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Readability Indices Do Not Say It All on a Text Readability

Readability Indices Do Not Say It All on a Text Readability Article Emilio Matricciani Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB), Politecnico di Milano, 20133 Milan, Italy; emilio.matricciani@polimi.it Abstract: We propose a universal readability index, G , applicable to any alphabetical language and related to cognitive psychology, the theory of communication, phonics and linguistics. This index also considers readers’ short-term-memory processing capacity, here modeled by the word interval I , namely, the number of words between two interpunctions. Any current readability formula does not consider I , but scatterplots of I versus a readability index show that texts with the same p p readability index can have very different I , ranging from 4 to 9, practically Miller ’s range, which refers to 95% of readers. It is unlikely that I has no impact on reading difficulty. The examples shown are taken from Italian and English Literatures, and from the translations of The New Testament in Latin and in contemporary languages. We also propose an extremely compact formula, relating the capacity of human short-term memory to the difficulty of reading a text. It should synthetically model human reading difficulty, a kind of “footprint” of humans. However, further experimental and multidisciplinary work is necessary to confirm our conjecture about the dependence of a readability index on a reader ’s short-term-memory capacity. Keywords: alphabetical languages; ARI; English literature; Flesch Reading Ease Index; GULPEASE; human footprint; Italian literature; Miller ’s Law; short-term capacity; universal readability index; word interval 1. Introduction First developed in the United States [1–9], readability formulae are applicable to any alphabetical language. They are based on the length of words and sentences, and therefore they allow the comparison of different texts automatically and objectively to assess the Citation: Matricciani, E. Readability Indices Do Not Say It All on a Text difficulty that readers may find in reading them. From the point of view of the writer, a Readability. Analytics 2023, 2, readability formula allows the design of the best possible match between readers and texts. 296–314. https://doi.org/10.3390/ Many readability formulae have been proposed for English [6], and only some for very few analytics2020016 languages [10]. In Reference [11] we have defined a global readability formula applicable to any Academic Editor: Jong-Min Kim alphabetical language, based on a calque of the readability formula used in Italian [12], Received: 2 February 2023 both for providing it for languages that have none, and also for estimating, on common Revised: 2 March 2023 grounds, the readability of texts belonging to different languages/translations. Accepted: 21 March 2023 In fact, because an “absolute” readability formula—i.e., a formula that provides Published: 30 March 2023 numerical indices related to a universal origin, such as “zero”—might not exist at all, the readability formula proposed in Reference [11] can be used to compare different texts, because what counts, in this comparison, is the difference between numerical values. In other words, differences give more insight than absolute values for the purpose of Copyright: © 2023 by the author. comparing texts [11]. Licensee MDPI, Basel, Switzerland. As the title of this article claims, any current readability formula, however, does not say This article is an open access article everything about a text readability, because it neglects the response of readers’ short-term distributed under the terms and conditions of the Creative Commons memory to the partial stimuli contained in a sentence, i.e., to how the words of a sentence Attribution (CC BY) license (https:// are punctuated, a process described by the word interval I [13]. All readability formula creativecommons.org/licenses/by/ neglect, in fact, the empirical connection between the shortterm memory capacity of 4.0/). Analytics 2023, 2, 296–314. https://doi.org/10.3390/analytics2020016 https://www.mdpi.com/journal/analytics Analytics 2023, 2 297 readers (approximately described by Miller ’s 7  2 law [14]) and the word interval I , which appears, at least empirically, justified and natural [11,13,15–17]. The purpose of this article is to propose a universal readability formula, applicable to any alphabetical language, which includes the effect of short-term memory capacity. We base this formula on the global readability formula defined in Reference [11], which we will modify by including the word interval I . After this Introduction, Section 2 revisits the classical readability formula of Italian and its relationship with the Flesch Reading Ease Index and the Automated Readability Index, largely used in English texts; Section 3 summarizes the relationship between the word interval (number of words between two interpunctions, modeling the short-term memory capacity [13]) and the number of words per sentence; examples are drawn from Italian [13] and English literature [17]; Section 4 defines and discusses our proposal of a