## Histogram: Length of Reasoning Chains in Tokens, Comparative Illusion vs. Control ### Overview The image is a histogram comparing the distribution of reasoning chain lengths (in tokens) for two sentence types: "Comparative Illusion" and "Control". The histogram displays the frequency (count) of different reasoning chain lengths, with overlaid curves representing the estimated probability density for each sentence type. ### Components/Axes * **Title:** Length of Reasoning Chains in Tokens, Comparative Illusion vs. Control * **X-axis:** Reasoning Chain Length (tokens) * Scale: 0 to 7000, with tick marks at 1000, 2000, 3000, 4000, 5000, 6000, and 7000. * **Y-axis:** Count * Scale: 0 to 50, with tick marks at 0, 10, 20, 30, 40, and 50. * **Legend:** Located in the top-right corner. * "Comparative Illusion": Represented by light blue bars and a blue curve. * "Control": Represented by light orange bars and an orange curve. ### Detailed Analysis The histogram shows the distribution of reasoning chain lengths for both "Comparative Illusion" and "Control" sentence types. * **Comparative Illusion (Light Blue):** * The distribution is centered around 4000 tokens. * The count for the 1000 token bin is approximately 13. * The count for the 2000 token bin is approximately 21. * The count for the 3000 token bin is approximately 33. * The count for the 4000 token bin is approximately 43. * The count for the 5000 token bin is approximately 30. * The count for the 6000 token bin is approximately 6. * The blue curve shows a unimodal distribution, peaking around 4000 tokens. * **Control (Light Orange):** * The distribution is centered around 1000 tokens. * The count for the 1000 token bin is approximately 48. * The count for the 2000 token bin is approximately 43. * The count for the 3000 token bin is approximately 33. * The count for the 4000 token bin is approximately 12. * The count for the 5000 token bin is approximately 2. * The count for the 6000 token bin is approximately 1. * The orange curve shows a unimodal distribution, peaking around 1000 tokens. ### Key Observations * The "Control" sentence type has a much shorter reasoning chain length compared to the "Comparative Illusion" sentence type. * The "Control" distribution is heavily skewed towards shorter chain lengths, with a peak around 1000 tokens. * The "Comparative Illusion" distribution is more spread out, with a peak around 4000 tokens. * There is some overlap between the two distributions in the 2000-3000 token range. ### Interpretation The data suggests that "Comparative Illusion" sentences require longer reasoning chains than "Control" sentences. This could indicate that understanding or processing "Comparative Illusion" sentences involves more complex cognitive processes or requires the consideration of more information. The difference in distribution shapes further supports this, with "Comparative Illusion" showing a broader range of reasoning chain lengths, possibly reflecting varying levels of complexity in processing these types of sentences. The "Control" sentences, on the other hand, appear to be processed with relatively shorter and more consistent reasoning chains.

## Histogram: Length of Reasoning Chains in Tokens, Comparative Illusion vs. Control ### Overview This image presents a histogram comparing the distribution of reasoning chain lengths (measured in tokens) for two sentence types: "Comparative Illusion" and "Control". The x-axis represents the reasoning chain length in tokens, and the y-axis represents the count (frequency) of reasoning chains within each length range. The data is visualized using stacked histograms with overlaid probability density curves. ### Components/Axes * **Title:** "Length of Reasoning Chains in Tokens, Comparative Illusion vs. Control" (Top-center) * **X-axis Label:** "Reasoning Chain Length (tokens)" (Bottom-center) * Scale: Ranges from approximately 0 to 7000 tokens. Markers are present at 1000, 2000, 3000, 4000, 5000, and 6000 tokens. * **Y-axis Label:** "Count" (Left-center) * Scale: Ranges from 0 to 50. * **Legend:** Located in the top-right corner. * "Sentence Type" * "Comparative Illusion" - Represented by a light blue color. * "Control" - Represented by a light orange color. * **Overlaid Curves:** Two smooth curves are overlaid on the histograms, representing the probability density functions for each sentence type. The blue curve corresponds to "Comparative Illusion", and the orange curve corresponds to "Control". ### Detailed Analysis The "Control" data (orange) is concentrated in the lower reasoning chain length range, peaking around 1000-1500 tokens. The count at the peak is approximately 48. The distribution decreases steadily as the reasoning chain length increases. The curve representing the control group is centered around 1200 tokens, with a rapid decline after 2000 tokens. The "Comparative Illusion" data (blue) is shifted towards higher reasoning chain lengths, peaking around 4000-4500 tokens. The count at the peak is approximately 42. The distribution is more spread out than the "Control" data, with a longer tail extending to higher reasoning chain lengths. The curve representing the comparative illusion