "No LLM" model. * The "LLM + Conf (Query)" model appears to have the most consistent performance, as indicated by the narrowest violin plot. * The "No LLM" model has the widest distribution of accuracy scores, suggesting the least consistent performance. * The red dashed line at y=0.3 serves as a visual reference point, highlighting the proportion of scores above this threshold for each model. ### Interpretation The data suggests that incorporating an LLM significantly improves accuracy on elementary math problems. Furthermore, using confidence measures, especially with query-based selection, can lead to more consistent performance. The "No LLM" model's wide distribution indicates that its accuracy is highly variable, while the "LLM + Conf (Query)" model's narrow distribution suggests a more reliable and predictable performance. The red line at 0.3 could represent a minimum acceptable accuracy level, and the plot shows how each model performs relative to this threshold.

\n ## Violin Plot: Elementary Math Accuracy ### Overview The image presents a violin plot comparing the accuracy of different approaches to solving elementary math problems. The x-axis represents the method used, and the y-axis represents the accuracy score. Five different methods are compared: "No LLM", "LLM", "LLM + Conf (Rand)", "LLM + Conf (Query)", and "LLM + Conf (CT)". A horizontal dashed red line is present at approximately 0.5 accuracy. ### Components/Axes * **Title:** "Elementary Math" (centered at the top) * **X-axis Label:** Method (categorical) with the following categories: * "No LLM" * "LLM" * "LLM + Conf (Rand)" * "LLM + Conf (Query)" * "LLM + Conf (CT)" * **Y-axis Label:** "Accuracy" (vertical, left side) with a scale ranging from approximately 0.2 to 1.0. * **Horizontal Line:** A dashed red line at approximately y = 0.5. ### Detailed Analysis The plot displays the distribution of accuracy scores for each method using violin plots. * **No LLM (Blue):** The violin plot is centered around an accuracy of approximately 0.8, with a range from roughly 0.6 to 1.0. The distribution is relatively narrow, indicating consistent performance. * **LLM (Orange):** The violin plot is centered around an accuracy of approximately 0.65, with a range from roughly 0.4 to 0.9. The distribution is wider than "No LLM", suggesting more variability. * **LLM + Conf (Rand) (Green):** The violin plot is centered around an accuracy of approximately 0.75, with a range from roughly 0.5 to 0.95. The distribution is similar in width to "No LLM". * **LLM + Conf (Query) (Red):** The violin plot is centered around an accuracy of approximately 0.7, with a range from roughly 0.45 to 0.95. The distribution is wider than "No LLM". * **LLM + Conf (CT) (Purple):** The violin plot is centered around an accuracy of approximately 0.85, with a range from roughly 0.6 to 1.0. The distribution is similar in width to "No LLM". ### Key Observations * The "No LLM" method has the highest median accuracy. * The "LLM" method has the lowest median accuracy. * Adding confidence information ("Conf") to the LLM improves accuracy compared to using the LLM alone, with "LLM + Conf (CT)" achieving the highest accuracy among the LLM-based methods. * All methods generally achieve accuracy above the 0.5 threshold indicated by the red dashed line. * The "LLM + Conf (Rand)" and "LLM + Conf (Query)" methods show a wider distribution of accuracy scores, indicating more variability in performance. ### Interpretation The data suggests that while Large Language Models (LLMs) alone perform worse than a "No LLM" baseline on elementary math problems, incorporating confidence information can significantly improve their accuracy. The method of obtaining confidence information appears to matter, with "LLM + Conf (CT)" performing best. The violin plots reveal the spread of the data, showing that some methods are more consistent than others. The red dashed line at 0.5 likely represents a threshold for acceptable performance, and all methods exceed this threshold. The variability in the "LLM + Conf (Rand)" and "LLM + Conf (Query)" methods could indicate sensitivity to the specific problem instances or the randomness inherent in those confidence estimation techniques. The data implies that LLMs can be valuable tools for solving elementary math problems, but their effectiveness is highly dependent on how confidence information is integrated.

