## Line Chart: Pass@k(%) vs. Number of Sample k ### Overview The image is a line chart comparing the performance of "critical tokens" and "random tokens" based on the "pass@k(%)" metric, as the "number of sample k" increases. The chart includes error bars for each data point, indicating variability. ### 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 30, 40, 50, 60, 70, 80, and 90. * **Legend:** Located in the center of the chart. * Red line with triangle markers: "critical tokens" * Purple line with plus markers: "random tokens" ### Detailed Analysis * **Critical Tokens (Red Line):** The line slopes upward, indicating an increase in "pass@k(%)" as the "number of sample k" increases. * At k=5, pass@k(%) is approximately 57% (with an error range of +/- 3%). * At k=15, pass@k(%) is approximately 75% (with an error range of +/- 3%). * At k=25, pass@k(%) is approximately 80% (with an error range of +/- 3%). * At k=30, pass@k(%) is approximately 83% (with an error range of +/- 3%). * At k=45, pass@k(%) is approximately 87% (with an error range of +/- 4%). * **Random Tokens (Purple Line):** The line also slopes upward, but at a shallower angle compared to the "critical tokens" line. * At k=5, pass@k(%) is approximately 30% (with an error range of +/- 3%). * At k=15, pass@k(%) is approximately 40% (with an error range of +/- 3%). * At k=25, pass@k(%) is approximately 42% (with an error range of +/- 3%). * At k=30, pass@k(%) is approximately 45% (with an error range of +/- 4%). * At k=45, pass@k(%) is approximately 47% (with an error range of +/- 6%). ### Key Observations * The "critical tokens" consistently outperform "random tokens" across all values of k. * The performance gap between "critical tokens" and "random tokens" widens as k increases, but the rate of increase slows down for both. * The error bars suggest that the variability in "pass@k(%)" is relatively consistent across different values of k for both token types. ### 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, whatever that may be. The increasing performance with higher k values suggests that increasing the number of samples improves the overall performance for both token types, but the effect is more pronounced for "critical tokens." The error bars provide a measure of confidence in these observations, suggesting that the observed differences are statistically significant. The flattening of the curves at higher k values might indicate a point of diminishing returns, where increasing the number of samples provides less and less improvement in performance.

\n ## Line Chart: Pass@k vs. Number of Sample k ### Overview This line chart compares the performance of "critical tokens" and "random tokens" based on the metric "pass@k (%)" as a function of the "number of sample k". The chart displays two lines with error bars, representing the mean and standard deviation of the pass@k metric for each token type at different values of k. ### 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 25% to 95%, with markers at 30%, 40%, 50%, 60%, 70%, 80%, and 90%. * **Legend:** Located in the top-right corner. * Red line with error bars: "critical tokens" * Purple line with error bars: "random tokens" * **Gridlines:** Horizontal and vertical gridlines are present to aid in reading values. ### Detailed Analysis **Critical Tokens (Red Line):** The red line representing "critical tokens" shows an upward trend. * At k = 0, pass@k is approximately 54% ± 6%. * At k = 10, pass@k is approximately 68% ± 5%. * At k = 20, pass@k is approximately 77% ± 4%. * At k = 30, pass@k is approximately 81% ± 4%. * At k = 40, pass@k is approximately 86% ± 4%. **Random Tokens (Purple Line):** The purple line representing "random tokens" also shows an upward trend, but is less steep than the red line. * At k = 0, pass@k is approximately 28% ± 4%. * At k = 10, pass@k is approximately 34% ± 4%. * At k = 20, pass@k is approximately 41% ± 4%. * At k = 30, pass@k is approximately 44% ± 4%. * At k = 40, pass@k is approximately 48% ± 6%. The error bars indicate the variability in the pass@k metric for each token type at each value of k. The error bars are relatively consistent in size across the range of k values. ### Key Observations * "Critical tokens" consistently outperform "random tokens" across all values of k. * The performance gap between "critical tokens" and "random tokens" widens as k increases. * The rate of improvement in pass@k decreases as k increases for both token types. * The error bars suggest that the variability in performance is relatively consistent across different values of k. ### Interpretation The data suggests that selecting "critical tokens" leads to significantly better performance (as measured by pass@k) compared to selecting "random tokens". This indicates that the "critical tokens" are more informative or relevant for the task being evaluated. As the number of samples (k) increases, the performance of both token types improves, but the advantage of using "critical tokens" becomes more pronounced. This could be because the "critical tokens" provide a stronger signal, allowing for more accurate predictions even with a limited number of samples. The consistent error bars suggest that the observed differences in performance are statistically significant and not due to random chance. The diminishing returns in performance as k increases suggest that there may be a point of diminishing returns where adding more samples does not significantly improve the pass@k metric. This could be due to the fact that the most informative tokens have already been selected, and adding more samples provides less additional information.