group is centered around 4200 tokens, with a slower decline after the peak. Here's a breakdown of approximate counts at key reasoning chain lengths: | Reasoning Chain Length (tokens) | Comparative Illusion (Count) | Control (Count) | |---|---|---| | 1000 | ~5 | ~45 | | 2000 | ~18 | ~20 | | 3000 | ~25 | ~32 | | 4000 | ~42 | ~10 | | 5000 | ~32 | ~5 | | 6000 | ~8 | ~2 | | 7000 | ~1 | ~0 | ### Key Observations * The "Comparative Illusion" sentence type generally requires longer reasoning chains than the "Control" sentence type. * The distributions are not symmetrical. Both distributions are skewed to the right, but the "Comparative Illusion" distribution is more heavily skewed. * There is some overlap between the two distributions, indicating that some "Control" sentences also require relatively long reasoning chains, and some "Comparative Illusion" sentences require shorter ones. * The peak of the "Control" distribution is significantly higher than the peak of the "Comparative Illusion" distribution, suggesting that shorter reasoning chains are more common in the "Control" condition. ### Interpretation The data suggests that processing "Comparative Illusion" sentences involves more extensive reasoning processes compared to processing "Control" sentences. This is evidenced by the shift in the distribution towards longer reasoning chain lengths. The longer reasoning chains required for "Comparative Illusion" sentences may be due to the inherent complexity of comparative statements, which often require more cognitive effort to evaluate and understand. The overlaid probability density curves visually reinforce this difference, showing a clear separation in the typical reasoning chain lengths for the two sentence types. The fact that the distributions overlap suggests that the effect is not absolute, and individual sentence characteristics can also influence reasoning chain length. The difference in peak heights indicates that shorter reasoning chains are more typical for control sentences, while longer chains are more typical for comparative illusion sentences. This could be due to the nature of the illusion itself requiring more steps to resolve.

## Histogram: Length of Reasoning Chains in Tokens, Comparative Illusion vs. Control ### Overview The image is a histogram comparing the distribution of reasoning chain lengths (measured in tokens) for two types of sentences: "Comparative Illusion" and "Control." The chart includes overlaid density curves to visualize the shape of each distribution. ### Components/Axes * **Title:** "Length of Reasoning Chains in Tokens, Comparative Illusion vs. Control" * **X-Axis:** Labeled "Reasoning Chain Length (tokens)". The scale runs from 0 to 7000, with major tick marks at every 1000-token interval (0, 1000, 2000, 3000, 4000, 5000, 6000, 7000). * **Y-Axis:** Labeled "Count". The scale runs from 0 to 50, with major tick marks at every 10-unit interval (0, 10, 20, 30, 40, 50). * **Legend:** Located in the top-right corner of the plot area. It is titled "Sentence Type" and contains two entries: * A blue square labeled "Comparative Illusion". * An orange square labeled "Control". * **Data Series:** 1. **Comparative Illusion (Blue Bars & Curve):** Represented by blue histogram bars and a blue density curve. 2. **Control (Orange Bars & Curve):** Represented by orange histogram bars and an orange density curve. * **Spatial Layout:** The histogram bars for the two series are overlaid, with the orange "Control" bars appearing behind the blue "Comparative Illusion" bars where they overlap, creating a grayish overlap region. The density curves are drawn on top of the bars. ### Detailed Analysis **Trend Verification & Data Points (Approximate):** * **Control (Orange) Distribution:** * **Trend:** The distribution is strongly right-skewed. It peaks sharply at the lower end of the token length scale and tapers off as length increases. * **Key Points (Estimated from bar heights):** * Bin ~500-1000 tokens: Count ~45 (Highest peak for Control). * Bin ~1000-1500 tokens: Count ~49 (Appears to be the global maximum for the entire chart). * Bin ~1500-2000 tokens: Count ~43. * Bin ~2000-2500 tokens: Count ~21. * Bin ~2500-3000 tokens: Count ~33. * Bin ~3000-3500 tokens: Count ~28. * Bin ~3500-4000 tokens: Count ~12. * Bin ~4000-4500 tokens: Count ~8. * Bin ~4500-5000 tokens: Count ~2. * Bins beyond 5000 tokens: Counts are very low, approaching 0. * **Comparative Illusion (Blue) Distribution:** * **Trend:** The distribution is more symmetric and bell-shaped compared to the Control, centered at a higher token length. It has a broader peak. * **Key Points (Estimated from bar heights):** * Bin ~500-1000 tokens: Count ~6. * Bin ~1000-1500 tokens: Count ~13. * Bin ~1500-2000 tokens: Count ~21. * Bin ~2000-2500 tokens: Count ~19. * Bin ~2500-3000 tokens: Count ~33. * Bin ~3000-3500 tokens: Count ~35. * Bin ~3500-4000 tokens: Count ~43 (Highest peak for Comparative Illusion). * Bin ~4000-4500 tokens: Count ~32. * Bin ~4500-5000 tokens: Count ~20. * Bin ~5000-5500 tokens: Count ~11. * Bin ~5500-6000 tokens: Count ~6. * Bin ~6000-6500 tokens: Count ~2. * Bin ~6500-7000 tokens: Count ~1. ### Key Observations 1. **Distinct Peaks:** The two distributions have clear, separate peaks. The Control group peaks between 1000-1500 tokens, while the Comparative Illusion group peaks between 3500-4000 tokens. 