## Stacked Bar Chart: R1-Qwen | MATH500 ### Overview The image is a stacked bar chart comparing the ratio of "Content Words" and "Function Words" across different performance percentiles (Top-10% to 90-100%) in the context of "R1-Qwen | MATH500". The x-axis represents the ratio in percentage, and the y-axis represents the performance percentile. ### Components/Axes * **Title:** R1-Qwen | MATH500 * **X-axis:** Ratio (%) - Ranges from 0 to 100. * **Y-axis:** Performance Percentiles - Categories are: Top-10%, 10-20%, 20-30%, 30-40%, 40-50%, 50-60%, 60-70%, 70-80%, 80-90%, 90-100%. * **Legend:** Located at the top-right of the chart. * Content Words (Dark Red) * Function Words (Light Gray with diagonal lines) ### Detailed Analysis The chart displays the ratio of content words and function words for different performance percentiles. The "Content Words" are represented by dark red bars, and "Function Words" are represented by light gray bars with diagonal lines. The bars are stacked, so the total length of each bar represents 100%. Here's a breakdown of the data for each percentile: * **Top-10%:** Content Words: 39.8%, Function Words: approximately 60.2% * **10-20%:** Content Words: 38.9%, Function Words: approximately 61.1% * **20-30%:** Content Words: 37.8%, Function Words: approximately 62.2% * **30-40%:** Content Words: 36.0%, Function Words: approximately 64.0% * **40-50%:** Content Words: 33.9%, Function Words: approximately 66.1% * **50-60%:** Content Words: 31.9%, Function Words: approximately 68.1% * **60-70%:** Content Words: 29.8%, Function Words: approximately 70.2% * **70-80%:** Content Words: 27.5%, Function Words: approximately 72.5% * **80-90%:** Content Words: 24.5%, Function Words: approximately 75.5% * **90-100%:** Content Words: 20.6%, Function Words: approximately 79.4% **Trends:** * The ratio of "Content Words" generally decreases as the performance percentile decreases (from Top-10% to 90-100%). * Conversely, the ratio of "Function Words" generally increases as the performance percentile decreases. ### Key Observations * The "Top-10%" percentile has the highest ratio of "Content Words" (39.8%). * The "90-100%" percentile has the lowest ratio of "Content Words" (20.6%). * The difference in "Content Words" ratio between the "Top-10%" and "90-100%" percentiles is significant (39.8% vs 20.6%). ### Interpretation The data suggests that higher-performing models (Top-10%) tend to use a higher proportion of "Content Words" compared to lower-performing models (90-100%). This could indicate that better models are more focused on the core content and meaning, while lower-performing models rely more on function words (e.g., prepositions, articles, conjunctions) which may contribute less to the semantic content. The trend highlights a potential correlation between the type of words used and the overall performance of the model in the MATH500 task. It is important to note that this is just one aspect of model performance, and other factors also play a significant role.

