## Chart: Pass@k vs. Number of Samples ### Overview The image is a line chart comparing the "pass@k(%)" metric for "critical tokens" and "random tokens" against the "number of sample k". The chart displays two lines, one red for "critical tokens" and one purple for "random tokens", with error bars indicating variability. The x-axis represents the number of samples, and the y-axis represents the pass@k percentage. ### Components/Axes * **X-axis:** "number of sample k" with tick marks at 10, 20, 30, and 40. * **Y-axis:** "pass@k(%)" with tick marks at 50, 55, 60, 65, 70, 75, 80, and 85. * **Legend:** Located in the bottom-right corner, it identifies the red line as "critical tokens" and the purple line as "random tokens". * **Gridlines:** Gray dashed lines are present for both x and y axes. ### Detailed Analysis * **Critical Tokens (Red):** The red line represents the "pass@k(%)" for "critical tokens". The trend is generally upward, with a steeper increase initially, followed by a plateau. * At k=5, pass@k(%) ≈ 71% ± 2% * At k=15, pass@k(%) ≈ 82% ± 2% * At k=30, pass@k(%) ≈ 85% ± 1% * At k=45, pass@k(%) ≈ 86% ± 1% * **Random Tokens (Purple):** The purple line represents the "pass@k(%)" for "random tokens". The trend is also upward, but less steep than the "critical tokens" line. * At k=5, pass@k(%) ≈ 51% ± 3% * At k=15, pass@k(%) ≈ 60% ± 3% * At k=30, pass@k(%) ≈ 62% ± 3% * At k=45, pass@k(%) ≈ 64% ± 3% ### Key Observations * The "pass@k(%)" is consistently higher for "critical tokens" compared to "random tokens" across all values of k. * The "pass@k(%)" for "critical tokens" increases rapidly initially, then plateaus. * The "pass@k(%)" for "random tokens" increases more gradually. * The error bars suggest more variability in the "random tokens" data, especially at lower values of k. ### Interpretation The data suggests that using "critical tokens" leads to a significantly higher "pass@k(%)" compared to using "random tokens". This indicates that "critical tokens" are more effective in achieving the desired outcome, as measured by the "pass@k(%)" metric. The initial rapid increase in "pass@k(%)" for "critical tokens" suggests that a relatively small number of samples is sufficient to achieve a substantial improvement, while further increasing the number of samples provides diminishing returns. The higher variability in the "random tokens" data may indicate that the performance is more sensitive to the specific set of random tokens used.

\n ## Line Chart: Pass@k vs. Number of Sample k ### Overview This line chart compares the "pass@k%" metric for two categories: "critical tokens" and "random tokens" as a function of the "number of sample k". Error bars are included for each data point, indicating the variability or confidence interval. ### Components/Axes * **X-axis:** "number of sample k". Scale ranges from approximately 0 to 45, with markers at 0, 10, 20, 30, and 40. * **Y-axis:** "pass@k%". Scale ranges from approximately 50% to 87%, with markers at 50%, 55%, 60%, 65%, 70%, 75%, 80%, and 85%. * **Data Series 1:** "critical tokens" - Represented by a red line with triangular markers and error bars. * **Data Series 2:** "random tokens" - Represented by a purple line with square markers and error bars. * **Legend:** Located in the bottom-right corner, clearly labeling each data series with its corresponding color. ### Detailed Analysis **Critical Tokens (Red Line):** The line representing "critical tokens" slopes generally upward, indicating an increasing "pass@k%" with increasing "number of sample k". * At k = 0, pass@k% is approximately 71% ± 4%. * At k = 10, pass@k% is approximately 78% ± 3%. * At k = 20, pass@k% is approximately 82% ± 2%. * At k = 30, pass@k% is approximately 84% ± 2%. * At k = 40, pass@k% is approximately 85% ± 2%. **Random Tokens (Purple Line):** The line representing "random tokens" also slopes upward, but at a slower rate than the "critical tokens" line. * At k = 0, pass@k% is approximately 52% ± 5%. * At k = 10, pass@k% is approximately 58% ± 4%. * At k = 20, pass@k% is approximately 61% ± 3%. * At k = 30, pass@k% is approximately 63% ± 3%. * At k = 40, pass@k% is approximately 65% ± 4%. ### Key Observations * "Critical tokens" consistently achieve a higher "pass@k%" than "random tokens" across all values of "number of sample k". * The difference in "pass@k%" between the two categories appears to be more pronounced at lower values of "number of sample k". * The error bars suggest that the "critical tokens" data has slightly less variability than the "random tokens" data. * Both lines appear to be approaching a plateau as "number of sample k" increases, suggesting diminishing returns. ### Interpretation The data suggests that using "critical tokens" leads to a significantly higher "pass@k%" compared to using "random tokens". This implies that selecting tokens based on their importance or criticality is a more effective strategy for achieving a desired level of performance (as measured by "pass@k%"). The diminishing returns observed at higher values of "number of sample k" suggest that there is a point beyond which increasing the sample size provides minimal improvement. The smaller error bars for "critical tokens" indicate a more consistent and reliable performance compared to "random tokens". This chart likely represents the results of an experiment evaluating different token selection strategies in a machine learning or natural language processing context, where "pass@k%" is a metric for evaluating the quality of generated outputs. The "number of sample k" likely refers to the number of candidate tokens considered during the generation process.

