Readability Formulas
Axis A computes ten standardized readability formulas and a consensus grade level. This reference details the linguistic features, mathematical inputs, and operational boundaries of each formula.
Linguistic Inputs
Section titled “Linguistic Inputs”Formulas compute weighted linear combinations of core structural and lexical features:
- Sentence length: Words per sentence ($W/S$). Longer sentences raise grade levels across nearly all formulas.
- Word complexity (syllables): Syllables per word ($S/W$) or percentage of polysyllabic words (3+ syllables). Used in Flesch, Flesch-Kincaid, Gunning Fog, and SMOG.
- Word complexity (characters): Characters or letters per 100 words ($L/100W$). Used in Coleman-Liau and ARI to eliminate syllable-counter inaccuracies.
- Lexical familiarity: Proportion of words outside a curated vocabulary list. Used in Dale-Chall (3,000 common words) and Spache (primary school vocabulary).
Understanding which feature a formula isolates helps guide text simplification:
- High Coleman-Liau: Shorten word lengths.
- High Gunning Fog: Replace multi-syllable jargon.
- High Dale-Chall: Swap specialized terms for familiar vocabulary.
Formula Reference
Section titled “Formula Reference”| Metric | Output Range | Linguistic Mechanics |
|---|---|---|
| Flesch Reading Ease | 0–100 (Higher = Simpler) | Combines words per sentence and syllables per word. Scores of 60–70 denote plain English; scores below 30 indicate legal/academic prose. |
| Flesch-Kincaid Grade | U.S. Grade Level | Re-expresses Flesch inputs as formal years of U.S. schooling. |
| Gunning Fog Index | U.S. Grade Level | Weights average sentence length against the percentage of complex words (≥3 syllables), heavily penalizing technical jargon. |
| SMOG Index | U.S. Grade Level | Counts polysyllabic words across a fixed 30-sentence sample. Standard in clinical, health, and consumer disclosures. |
| Coleman-Liau Index | U.S. Grade Level | Computes letters per 100 words and sentences per 100 words, relying entirely on character counts. |
| Automated Readability (ARI) | U.S. Grade Level | Computes characters per word and words per sentence. |
| Linsear Write | U.S. Grade Level | Developed for technical manuals; classifies words into easy (1–2 syllables) and hard (3+ syllables) categories over 100-word blocks. |
| Dale-Chall Score | 0.0–10.0+ | Measures the percentage of words outside a 3,000-word common vocabulary list alongside sentence length. |
| Spache Score | Primary Grade Level | Evaluates unfamiliar vocabulary and sentence length calibrated specifically for primary school texts (grades 1–4). |
| Text Standard | Consensus Grade | Aggregate consensus grade level across all applicable formulas. |
Consensus Grade Robustness
Section titled “Consensus Grade Robustness”Individual formulas diverge based on their structural weights. On the golden-set legal sample, Coleman-Liau reports 14.7 while Linsear Write reports 25.0 on identical text, representing a ten-grade spread. Reporting any single formula in isolation risks selective interpretation.
text_standard computes the consensus across formulas, dampening the variance of individual metrics. Cite text_standard as the primary grade level, noting specific formula outliers when diagnostic context is required.
Operational Boundaries
Section titled “Operational Boundaries”Inverse Scaling in Reading Ease
Section titled “Inverse Scaling in Reading Ease”flesch_reading_ease scales inversely with grade levels: higher scores indicate simpler reading. The golden-set primary sample scores 106.9 on Reading Ease alongside a negative grade level, whereas the academic sample scores -29.8 with a grade level exceeding 22. Always specify the direction of scale when citing metrics.
Sample Size Requirements
Section titled “Sample Size Requirements”Readability formulas require sufficient sample volume for stable lexical ratios. For passages under 100 words, docstats marks metrics as low confidence, and the Spache formula may return null. See Troubleshooting.
Structural Diagnostics
Section titled “Structural Diagnostics”Raw structural statistics (word_count, sentence_count, syllable_count) identify the causes of high grade levels:
- Sentence length: The golden-set legal sample averages 36 words per sentence (109 words across 3 sentences). Splitting compound sentences provides the fastest reduction in grade level.
- Syllable density: The academic sample contains 358 syllables in 140 words (2.56 syllables/word). Replacing multi-syllable terms directly lowers Flesch-Kincaid and SMOG scores.
For mapping scores to audience targets, see Interpreting Scores.