\n ## Bar Chart: Content vs. Function Word Ratio in MATH500 (R1-Qwen) ### Overview This is a horizontal bar chart displaying the ratio of content words to function words across different percentage ranges within a text corpus (MATH500) analyzed by R1-Qwen. The chart shows how the proportion of content words varies as you consider increasingly frequent words in the text. ### Components/Axes * **Title:** R1-Qwen | MATH500 * **X-axis:** Ratio (%) - Scale ranges from 0 to 100. * **Y-axis:** Percentage ranges of words, labeled as follows (from top to bottom): * 90-100% * 80-90% * 70-80% * 60-70% * 50-60% * 40-50% * 30-40% * 20-30% * 10-20% * Top 10% * **Legend:** * Content Words (Dark Red) * Function Words (Light Gray) ### Detailed Analysis The chart consists of horizontal bars representing the ratio of content words for each percentage range. The function word portion is represented by the remaining space to 100%. Here's a breakdown of the data points, reading from top to bottom: * **90-100%:** Content Words: 20.6% * **80-90%:** Content Words: 24.5% * **70-80%:** Content Words: 27.5% * **60-70%:** Content Words: 29.8% * **50-60%:** Content Words: 31.9% * **40-50%:** Content Words: 33.9% * **30-40%:** Content Words: 36.0% * **20-30%:** Content Words: 37.8% * **10-20%:** Content Words: 38.9% * **Top 10%:** Content Words: 39.8% The trend is clearly upward. As the percentage range decreases (i.e., considering more frequent words), the ratio of content words increases. The increase is not linear, but appears to be slowing down as you approach the top 10%. ### Key Observations * The ratio of content words is lowest in the 90-100% range (20.6%) and highest in the top 10% range (39.8%). * The difference between the lowest and highest content word ratios is 19.2%. * The increase in content word ratio is most pronounced between the 90-100% and 20-30% ranges. ### Interpretation This chart demonstrates that the most frequent words in the MATH500 corpus (as analyzed by R1-Qwen) tend to be function words (articles, prepositions, conjunctions, etc.), while less frequent words are more likely to be content words (nouns, verbs, adjectives, etc.). This is a common characteristic of natural language, where a small set of function words accounts for a large proportion of the total word count. The increasing trend suggests that as you focus on the core vocabulary of the corpus, the proportion of meaningful content words increases. This could be useful for tasks like keyword extraction or topic modeling, where identifying the most important content words is crucial. The slowing increase towards the top 10% might indicate that even the most frequent content words are still relatively rare compared to function words, or that the top 10% contains a mix of both content and function words. Further analysis would be needed to determine the exact composition of the top 10% of words.

## Horizontal Bar Chart: R1-Qwen | MATH500 ### Overview This image is a horizontal bar chart comparing the percentage ratio of "Content Words" versus "Function Words" across different performance percentile groups for a model or system identified as "R1-Qwen" on the "MATH500" benchmark. The chart visualizes how the composition of language (content vs. function words) changes as performance improves. ### Components/Axes * **Chart Title:** "R1-Qwen | MATH500" (centered at the top). * **Y-Axis (Vertical):** Represents performance percentile groups. The categories, from top to bottom, are: * 90-100% * 80-90% * 70-80% * 60-70% * 50-60% * 40-50% * 30-40% * 20-30% * 10-20% * Top-10% * **X-Axis (Horizontal):** Labeled "Ratio (%)". It is a linear scale from 0 to 100, with major tick marks at 0, 20, 40, 60, 80, and 100. * **Legend:** Positioned in the top-right corner of the chart area. * A dark red square is labeled "Content Words". * A light gray square is labeled "Function Words". * **Data Series:** Each horizontal bar is a stacked bar representing 100% of the words for that percentile group. The bar is divided into two segments: * The left segment (dark red) represents the percentage of "Content Words". * The right segment (light gray) represents the percentage of "Function Words". * The exact percentage for each segment is printed inside or adjacent to its respective bar segment. ### Detailed Analysis The chart presents the following data points for each percentile group (Content Words % / Function Words %): 1. **90-100%:** 20.6% / 79.4% 2. **80-90%:** 24.3% / 75.7% 3. **70-80%:** 27.5% / 72.5% 4. **60-70%:** 29.8% / 70.2% 5. **50-60%:** 31.9% / 68.1% 6. **40-50%:** 33.9% / 66.1% 7. **30-40%:** 36.0% / 64.0% 8. **20-30%:** 37.8% / 62.2% 9. **10-20%:** 38.9% / 61.1% 10. **Top-10%:** 39.8% / 60.2% **Trend Verification:** * **Content Words (Dark Red):** The length of the dark red segment and its labeled percentage show a clear and consistent **upward trend** as we move down the