universal readability index; Section 5 proposes a synthetic readability index of humans, a kind of “footprint” that links human short-term memory to reading difficulty; finally Section 6 draws a conclusion and suggests future work. 2. A Readability Formula for Alphabetical Languages The observation that differences are more important than absolute values in using readability formulae [13] justifies the development of a readability formula that can be used to compare texts, even those written in different languages [15]. For most languages, in fact, no readability formula has been defined, and only few adapt English formulae to their texts [10,18]. The proposed formula, of course, does not exclude using other readability formulae specifically devised for a language—e.g., the large choice for English—[4,6] but it allows the comparison, on the same ground, of the readability of texts written in any language and in translation. For this purpose, we have proposed in Reference [11] to adopt, as a reference, the readability formula developed for Italian, known by the acronym GULPEASE [12]: G = 89 10C + 300/P (1) P F In Equation (1) C is the number of characters per word, and P , is the number P F of words per sentence. Notice that, like all readability formulae, Equation (1) does not contain any reference to interpunctions (besides, of course, full stops, question marks and exclamation marks, which determine the length of sentences), and therefore it does not consider the parameter very likely linked to the short–term memory capacity, namely the word interval I [13]. G can be interpreted as a readability index by considering the number of years of school attended in Italy’s school system (see Reference [12]), as shown in Figure 1. The larger G, the more readable the text for any number of school years. The continuous lines shown in Figure 1 divide the quadrant into areas of the same performance of texts, such as “almost unintelligible”, “very difficult”, etc. For example, the area labelled “easy” indicates all combinations of values of G and school years required to declare a text “easy” to read. In all cases, it is shown that, as the number of school years of the reader increases, the readability index he/she can tolerate decreases. In Reference [11] we have shown, for Italian literature, that the term 10C varies very little from text to text and across seven centuries, while the term 300/P varies very much and, in practice, determines the value of the readability index. Equation (1) says that a text is more difficult to read if P is large, i.e., if sentences are long, and if C is large, i.e., if words are long. In other words, a text is easier to read if it contains short words and short sentences, a result that is predicted by any known readability formula and should be true, of course, in any language. Analytics 2023, 2, FOR PEER REVIEW 3 Analytics 2023, 2 298 Figure 1. Readability index, G, of Italian (GULPEASE, see Reference [12]), as a function of the number Figure 1. Readability index, 𝐺 , of Italian (GULPEASE, see Reference [12]), as a function of the of school years attended in Italy. The continuous lines divide the quadrant into areas of the same number of school years attended in Italy. The continuous lines divide the quadrant into areas of the performance of texts. Elementary school lasts 5 years, junior high school lasts 3 years, and high school same performance of texts. Elementary school lasts 5 years, junior high school lasts 3 years, and high lasts 5 years. Children stay at school till they are 19 years old. For comparison, the green vertical axis school lasts 5 years. Children stay at school till they are 19 years old. For comparison, the green on the right refers to the Flesh Reading Ease index. vertical axis on the right refers to the Flesh Reading Ease index. In Reference [11], we have proposed the adoption of Equation (1) also for the other Equation (1) says that a text is more difficult to read if 𝑃 is large, i.e., if sentences languages, such as those listed in Table 1, by scaling the constant 10 according to the ratio are long, and if 𝐶 is large, i.e., if words are long. In other words, a text is easier to read if between the average number of characters per word in Italian, < C > = 4.48 and the p,I TA it contains short words and short sentences, a result that is predicted by any known average number of characters per word in another language, e.g., < C > = 4.24 for p,EN G readability formula and should be true, of course, in any language. English. The rationale for this choice is that C is a parameter typical of a language which, In Reference [11], we have proposed the adoption of Equation (1) also for the other if not scaled, would bias G without really quantifying the change in reading difficulty of languages, such as those listed in Table 1, by scaling the constant 10 according to the ratio readers, who are surely accustomed to reading, in their language, shorter or longer words, between the average number of characters per word in Italian, <𝐶 > = 4.48 and the on average, than those found in Italian. This scaling, therefore, avoids changing G for the average number of characters per word in another language, e.g., <𝐶 > = 4.24 for