2. **Shift in Central Tendency:** The central mass of the Comparative Illusion distribution is shifted significantly to the right (higher token counts) compared to the Control distribution. 3. **Difference in Spread:** The Control distribution is more concentrated at the lower end (narrower spread in the high-frequency region), while the Comparative Illusion distribution is more spread out across a wider range of token lengths. 4. **Overlap Region:** There is a substantial overlap between the two distributions, particularly in the range of approximately 2000 to 4500 tokens, where both sentence types have meaningful counts. 5. **Tail Behavior:** The Control distribution has a long, thin tail extending to the right, but with very low counts beyond 5000 tokens. The Comparative Illusion distribution also has a right tail, but it is more substantial, with non-negligible counts up to about 6000 tokens. ### Interpretation The data suggests a fundamental difference in the cognitive processing or linguistic complexity between "Comparative Illusion" sentences and "Control" sentences. The significantly longer reasoning chains (higher token counts) for Comparative Illusion sentences indicate they likely require more steps of logical deduction, working memory maintenance, or resolution of ambiguity to be understood or evaluated. The Control sentences, peaking at much shorter lengths, appear to be processed more directly. The overlap between the distributions is crucial; it shows that not all Comparative Illusion sentences are long, and not all Control sentences are short. However, the clear separation of the peaks demonstrates a strong systematic effect. This pattern is consistent with the hypothesis that comparative illusions (e.g., "More people have been to Russia than I have") create a specific type of processing difficulty that manifests as extended reasoning chains, distinguishing them from syntactically similar but semantically straightforward control sentences. The broader spread of the Comparative Illusion data may reflect variability in how strongly the illusion affects different specific sentences or individuals.

## Histogram: Length of Reasoning Chains in Tokens, Comparative Illusion vs. Control ### Overview The image is a comparative histogram showing the distribution of reasoning chain lengths (in tokens) for two groups: "Comparative Illusion" (blue) and "Control" (orange). Overlaid on the histograms are kernel density estimation (KDE) curves, which smooth the distributions. The x-axis represents reasoning chain length (tokens), and the y-axis represents count. The Control group’s data is concentrated at shorter chain lengths, while the Comparative Illusion group’s data spans longer chain lengths with greater variability. ### Components/Axes - **Title**: "Length of Reasoning Chains in Tokens, Comparative Illusion vs. Control" - **X-axis**: "Reasoning Chain Length (tokens)" (range: 0–7000) - **Y-axis**: "Count" (range: 0–50) - **Legend**: - Blue: Comparative Illusion - Orange: Control - **Placement**: - Legend: Top-right corner - Histograms: Side-by-side (orange on the left, blue on the right) - KDE curves: Overlaid on respective histograms ### Detailed Analysis - **Control Group (Orange)**: - Histogram peaks sharply at **1000–2000 tokens**, with a KDE curve showing a narrow, high-density peak. - Counts drop rapidly beyond 2000 tokens, with minimal presence above 3000 tokens. - Example: ~45 counts at 1000 tokens, ~30 counts at 2000 tokens, and near-zero counts above 5000 tokens. - **Comparative Illusion Group (Blue)**: - Histogram peaks at **3000–4000 tokens**, with a broader, flatter distribution. - KDE curve shows a wider spread, with counts extending up to ~6000 tokens. - Example: ~35 counts at 3000 tokens, ~25 counts at 4000 tokens, and ~5 counts at 6000 tokens. ### Key Observations 1. **Control Group**: Dominated by short reasoning chains (1000–2000 tokens), suggesting simpler or more direct reasoning processes. 2. **Comparative Illusion Group**: Longer and more variable reasoning chains (3000–6000 tokens), indicating increased complexity or uncertainty in reasoning. 3. **KDE Curves**: Confirm the Control group’s narrow distribution vs. the Comparative Illusion group’s broader spread. 4. **Outliers**: No extreme outliers in either group, but the Comparative Illusion group shows a tail extending to ~6000 tokens. ### Interpretation The data suggests that **Comparative Illusion** introduces cognitive complexity, leading to longer reasoning chains compared to the Control group. The Control group’s shorter chains may reflect more straightforward decision-making, while the Comparative Illusion group’s variability could stem from conflicting information or heightened uncertainty. The KDE curves reinforce these trends, highlighting the Control group’s efficiency and the Comparative Illusion group’s resource-intensive processing. This aligns with theories that illusory comparisons require deeper cognitive engagement to resolve ambiguities.