## Violin Plot: Elementary Math Accuracy by LLM Configuration ### Overview The image displays a violin plot titled "Elementary Math," comparing the distribution of accuracy scores across five different experimental conditions involving Large Language Models (LLMs). The plot visualizes the probability density of the data at different values, with wider sections representing a higher frequency of data points. ### Components/Axes * **Chart Title:** "Elementary Math" (centered at the top). * **Y-Axis:** Labeled "Accuracy." The scale runs from 0.0 to 1.0, with major tick marks at 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0. * **X-Axis:** Contains five categorical labels corresponding to the experimental conditions: 1. No LLM 2. LLM 3. LLM + Conf (Rand) 4. LLM + Conf (Query) 5. LLM + Conf (CT) * **Data Series:** Five distinct colored violin plots, one for each category on the x-axis. The colors are (from left to right): blue, orange, green, red, and purple. * **Reference Line:** A horizontal red dashed line is drawn across the plot at approximately y = 0.3. * **Internal Markers:** Each violin contains three horizontal black lines, representing the quartiles (25th, 50th/median, and 75th percentiles) of the distribution. ### Detailed Analysis The analysis proceeds from left to right, matching the x-axis order. 1. **No LLM (Blue Violin):** * **Trend/Shape:** The distribution is broad and somewhat symmetric, with the widest section (highest density) centered around the median. It tapers smoothly towards both the high and low accuracy extremes. * **Key Values:** The median (middle black line) is approximately **0.70**. The interquartile range (IQR, distance between the top and bottom black lines) spans roughly from **0.55 to 0.85**. The full range extends from near **0.15** to **1.0**. 2. **LLM (Orange Violin):** * **Trend/Shape:** This distribution is narrower and more concentrated than the "No LLM" case. It is slightly skewed towards lower accuracy, with a longer tail extending downward. * **Key Values:** The median is lower, at approximately **0.60**. The IQR is tighter, spanning from about **0.50 to 0.75**. The range extends from approximately **0.25** to **0.95**. 3. **LLM + Conf (Rand) (Green Violin):** * **Trend/Shape:** This distribution is distinctly **bimodal**, with two clear peaks (widest sections). One peak is in the lower accuracy range (~0.4), and another is in the higher range (~0.8). This suggests two subgroups within the data. * **Key Values:** The median line sits between the two modes, at approximately **0.65**. The IQR is wide, from about **0.45 to 0.85**. The overall range is very broad, from near **0.10** to **1.0**. 4. **LLM + Conf (Query) (Red Violin):** * **Trend/Shape:** This is the tallest and narrowest violin, indicating a highly concentrated distribution. The density is sharply peaked around the median, with very thin tails. * **Key Values:** The median is high, at approximately **0.70**. The IQR is very narrow, spanning from about **0.65 to 0.75**. The range is also constrained, from roughly **0.30** to **0.90**. 5. **LLM + Conf (CT) (Purple Violin):** * **Trend/Shape:** This distribution is broad and appears slightly right-skewed (longer tail towards higher accuracy). It has a wide, dense central region. * **Key Values:** This condition shows the highest median, at approximately **0.80**. The IQR spans from about **0.70 to 0.90**. The range extends from near **0.20** to **1.0**. ### Key Observations * **Baseline Reference:** The red dashed line at **Accuracy ≈ 0.3** likely represents a baseline performance level, such as random guessing or a simple heuristic. All distributions are predominantly above this line. * **Impact of Confidence Calibration:** Adding confidence calibration ("Conf") generally shifts the median accuracy upward compared to the base "LLM" condition, except for the "Rand" variant which shows high variance. * **Highest & Most Consistent Performance:** The **"LLM + Conf (CT)"** condition achieves the highest median accuracy (~0.80) and maintains a broad, high-performing distribution. The **"LLM + Conf (Query)"** condition shows the most consistent (least variable) performance, with the narrowest spread. * **Bimodal Anomaly:** The **"LLM + Conf (Rand)"** condition is a clear outlier in shape, exhibiting a bimodal distribution. This