only reason that a language has, on average, words shorter or longer than , Italian. In any English. The rationale for this choice is that 𝐶 is a parameter typical of a language which, case, as recalled above, C affects a readability formula much less than P [13]. if not scaled, would bias 𝐺 without really quantifying the change in reading difficulty of readers, who are surely accustomed to reading, in their language, shorter or longer words, Table 1. Values of C and k of Equations (2) and (3) in the New Testament texts in the indicated on average, than those found in Italian. This scaling, therefore, avoids changing 𝐺 for the languages. Languages are listed according to their language family (see Reference [11]). only reason that a language has, on average, words shorter or longer than Italian. In any Language Language Family C k case, as recalled above, 𝐶 affects a readability formula much less than 𝑃 [13]. Greek Hellenic 4.86 0.92 Latin Italic 5.16 0.87 Table 1. Values of 𝐶 and 𝑘 of Equations (2) and (3) in the New Testament texts in the indicated Esperanto Constructed 4.43 1.01 languages. Languages are listed according to their language family (see Reference [11]). French Romance 4.20 1.07 Language Language Family 𝑪 𝒌 Italian Romance 4.48 1.00 Portuguese Romance 4.43 1.01 Greek Hellenic 4.86 0.92 Romanian Romance 4.34 1.03 Latin Italic 5.16 0.87 Spanish Romance 4.30 1.04 Esperanto Constructed 4.43 1.01 Danish Germanic 4.14 1.08 French Romance 4.20 1.07 English Germanic 4.24 1.06 Finnish Germanic 5.90 0.76 Italian Romance 4.48 1.00 German Germanic 4.68 0.96 Portuguese Romance 4.43 1.01 Analytics 2023, 2 299 Table 1. Cont. Language Language Family C k Icelandic Germanic 4.34 1.03 Norwegian Germanic 4.08 1.10 Swedish Germanic 4.23 1.06 Bulgarian BaltoSlavic 4.41 1.02 Czech BaltoSlavic 4.51 0.99 Croatian BaltoSlavic 4.39 1.02 Polish BaltoSlavic 5.10 0.88 Russian BaltoSlavic 4.67 0.96 Serbian BaltoSlavic 4.24 1.06 Slovak BaltoSlavic 4.65 0.96 Ukrainian BaltoSlavic 4.56 0.98 Estonian Uralic 4.89 0.92 Hungarian Uralic 5.31 0.84 Albanian Albanian 4.07 1.10 Armenian Armenian 4.75 0.94 Welsh Celtic 4.04 1.11 Basque Isolate 6.22 0.72 Hebrew Semitic 4.22 1.06 Cebuano Austronesian 4.65 0.96 Tagalog Austronesian 4.83 0.93 Chichewa NigerCongo 6.08 0.74 Luganda NigerCongo 6.23 0.72 Somali AfroAsiatic 5.32 0.84 Haitian French Creole 3.37 1.33 Nahuatl UtoAztecan 6.71 0.67 On the other hand, we have maintained the constant 300 because P depends signifi- cantly on author ’s style [13,15], not on language. Finally, notice that the constant 89 sets just the absolute ordinate scale, and therefore it has no impact on comparisons [13]. In conclusion, in Reference [11] we have defined a global readability index applicable to texts written in a language as: G = 89 10kC + 300/P (2) P F with k = < C >/< C > (3) P,I TA P By using Equations (2) and (3), we force the average value of 10  C of any language to be equal to that found in Italian, namely 10  4.48. Table 1 reports for Greek, Latin and 35 contemporary languages, the average values of C [11] and the calculated values of the constant k of Equation (3). For example, for English texts, C of a sample text is multiplied by 10.6, instead of 10; for Nahuatl (longer words), C is multiplied by 6.7, and for Haitian (shorter words) by 13.3. Notice that k seems to be a stable factor. For example, in the sample of the English literature studied in Reference [17], we have found < C > = 4.23 (instead of the 4.24 P,EN G of Table 1). Now, because the value found in the Italian literature [13] is < C > = 4.67, P,I TA therefore k = 4.67/4.23 = 1.10, instead of the k = 4.48/4.24 = 1.06 of Table 1. As recalled above, all readability formulae substantially tell the same story, and therefore they should be very similar and it is very likely that any one of them can be obtained from another. We illustrate this fact with an example. Because English is the language that has more readability formulae than any other language, let us compare G to the most classical English readability formula proposed and amply discussed by Flesch [1,2], known as the Flesch Reading Ease (RE) formula: RE = 206.8 1.015w 84.6s (4) Analytics 2023, 2, FOR PEER REVIEW 5 As recalled above, all readability formulae substantially tell the same story, and therefore they should be very similar and it is very likely that any one of them can be obtained from another. We illustrate this fact with an example. Because English is the language that has more readability formulae than any other language, let us compare 𝐺 to the most classical English readability formula proposed and amply discussed by Flesch [1,2], known as the Flesch Reading Ease ( ) formula: Analytics 2023, 2 300 = 206.8 − 1.015𝑤 − 84.6𝑠 (4) In Equation (4), 𝑤 is the average number of words per sentence, and 𝑠 is the average number of syllables per word. Because the number of characters per word is, on In Equation (4), w is the average number of words per sentence, and s is the average average, proportional to the number of syllables per word, the parameter 𝑠 paralles 𝐶 number of syllables per word. Because the number of characters per word is, on average, and, of course, 𝑤= 𝑃 . proportional to the number of syllables