Interpreting Classical Readability Formulae as Surprisal
Yo Ehara
Abstract
Surprisal correlates closely with reading time and is widely used in the cognitive science of reading as a measure of processing difficulty. Classical readability formulae such as the Flesch-Kincaid Grade Level, which scores a text by the US school grade expected to read it, are the most widely used practical measures of text difficulty, but their coefficients are unitless regression constants and have never been related to surprisal. We theoretically show that these coefficients can be interpreted as surprisal: every formula of the classical additive shape is a sum of per-word and per-sentence cross-entropies under a memoryless model, and once that model is tied to the language through Zipf's law and the formula is read as its linearisation at a reference text, each coefficient factorises into a scale in grades per nat, a Zipf slope and a reference sentence length. A grade thereby becomes a quantity in surprisal. Because the formula outputs grades and children's vocabulary has been measured by grade, the interpretation recovers the vocabulary growth the formula implicitly assumes and lets it be checked against development: holding sentence length fixed, the word coefficient of the Flesch-Kincaid formula converts to 0.18 nat of unigram surprisal per word per grade (0.15 to 0.19 across the regression-derived formulae), and normative root-word vocabulary growth (Biemiller and Slonim, 2001), converted through Zipf's law, gives 0.16 nat per year at the frontier of the known vocabulary. The word coefficient is one year of vocabulary growth, and the grade is a vocabulary scale. The identification also yields a diagnostic (the sentence-length terms are not those of a memoryless surprisal) and extends to a Japanese formula.