y-axis from the highest performance group (90-100%) to the lowest (Top-10%). The value increases from 20.6% to 39.8%. * **Function Words (Light Gray):** Conversely, the length of the light gray segment and its percentage show a consistent **downward trend** over the same progression. The value decreases from 79.4% to 60.2%. ### Key Observations 1. **Inverse Relationship:** There is a perfect inverse relationship between the two word categories. As the percentage of Content Words increases, the percentage of Function Words decreases by the same amount, summing to 100% for each bar. 2. **Monotonic Change:** The change in percentages is monotonic and nearly linear across the performance groups. There are no outliers or reversals in the trend. 3. **Magnitude of Shift:** The shift is substantial. The Top-10% group uses nearly double the proportion of Content Words (39.8%) compared to the 90-100% group (20.6%). 4. **Labeling Precision:** All data points are explicitly labeled with one decimal place of precision. ### Interpretation This chart suggests a strong correlation between a model's performance on the MATH500 benchmark and the lexical composition of its outputs. The data demonstrates that **higher-performing models (those in the 90-100% percentile) rely significantly more on Function Words** (e.g., articles, prepositions, conjunctions) and less on Content Words (e.g., nouns, verbs, adjectives carrying substantive meaning). This pattern could imply several things about the nature of high-performance reasoning on this math benchmark: * **Efficiency and Abstraction:** Top models may generate more concise, logically structured explanations where function words are crucial for connecting steps and maintaining grammatical coherence, while the core mathematical content is expressed with fewer, more potent content words. * **Reasoning Style:** Lower-performing models might "pad" their responses with more descriptive content words, potentially indicating less focused or less efficient reasoning chains. * **Benchmark Characteristics:** The trend might reflect the specific language patterns rewarded by the MATH500 evaluation, where clarity of logical flow (aided by function words) is paramount. The consistent, monotonic nature of the trend across all ten performance brackets strongly indicates that this is a fundamental characteristic of the model's behavior on this task, not a statistical anomaly. The chart effectively argues that linguistic style, as measured by this simple content/function word ratio, is a key differentiator of performance.

## Bar Chart: R1-Qwen | MATH500 ### Overview The chart compares the ratio of Content Words and Function Words across performance tiers (90-100% to Top-10%) for the R1-Qwen model on the MATH500 dataset. Content Words are represented by red bars, while Function Words use gray diagonal stripes. The x-axis shows percentages (0-100%), and the y-axis categorizes performance ranges. ### Components/Axes - **X-Axis**: Ratio (%) (0 to 100, linear scale) - **Y-Axis**: Performance tiers (90-100%, 80-90%, ..., Top-10%) - **Legend**: - Red: Content Words - Gray (diagonal stripes): Function Words - **Data Labels**: Percentages embedded in red bars (e.g., "20.6%", "39.8%") ### Detailed Analysis 1. **Content Words (Red Bars)**: - **90-100%**: 20.6% - **80-90%**: 24.5% - **70-80%**: 27.5% - **60-70%**: 29.8% - **50-60%**: 31.9% - **40-50%**: 33.9% - **30-40%**: 36.0% - **20-30%**: 37.8% - **10-20%**: 38.9% - **Top-10%**: 39.8% 2. **Function Words (Gray Bars)**: - Calculated as 100% minus Content Words for each tier: - **90-100%**: 79.4% - **80-90%**: 75.5% - **70-80%**: 72.5% - **60-70%**: 70.2% - **50-60%**: 68.1% - **40-50%**: 66.1% - **30-40%**: 64.0% - **20-30%**: 62.2% - **10-20%**: 61.1% - **Top-10%**: 60.2% ### Key Observations - **Inverse Relationship**: Content Words increase monotonically (20.6% → 39.8%) as performance tiers decrease, while Function Words decrease correspondingly (79.4% → 60.2%). - **Steepest Gradient**: The largest shift occurs between 90-100% and 80-90% tiers (Content Words +3.9%, Function Words -3.9%). - **Top-10% Dominance**: Content Words reach their peak (39.8%) in the highest-performing tier, suggesting a correlation between content-focused language and performance. ### Interpretation The data implies that higher-performing tiers (e.g., Top-10%) rely more heavily on Content Words, which may reflect domain-specific knowledge or problem-solving relevance. Function Words dominate in lower tiers, potentially indicating generic or structural language use. This trend could highlight how R1-Qwen prioritizes substantive content over functional phrasing in high-stakes scenarios, aligning with its optimization for mathematical reasoning tasks. The consistent inverse relationship across all tiers suggests a systematic design choice favoring content density in high-performance outputs.