per word, the parameter s paralles C and, of How Equation (4) quantifies the degree of difficulty was defined by Flesch himself course, w = P . [1,2], and its values are reported in the vertical scale of Figure 1 (right ordinate scale), for How Equation (4) quantifies the degree of difficulty was defined by Flesch himself [1,2], comparison with 𝐺 (left ordinate scale). Figure 2 shows the scatterplot between the and its values are reported in the vertical scale of Figure 1 (right ordinate scale), for values calculated with the global readability index 𝐺 , Equation (2), versus those comparison with G (left ordinate scale). Figure 2 shows the scatterplot between the values calculated with , Equation (4), according to WinWord, in novels from English literature calculated with the global readability index G, Equation (2), versus those calculated with [17], Table 2. RE, Equation (4), according to WinWord, in novels from English literature [17], Table 2. Figure 2. Flesch Reading Ease (RE) index, Equation (4), versus the global index G, Equation (2), for Figure 2. Flesch Reading Ease (RE) index, Equation (4), versus the global index 𝐺 , Equation (2), for the novels of the English Literature listed in Table 2. Robinson Crusoe, cyan “o”; Pride and Prejudice, the novels of the English Literature listed in Table 2. Robinson Crusoe, cyan “o”; Pride and Prejudice, black “o”; Vanity Fair, blue “o”; Alice’s Adventures in Wonderland, magenta “o”; Treasure Island, green black “o”; Vanity Fair, blue “o”; Alice’s Adventures in Wonderland, magenta “o”; Treasure Island, green “o”; Adventures of Huckleberry Finn, red “+”; Peter Pan, blue “+”; The Sun Also Rises, green “+”; A “o”; Adventures of Huckleberry Finn, red “+”; Peter Pan, blue “+”; The Sun Also Rises, green “+”; A Farewell to Arms, black “+”. Farewell to Arms, black “+”. Table 2. Novels from English literature. Deep-language parameters C , P , I , G and universal P F P Table 2. Novels from English literature. Deep-language parameters 𝐶 , 𝑃 , 𝐼 , 𝐺 and universal readability index G , the latter discussed in Section 4. Novels are listed according to the year readability index 𝑮 , the latter discussed in Section 4. Novels are listed according to the year of of publication. publication. Literary Work C P I G G p F P U Literary Work 𝑷 𝑰 𝑮 𝑮 𝑭 𝑷 𝑼 Matthew King James translation (1611) 4.27 23.51 5.91 55.14 55.86 Matthew King James translation (1611) 4.27 23.51 5.91 55.14 55.86 Robinson Crusoe (D. Defoe, 1719) 3.94 57.75 7.12 50.84 42.22 Robinson Crusoe (D. Defoe, 1719) 3.94 57.75 7.12 50.84 42.22 Pride and Prejudice (J. Austen, 1813) 4.40 24.86 7.16 52.79 43.89 Wuthering Heights (E. Brontë, 1845–1846) 4.27 25.82 5.97 53.65 53.89 Vanity Fair (W. Thackeray, 1847–1848) 4.63 25.74 6.73 49.75 44.10 David Copperfield (C. Dickens, 1849–1850) 4.04 24.40 5.61 56.68 59.66 Moby Dick (H. Melville, 1851) 4.52 31.18 6.45 49.11 45.66 The Mill on The Floss (G. Eliot, 1860) 4.29 28.03 7.09 52.70 44.32 Alice’s Adventures in Wonderland (L. Carroll, 1865) 3.96 30.92 5.79 56.14 57.76 𝑅𝐸 𝑅𝐸 𝑅𝐸 Analytics 2023, 2 301 Table 2. Cont. Literary Work C P I G G p F P U Little Women (L.M. Alcott, 1868–1869) 4.18 21.08 6.30 57.31 54.99 Treasure Island (R. L. Stevenson, 1881–1882) 4.02 21.89 6.05 58.78 58.39 Adventures of Huckleberry Finn (M. Twain, 1884) 3.85 24.89 6.63 59.01 54.14 Three Men in a Boat (J.K. Jerome, 1889) 4.25 13.71 6.14 64.19 63.13 The Picture of Dorian Gray (O. Wilde, 1890) 4.19 16.56 6.29 62.83 60.58 The Jungle Book (R. Kipling, 1894) 4.11 21.52 7.15 57.95 49.14 The War of the Worlds (H.G. Wells, 1897) 4.38 20.85 7.67 55.31 42.48 The Wonderful Wizard of Oz (L.F. Baum, 1900) 4.02 20.55 7.63 59.38 46.85 The Hound of The Baskervilles (A.C. Doyle, 1901–1902) 4.15 17.79 7.83 60.27 46.16 Peter Pan (J.M. Barrie, 1902) 4.12 18.20 6.35 60.53 57.85 A Little Princess (F.H. Burnett, 1902–1905) 4.18 16.38 6.80 61.57 55.45 Martin Eden (J. London, 1908–1909) 4.32 16.94 6.76 59.38 53.50 Women in love (D.H. Lawrence, 1920) 4.26 13.71 5.22 63.98 70.02 The Secret Adversary (A. Christie, 1922) 4.28 11.02 5.52 69.08 72.76 The Sun Also Rises (E. Hemingway, 1926) 3.92 10.70 6.02 72.58 72.45 A Farewell to Arms (H. Hemingway,1929) 3.94 10.12 6.80 73.17 66.99 Of Mice and Men (J. Steinbeck, 1937) 4.02 9.67 5.61 74.20 77.24 We can notice a fair agreement between the two indices, with a correlation coefficient of 0.850. The bias could be compensated by downscaling RE. The attribution of the grade level G L in the USA school system was defined by Kincaid et al. [3], by using the same parameters w and s. The grade level is similar to that attributed to G. Another readability formula, the Automated Readability Index (ARI), was also defined by Kincaid et al. ii for specific military documents [3]. It is fully related to G because it depends on the same parameters, C and P : P F AR I = 4.71C + 0.5P 21.43 (5) p F As AR I increases, the age of required readers increases too. Figure 3 shows the scatterplot between the global G, Equation (2), and AR I, for the the same English novels considered in Figure 2. We can see a very tight relationship for fixed C . In conclusion, the global readability formula, Equation (2), provides a readability index that can be directly