indicates that the random confidence calibration method produces highly inconsistent results, splitting performance into a low-accuracy group and a high-accuracy group. * **Performance Hierarchy (by Median):** LLM + Conf (CT) > LLM + Conf (Query) ≈ No LLM > LLM + Conf (Rand) > LLM. ### Interpretation This chart demonstrates the effect of different LLM augmentation strategies on performance in elementary math tasks. The core finding is that **structured confidence calibration methods (Query and CT) can improve both the median accuracy and the reliability (consistency) of LLM outputs** compared to using a base LLM alone or using an unstructured (Random) calibration method. * The **"No LLM"** baseline performs surprisingly well, suggesting the task may have patterns that are accessible without model assistance, or that the "LLM" being tested is not specialized for this domain. * The poor and inconsistent performance of **"LLM + Conf (Rand)"** highlights that the *method* of confidence calibration is critical; a naive random approach introduces noise and bifurcates outcomes. * The tight distribution of **"LLM + Conf (Query)"** suggests it makes the model's performance highly predictable, which is valuable for deployment where reliability is key. * The high median of **"LLM + Conf (CT)"** suggests it is the most effective method for boosting raw accuracy, though with slightly more variability than the Query method. * The red baseline at 0.3 provides crucial context, showing that even the worst-performing condition (the lower mode of the Rand method) still generally outperforms a minimal baseline. In summary, the data argues for the implementation of specific, structured confidence calibration techniques (particularly CT and Query variants) to enhance and stabilize LLM performance on elementary math problems, moving beyond both unaugmented models and simplistic calibration approaches.

## Violin Plot: Elementary Math Accuracy Comparison ### Overview The image displays a violin plot comparing the accuracy distributions of five different model configurations in elementary math tasks. The plot uses color-coded distributions to visualize performance variations across model architectures and confidence mechanisms. ### Components/Axes - **X-axis (Categories)**: - No LLM (blue) - LLM (orange) - LLM + Conf (Rand) (green) - LLM + Conf (Query) (red) - LLM + Conf (CT) (purple) - **Y-axis (Accuracy)**: Ranges from 0.0 to 1.0 with horizontal gridlines at 0.2, 0.4, 0.6, 0.8 - **Reference Line**: Red dashed line at y=0.3 - **Color Legend**: Directly mapped to x-axis categories (no separate legend box) ### Detailed Analysis 1. **No LLM (Blue)**: - Distribution peaks between 0.7-0.8 - Narrowest distribution with minimal spread - Median accuracy ~0.75 2. **LLM (Orange)**: - Bimodal distribution with peaks at ~0.5 and ~0.6 - Wider spread than No LLM - Median accuracy ~0.55 3. **LLM + Conf (Rand) (Green)**: - Single peak at ~0.6 - Moderate spread with slight dip at 0.4-0.5 - Median accuracy ~0.6 4. **LLM + Conf (Query) (Red)**: - Double-peaked distribution at ~0.6 and ~0.7 - Significant spread between 0.4-0.8 - Median accuracy ~0.65 5. **LLM + Conf (CT) (Purple)**: - Tallest peak at ~0.75 - Narrowest distribution among confidence-enhanced models - Median accuracy ~0.78 ### Key Observations - All models exceed the 0.3 accuracy threshold (red dashed line) - LLM + Conf (CT) shows highest median accuracy (0.78) and tightest distribution - No LLM and LLM + Conf (Rand) have the lowest variability - LLM + Conf (Query) demonstrates the widest performance spread - Confidence mechanisms generally improve performance over base LLM ### Interpretation The data suggests that confidence-enhanced models (particularly CT) achieve superior accuracy in elementary math tasks compared to base LLM configurations. The CT method appears most effective at maintaining high accuracy with minimal performance variance. The red threshold line indicates a performance floor, with all models demonstrating capability above this level. The widening distributions in confidence-enhanced models suggest increased computational complexity or parameter tuning requirements. Notably, the Query confidence method shows significant performance variability, potentially indicating sensitivity to input formatting or question complexity.