## Line Chart: pass@k(%) vs. number of sample k ### Overview The image is a line chart with error bars, plotting the performance metric "pass@k(%)" against the "number of sample k". It compares two distinct methods or conditions: "critical tokens" and "random tokens". The chart demonstrates how the pass rate changes as the number of samples (k) increases for each condition. ### 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 approximately 25 to 95. * **Major Tick Marks:** 30, 40, 50, 60, 70, 80, 90. * **Legend:** * **Position:** Centered within the plot area, slightly to the right. * **Entry 1:** Red line with upward-pointing triangle markers, labeled `critical tokens`. * **Entry 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 1: critical tokens (Red line, triangle markers)** * **Trend:** The line shows a steep, concave-down increase. The rate of improvement in pass@k is highest for small k and gradually diminishes as k increases. * **Data Points (Approximate with Error Bar Ranges):** * k ≈ 5: pass@k ≈ 55% (Error bar range: ~53% to ~57%) * k ≈ 8: pass@k ≈ 67% (Error bar range: ~65% to ~70%) * k ≈ 15: pass@k ≈ 75% (Error bar range: ~72% to ~78%) * k ≈ 23: pass@k ≈ 79% (Error bar range: ~76% to ~83%) * k ≈ 31: pass@k ≈ 83% (Error bar range: ~80% to ~87%) * k ≈ 47: pass@k ≈ 87% (Error bar range: ~83% to ~91%) **Data Series 2: random tokens (Purple line, star markers)** * **Trend:** The line shows a steady, nearly linear increase. The slope is positive but significantly shallower than the "critical tokens" series. * **Data Points (Approximate with Error Bar Ranges):** * k ≈ 5: pass@k ≈ 29% (Error bar range: ~26% to ~32%) * k ≈ 8: pass@k ≈ 34% (Error bar range: ~30% to ~38%) * k ≈ 15: pass@k ≈ 39% (Error bar range: ~35% to ~43%) * k ≈ 23: pass@k ≈ 42% (Error bar range: ~38% to ~47%) * k ≈ 31: pass@k ≈ 44% (Error bar range: ~39% to ~49%) * k ≈ 47: pass@k ≈ 47% (Error bar range: ~41% to ~53%) ### Key Observations 1. **Performance Gap:** There is a substantial and consistent performance gap between the two methods. "Critical tokens" achieves a pass@k rate approximately 25-40 percentage points higher than "random tokens" across all measured values of k. 2. **Diminishing Returns:** The "critical tokens" series exhibits clear diminishing returns. The gain from k=5 to k=8 (~12%) is much larger than the gain from k=31 to k=47 (~4%). The "random tokens" series shows more constant, linear returns. 3. **Error Bar Magnitude:** The error bars (representing uncertainty or variance) for both series appear to increase slightly in absolute terms as k increases. The relative uncertainty (error bar size compared to the mean value) seems more stable. 4. **Convergence:** The two lines are not converging. The absolute difference in pass@k between them remains large even at the highest k value shown (k≈47). ### Interpretation This chart likely comes from a machine learning or natural language processing context, evaluating a model's ability to generate correct outputs (pass@k) given a certain number of sampling attempts (k). The "critical tokens" method appears to be a targeted or informed strategy for guiding generation, while "random tokens" represents a baseline or unguided approach. The data strongly suggests that **strategically selecting or influencing "critical tokens" is a far more effective strategy for improving model performance than relying on random sampling.** The steep initial rise for critical tokens indicates that even a small number of guided samples yields a high probability of success. The persistent gap shows that the advantage of the guided method does not diminish with more attempts; it maintains a superior efficiency. The linear trend for random tokens is characteristic of a baseline where each additional sample provides a roughly equal, independent chance of success. The chart effectively argues for the value of the "critical tokens" intervention.

## Line Graph: Pass@k Performance Comparison ### Overview The image displays 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 (%)" (percentage of successful outcomes), while the x-axis represents "number of sample k" (sample size). Two data series are plotted with distinct markers and error bars, showing performance trends as sample size increases. ### Components/Axes - **X-axis**: "number of sample k" (ranges from 5 to 45 in increments of 10) - **Y-axis**: "pass@k (%)" (ranges from 30% to 90% in increments of 10) - **Legend**: Located in the top-right corner, with: - Red triangles (▲) labeled "critical tokens" - Purple stars (★) labeled "random tokens" - **Error Bars**: Vertical lines extending from each data point, indicating measurement uncertainty. ### Detailed Analysis #### Critical Tokens (Red) - **Trend**: Steep upward trajectory from ~55% at k=5 to ~85% at k=45. - **Data Points**: - k=5: 55% ±3% - k=15: 75% ±2% - k=25: 80% ±2% - k=35: 82% ±1% - k=45: 85% ±2% - **Error Bars**: Consistently ±1–3%, smallest at k=35. #### Random Tokens (Purple) - **Trend**: Gradual upward slope from ~30% at k=5 to ~47% at k=45. - **Data Points**: - k=5: 30% ±3% - k=15: 38% ±2% - k=25: 42% ±3% - k=35: 44% ±2% - k=45: 47% ±3% - **Error Bars**: Larger variability (±2–3%) compared to critical tokens. ### Key Observations 1. **Performance Gap**: Critical tokens consistently outperform random tokens across all sample sizes (e.g., 55% vs. 30% at k=5; 85% vs. 47% at k=45). 2. **Error Margin**: Critical tokens exhibit tighter error bars, suggesting more reliable measurements. 3. **Diminishing Returns**: Both strategies show slowing growth in pass@k as k increases, but critical tokens maintain a higher plateau. ### Interpretation The data demonstrates that critical token selection significantly improves performance reliability compared to random selection. The steeper slope and smaller error margins for critical tokens suggest they are more effective at capturing relevant information in larger datasets. The diminishing returns for both strategies imply that increasing sample size beyond a certain point yields diminishing benefits, but critical tokens retain a clear advantage. This could reflect their ability to prioritize high-impact tokens in applications like NLP or recommendation systems.