scaled to AR I and approximately also to RE. For this reason, we continue studying G, which we will modify by introducing the word interval I to obtain the universal readability formula/index mentioned above. To do so we need to recall, in the next section, some fundamental knowledge on I . P Analytics 2023, 2, FOR PEER REVIEW 7 Analytics 2023, 2 302 Figure 3. Automated Readability Index (ARI), Equation (5), versus the global index G, Equation (3), Figure 3. Automated Readability Index (ARI), Equation (5), versus the global index 𝐺 , Equation (3), for the novels of English literature listed in Table 2. The continuous lines assume constant values of for the novels of English literature listed in Table 2. The continuous lines assume constant values of C . Robinson Crusoe, cyan “o”; Pride and Prejudice, black “o”; Vanity Fair, blue “o”; Alice’s Adventures 𝐶 . Robinson Crusoe, cyan “o”; Pride and Prejudice, black “o”; Vanity Fair, blue “o”; Alice’s Adventures in Wonderland, magenta “o”; Treasure Island, green “o”; Adventures of Huckleberry Finn, red “+”; Peter in Wonderland, magenta “o”; Treasure Island, green “o”; Adventures of Huckleberry Finn, red “+”; Peter Pan, blue “+”; The Sun Also Rises, green “+”; A Farewell to Arms, black “+”. Pan, blue “+”; The Sun Also Rises, green “+”; A Farewell to Arms, black “+”. 3. Word Interval and Short-Term Memory In conclusion, the global readability formula, Equation (2), provides a readability As we have discussed in References [11,13,15], the word interval I ¯namely the number index that can be directly scaled to and approximately also to . For this reason, of words per interpunctions—varies in the same range of the short-term memory capacity- we continue studying 𝐺 , which we will modify by introducing the word interval 𝐼 to given by Miller ’s 7  2 law [14], a range that includes 95% of all cases, and very likely obtain the universal readability formula/index mentioned above. To do so we need to the two ranges are deeply related because interpunctions organize small portions of more recall, in the next section, some fundamental knowledge on 𝐼 . complex arguments (which make a sentence) in short chunks of text, which are the natural input to short-term memory [19–27]. Moreover, I , drawn against the number of words per 3. Word Interval and Short-Term Memory sentence, P , tends to approach a horizontal asymptote as P increases, and this occurs both F F in ancient As we have classical disc languages ussed in (Gr References eek and Latin) [11,13,15], th and in contemporary e word interval languages, 𝐼 —namely the as shown in References [11,13] by studying translations of the New Testament books from Greek. In number of words per interpunctions—varies in the same range of the short-term memory other words, even if sentences get longer, I cannot get larger than about the upper limit capacity-given by Miller’s 7±2 law [14], a ra pnge that includes 95% of all cases, and very of Millers’ law (namely 9), because of the constraints imposed by the short-term memory likely the two ranges are deeply related because interpunctions organize small portions capacity of readers and writers, as well. of more complex arguments (which make a sentence) in short chunks of text, which are The average value of I can be empirically related to the average value of P according the natural input to short-term p memory [19–27]. Moreover, 𝐼 , drawn against the F number to the non-linear relationship [13]: of words per sentence, 𝑃 , tends to approach a horizontal asymptote as 𝑃 increases, and this occurs both in ancient classical languages (Greek and Latin) and in contemporary (<P >1) (P 1) languages, as shown in References [11,13] by studying translations of the New Testament Fo < I >= ( I 1)  1 e + 1 (6) P P¥ books from Greek. In other words, even if sentences get longer, 𝐼 cannot get larger than about the upper limit of Millers’ law (namely 9), because of the constraints imposed by where I gives the horizontal asymptote, and P gives the value of < P > at which the P¥ Fo F the short-term memory capacity of readers and writers, as well. exponential falls at 1/e of its maximum value. The average value of 𝐼 can be empirically related to the average value of 𝑃 Equation (6) is a good average mathematical model for Italian literature [13] and also according to the non-linear relationship [13]: for Greek, Latin and contemporary languages [11,15]. Reference [11] reports the values of I and P for each language considered. P¥ Fo <𝐼 >= 𝐼 −1 ×1−𝑒 +1 (6) Presently, we have carried out the same analysis as for the large corpus of Italian literature [13] for a smaller but useful corpus of the English literature recently studied in 𝑅𝐸 𝐴𝑅𝐼 Analytics 2023, 2, FOR PEER REVIEW 8 where 𝐼 gives the horizontal asymptote, and 𝑃 gives the value of <𝑃 > at which the exponential falls at 1/𝑒 of its maximum value. Equation (6) is a good average mathematical model for Italian literature [13] and also for Greek, Latin and contemporary languages [11,15]. Reference [11] reports the values of Analytics 2023, 2 303 𝐼 and 𝑃 for each language considered. Presently, we have carried out the same analysis as for the large corpus of Italian literature [13] for a smaller but useful corpus