## Line Chart: pass@k(%) vs. number of sample k ### Overview The image is a line chart comparing the performance of two methods, "critical tokens" and "random tokens," across different sample sizes (k). The performance metric is "pass@k(%)" which likely represents the percentage of problems solved correctly when given k samples. The chart includes error bars for each data point, indicating variability or confidence intervals. ### Components/Axes * **Chart Type:** Line chart with error bars. * **X-Axis:** * **Label:** "number of sample k" * **Scale:** Linear scale from approximately 0 to 50. * **Major Tick Marks:** 10, 20, 30, 40. * **Y-Axis:** * **Label:** "pass@k(%)" * **Scale:** Linear scale from 50 to approximately 88. * **Major Tick Marks:** 50, 55, 60, 65, 70, 75, 80, 85. * **Legend:** * **Location:** Bottom-right corner of the plot area. * **Series 1:** Red line with upward-pointing triangle markers, labeled "critical tokens". * **Series 2:** Purple (magenta) line with star markers, labeled "random tokens". * **Grid:** Dashed gray grid lines are present for both major x and y ticks. ### Detailed Analysis **Data Series: critical tokens (Red line, triangle markers)** * **Trend:** The line shows a clear upward trend, with a steep initial increase that gradually flattens as k increases. * **Data Points (Approximate):** * k ≈ 5: pass@k ≈ 71% (Error bar range: ~70% to ~73%) * k ≈ 10: pass@k ≈ 78% (Error bar range: ~76% to ~80%) * k ≈ 15: pass@k ≈ 82% (Error bar range: ~80.5% to ~83.5%) * k ≈ 20: pass@k ≈ 84% (Error bar range: ~82.5% to ~85.5%) * k ≈ 30: pass@k ≈ 85% (Error bar range: ~83.5% to ~86.5%) * k ≈ 45: pass@k ≈ 86% (Error bar range: ~85% to ~87.5%) **Data Series: random tokens (Purple line, star markers)** * **Trend:** The line also shows an upward trend, but it is consistently lower than the "critical tokens" series. The slope is more gradual throughout. * **Data Points (Approximate):** * k ≈ 5: pass@k ≈ 51% (Error bar range: ~48% to ~54%) * k ≈ 10: pass@k ≈ 57% (Error bar range: ~53.5% to ~60%) * k ≈ 15: pass@k ≈ 60% (Error bar range: ~57% to ~63%) * k ≈ 20: pass@k ≈ 61.5% (Error bar range: ~58.5% to ~64.5%) * k ≈ 30: pass@k ≈ 62.5% (Error bar range: ~59.5% to ~65.5%) * k ≈ 45: pass@k ≈ 64% (Error bar range: ~60.5% to ~67.5%) ### Key Observations 1. **Performance Gap:** There is a significant and consistent performance gap between the two methods. "Critical tokens" outperforms "random tokens" at every measured value of k. 2. **Diminishing Returns:** Both curves show diminishing returns. The largest performance gains for both methods occur when increasing k from 5 to 15. The improvement per additional sample becomes smaller as k grows larger. 3. **Error Bar Comparison:** The error bars for the "random tokens" series appear visually larger (especially at k=5 and k=45) than those for the "critical tokens" series. This suggests that the performance of the random method is more variable or less certain. 4. **Convergence:** The two lines do not appear to be converging. The absolute difference in pass@k between the two methods remains roughly constant (around 20-22 percentage points) across the range of k shown. ### Interpretation This chart demonstrates the superior effectiveness of a "critical tokens" strategy over a "random tokens" strategy for the task measured by pass@k. The data suggests that intelligently selecting or focusing on "critical" tokens leads to a much higher success rate than using randomly selected tokens, regardless of the number of samples (k) considered. The consistent gap indicates that the advantage of the critical token approach is fundamental and not merely a function of sample size. The larger error bars for the random method imply it is less reliable. The diminishing returns for both methods are typical in sampling-based approaches, but the critical token method reaches a high level of performance (over 80%) with a relatively small number of samples (k=15), making it potentially more efficient. The chart provides strong empirical evidence for prioritizing the identification and use of critical tokens in this context.

## Line Graph: Pass@k Performance Comparison ### Overview The image depicts a line graph comparing the performance of two token selection strategies ("critical tokens" and "random tokens") across varying sample sizes (k). The y-axis represents "pass@k(%)", while the x-axis shows the "number of sample k". Error bars indicate measurement uncertainty for each data point. ### Components/Axes - **X-axis**: "number of sample k" (ranges from 10 to 40 in increments of 10) - **Y-axis**: "pass@k(%)" (ranges from 50% to 85% in 5% increments) - **Legend**: Located at bottom-right, with: - Red triangles: "critical tokens" - Purple stars: "random tokens" - **Error bars**: Vertical lines with caps at both ends, representing ± uncertainty for each data point ### Detailed Analysis **Critical Tokens (Red):** - At k=10: 70% ±3% (error bar spans 67–73%) - At k=20: 78% ±2% (66–80%) - At k=30: 82% ±1% (81–83%) - At k=40: 85% ±2% (83–87%) **Random Tokens (Purple):** - At k=10: 50% ±5% (45–55%) - At k=20: 58% ±4% (54–62%) - At k=30: 62% ±3% (59–65%) - At k=40: 64% ±4% (60–68%) ### Key Observations 1. **Performance Gap**: Critical tokens consistently outperform random tokens by 16–21 percentage points across all k values. 2. **Error Trends**: - Random tokens show larger error margins (4–5%) compared to critical tokens (1–3%). - Error margins for critical tokens decrease as k increases. 3. **Saturation Point**: Both strategies plateau near k=30–40, with diminishing returns in performance gains. ### Interpretation The data demonstrates that critical token selection significantly improves performance reliability compared to random selection. The narrowing performance gap at higher k values suggests diminishing returns for both strategies, but critical tokens maintain a clear advantage. The smaller error margins for critical tokens indicate more consistent results, making them preferable for applications requiring stable performance. The plateau observed at k≥30 implies that increasing sample size beyond this point yields minimal practical benefits for either strategy.