of the English literature recently studied in Refer Reference [17], and ence [17], and have havecalculated calculated the the best-fit best-fitvalues values of of Equation (6 Equation (6).). Figur Figuree4 4 shows shows the the scatter scatter plo plottof of I 𝐼 versus versus P𝑃 (values (values ca calculated lculated for for each ch each chapter) apter) and and the the b best-fit est-fit c curve, urve,with with I𝐼 = =6 6.70 .70and and P 𝑃 ==6 6.78.7 ,8 to, be to b compar e comed pare with d with I = 𝐼 7.37 =7 and .37 P and = 𝑃10.22 =1 of0the .22 of Italian the P¥ Fo P¥ Fo literature, whose curve is also drawn. Italian literature, whose curve is also drawn. Figure 4. Scatter plot of the word interval, I , and the number of words per sentence, P , for all P F Figure 4. Scatter plot of the word interval, 𝐼 , and the number of words per sentence, 𝑃 , for all samples of the novels considered in the English literature, Table 2. The continuous black line refers to samples of the novels considered in the English literature, Table 2. The continuous black line refers the best fit given by Equation (6). The green line refers to the Italian literature [13]. Miller ’s bounds to the best fit given by Equation (6). The green line refers to the Italian literature [13]. Miller’s bounds are given by I = 7  2. are given by 𝐼 =7 ∓ 2. Notice that the constants of the English literature differ from those reported in Notice that the constants of the English literature differ from those reported in Reference [17] ( I = 6.57, P = 4.16) for the same literary corpus, because the latter P¥ Fo Reference [17] (𝐼 =6.57, 𝑃 =4.16) for the same literary corpus, because the latter were the results of fitting Equation (6) to the average values of I and P , not to the values P F were the results of fitting Equation (6) to the average values of 𝐼 and 𝑃 , not to the values of I and P obtained by considering the samples (a sample for each chapter), which give P F of 𝐼 and 𝑃 obtained by considering the samples (a sample for each chapter), which give the scatterplot drawn in Figure 4. The different values are due, of course, to the non-linear the scatterplot drawn in Figure 4. The different values are due, of course, to the non-linear best fit. best fit. Now, as we have recalled in Section 2, any readability index is practically a function only Now, as we have recalled in Section 2, any readability index is practically a function of P . Readability formulae do not consider I , but the scatterplots of I versus G show an p p only of 𝑃 . Readability formulae do not consider 𝐼 , but the scatterplots of 𝐼 versus 𝐺 interesting story: texts with the same G do not show the same I . In other words, according show an interesting story: texts with the same 𝐺 do not show the same 𝐼 . In other words, to the theory of readability formulae, a text with a given index should be readable with the according to the theory of readability formulae, a text with a given index should be same effort both by readers who display a powerful short-term memory processing capacity readable with the same effort both by readers who display a powerful short-term memory (large I ) and by readers who do not (small I ). For example, for G = 60 (“easy/standard” p p processing capacity (large 𝐼 ) and by readers who do not (small 𝐼 ). For example, for 𝐺= texts for readers with 8 years of school, Figure 1), Figure 5 shows th at I can vary from 4 to 60 (“easy/standard” texts for readers with 8 years of school, Figure 1), Figure 5 shows that 9. This is practically Miller ’s range, which refers to 95% of readers [14]. We think that these readers should be distinguished, and therefore, our aim is to propose, in the next section, a possible “universal” readability index, G , based on G, which includes I . U p Analytics 2023, 2, FOR PEER REVIEW 9 𝐼 can vary from 4 to 9. This is practically Miller’s range, which refers to 95% of readers [14]. We think that these readers should be distinguished, and therefore, our aim is to Analytics 2023, 2 304 propose, in the next section, a possible “universal” readability index, 𝐺 , based on 𝐺 , which includes 𝐼 . (a) (b) Figure 5. Scatterplots of the readability index, 𝐺 , versus the word interval, 𝐼 . (a): Italian Literature Figure 5. Scatterplots of the readability index, G, versus the wor d interval, I . (a): Italian Litera- [11]; (b): English Literature, Table 2. Miller’s bounds are given by 𝐼 =7 ∓ 2. ture [11]; (b): English Literature, Table 2. Miller ’s bounds are given by I = 7  2. 4. A Universal Readability Formula 4. A Universal Readability Formula We suppose that the global readability index G should be modified by introducing We suppose that the global readability index 𝐺 should be modified by introducing a function that depends linearly on I . Our hypothesis is based on Miller ’s law, which a function that depends linearly on 𝐼 . Our hypot P hesis is based on Miller’s law, which quantifies linearly the processing capacity of the short-term memory. Moreover, the function quantifies linearly the processing capacity of the short-term memory. Moreover, the should not change the global value for a reader with an “average” processing short-term memory capacity. For words, this average is not 7, but about 6 [1,28]; therefore, in the following we assume this latter value. Notice that 6.03 is the average value of I (standard deviation 1.11) of the data listed in Table 2 of Reference [11], a further indication of its barycentric value. Analytics 2023, 2 305 We write our proposed universal readability formula as: DG G = G I 6 (7) ( ) U P D I where G is given by Equation (2). DG We assume that the numerical value of the discrete derivative is given by: D I DG G G max min = (8) D I I I P P,max P,min In Equation (8), the numerical values are the maximum and minimum averages found in the Italian literature—see Reference [13], whose oldest texts (seven centuries old, e.g., Boccaccio’s Decameron) are still read today in Italian high schools with a reasonable effort, a possibility not available in other Western languages. From [13], we calculate: DG 69.84 49.54 = = 6.15  6.00 (9) D I 8.24 4.94 Therefore, the proposed universal readability formula is given by G = G 6( I 6) (10) U P Equation (10) sets G = G for I = 6; G < G for I > 6 and G > G for I < 6. In U P U P U P other words, if a text with a given G, has a small word interval I , then it should be read more easily than a text with the same G, but larger I . For example, texts with G = 60 would be transformed in Miller ’s range of 5 to 9 to G = 66 for I = 5 and in G = 42 U P U for I = 9, and therefore, in the first case, the text considered “easy” after 8 years of school (Figure 1), is considered “easy” to read but only after 7.2 years of school; in the second case, the text would be considered “easy”, but only after about 13.2 years of school. The meaningful difference between the two indices is therefore very large: 66 42 = 24, corresponding to 13.2 7.2 = 5 years of school. This significant difference would be lost in the original formula of Equation (2), or in any other readability formula. Figure 6 shows the scatterplots between G and I (blue circles) for the samples U P concerning the literary texts considered in Italian [13] and in English Literatures, in Table 2. Compared to the scatterplots of Figure 5 (redrawn in Figure 6 with red circles), the difference between G and G is evident: the linear dependence of G on I , according to Equation (10), U U P spreads the values around a line and introduces significant correlation coefficients, 0.9016 for Italian, and 0.7730 for English. The regression line: G = a I + b (11) U P is very similar in the two languages: G = 9.47I + 115.71 (12) U,I TA P G = 8.88I + 111.64 (13) U,EN G P This result indicates that Equation (11) might be “universal”. Finally, some specific examples concerning novels taken from Italian and English literatures will further illustrate the relationship between G and G . U Analytics 2023, 2, FOR PEER REVIEW 11 Analytics 2023, 2 306 (a) (b) Figure 6. Scatt Figure er plot 6. Scatter of theplot readability in of the readability dex, 𝐺 index, , versus the G , versus word interval, the word interval, 𝐼 (blue Icirc (blue les).cir The cles). The red U P red circles refer to the scatterplots of Figure 5. (a): Italian Literature [13]; (b): English Literature, circles refer to the scatterplots of Figure 5. (a): Italian Literature [13]; (b): English Literature, Table 1. Table 1. Miller’s bounds are given by 𝐼 =7 ∓ 2. Miller ’s bounds are given by I = 7  2. Finally, some specific examples concerning novels taken from Italian and English Table 3 shows how the readability index is modified from G to G for some Italian literatures will further illustrate the relationship between 𝐺 and 𝐺 . novels written from the XIV to the XX century [13]. For example, it is interesting to notice Table how 3 sho Gw is s how th transformed e readability into G inde for the x is mod two novels ified from written 𝐺 to by 𝐺 Alessandr for some It o Manzoni. alian novels written from the XIV to the XX century [13]. For example, it is interesting to notice how 𝐺 is transformed into 𝐺 for the two novels written by Alessandro Manzoni. Analytics 2023, 2 307 Table 3. Novels from Italian Literature [13]. Average deep-language parameters C , P , I , and G P F P and corresponding universal readability index, G . Novels are listed according to the alphabetical order of the author‘s name. Novel C P I G G P F P U Anonymous (I Fioretti di San Francesco, XIV Century) 4.65 37.70 8.24 50.70 37.26 Boccaccio Giovanni (Decameron, XIV) 4.48 44.27 7.79 51.18 40.44 Buzzati Dino (Il deserto dei tartari, XX) 5.10 17.75 6.63 55.27 51.49 Calvino Italo (Marcovaldo, XX) 4.74 17.60 6.59 59.19 55.65 Cassola Carlo (La ragazza di Bube, XX) 4.48 11.93 5.64 69.84 72.00 Collodi Carlo (Pinocchio, XIX) 4.60 16.92 6.19 61.57 60.43 Deledda Grazia (Canne al vento, XX) 4.51 15.08 6.06 64.39 64.03 D’Annunzio Gabriele (Le novelle delle Pescara, XX) 4.91 17.99 6.38 58.16 55.88 Eco Umberto (Il nome della rosa, XX) 4.81 21.08 7.46 55.78 47.02 Fogazzaro (Piccolo mondo antico, XIX-XX) 4.79 16.08 6.10 61.46 60.86 Gadda (Quer pasticciaccio brutto . . . XX) 4.76 18.43 4.98 58.24 64.36 Machiavelli Niccolò (Il principe, XV-XVI) 4.71 40.17 6.45 49.54 46.84 Manzoni Alessandro (Fermo e Lucia, XIX) 4.75 30.98 7.17 51.72 44.70 Manzoni Alessandro (I promessi sposi, XIX) 4.60 24.83 5.30 56.00 60.20 Moravia Alberto (La ciociara, XX) 4.56 29.93 7.28 53.52 45.84 Pavese Cesare (La luna e i falò, XX) 4.47 17.83 6.83 61.90 56.92 Pirandello Luigi (Il fu Mattia Pascal) 4.63 14.57 4.94 63.94 70.30 Svevo Italo (Senilità, XX) 4.86 16.04 7.75 59.39 48.89 Tomasi di Lampedusa (Il gattopardo, XX) 4.99 26.42 7.90 50.72 39.32 Verga (I Malavoglia, XIX-XX) 4.46 20.45 6.82 59.34 54.42 Alessandro Manzoni (Milan 1785, Milan 1873), one of the most studied Italian novel- ist in Italian high schools (Licei) and universities, in 1827 published Fermo e Lucia (Fermo and Lucia), a text that scholars of Italian Literature—and Manzoni himself—consider the “first” version of his masterpiece I Promessi Sposi (The Betrothed, available in a new English translation [29]) published in the years 1840–1842. According to scholars of Italian litera- ture [30–33], the two versions differ very much, both in story structure and characters and, as far as we are here concerned, also in style and language; therefore, it is interesting to see how much the author transformed (mathematically) Fermo e Lucia into I Promessi Sposi, a study partially carried out in References [13,15]. As far as readability is concerned, from Table 3 we notice a large improvement in I Promessi Sposi, compared to Fermo e Lucia, if differences are considered. In fact, G = 51.72 in Fermo e Lucia and G = 56.00 in I Promessi Sposi, a difference of only 4.28 units, leading to a decrease in school years (for “easy” reading, Figure 1) of only about 0.8 years. This difference does not justify the reading difficulty of the two texts discussed by scholars of Italian literature [30–33]. However, if we consider G , then the difference is quite large, very likely measuring the relative reading difficulty, because G ranges from 44.70 to 60.20, a difference of 15.5 units leading to a decrease in school years (for “easy” reading, Figure 1) from 11.8 (Fermo e Lucia) to only 8 (I Promessi Sposi), well justified by scholars of Italian literature [30–33]. In conclusion, G is a better estimate than G in assessing the difference in reading difficulty between these two very studied novels. Table 2 shows also how the readability index is modified from G to G in some English novels. As we can read from Table 2, in Robinson Crusoe the readability index decreases from 50.84 to 42.22, therefore passing from about 10.3 to 12.4 years of school for “easy” reading Analytics 2023, 2 308 Analytics 2023, 2, FOR PEER REVIEW(Figur e 1). For Hemingway’s novels, The Sun Also Rises is more readable (72.45) 13 than A Farewell to Arms (66.99); the order given by G, i.e., 72.58 and 73.17, respectively, is reversed, therefore reducing the number of years of school required for “easy” reading by 1 (Figure 1). The Hound of The Baskervilles changes its readability index from 60.27 to 46.16, therefore reversed, therefore reducing the number of years of school required for “easy” reading by 1 (Figure 1). The Hound of The Baskervilles changes its readability index from 60.27 to 46.16, passing from 8 to 11.5 years of school for “easy” reading (Figure 1). therefore passing from 8 to 11.5 years of school for “easy” reading (Figure 1). In conclusion, by introducing the word interval I in the definition of a readability In conclusion, by introducing the word interval 𝐼 in the definition of a readability index, as in Equation (10), readability differences in texts are more “fine-tuned” for readers. index, as in Equation (10), readability differences in texts are more “fine-tuned” for readers. 5. A “Footprint” of Humans As already recalled, in Reference [11] we have studied the translation of the New 5. A “Footprint” of Humans Testament from Greek to Latin and to contemporary languages. For all these translations, As already recalled, in Reference [11] we have studied the translation of the New we have recently calculated the scatterplots between G and I , and between G and P U Testament from Greek to Latin and to contemporary languages. For all these translations, I , with results very similar to those shown in Figure 6. Some specific examples are we have recently calculated the scatterplots between 𝐺 and 𝐼 , and between 𝐺 and 𝐼 , reported in Appendix A. Similarly, we have calculated the linear best fit between G with results very similar to those shown in Figure 6. Some specific examples are reported and I . Appendix B lists the values of the constants a and b of Equation (11) for each in Appendix A. Similarly, we have calculated the linear best fit between 𝐺 and 𝐼 . translation/language. Appendix B lists the values of the constants 𝑎 and 𝑏 of Equation (11) for each translThis ation/ set langua of values ge. are useful because they could be used to compare texts written in any language. This seFor t of v example, alues are useful bec in David Copperfield ause they could be , G is estimated used to compare tex to be 61.82 ts written in with Equation (13) any language. For example, in David Copperfield, 𝐺 is estimated to be 61.82 with Equation and 66.02 with the values of Appendix B (the experimental average value is 59.66, Table 2); (13) and 66.02 with the values of Appendix B (the experimental average value is 59.66, in The Hound of The Baskervilles, G is estimated to be 42.11 with Equation (13) and 48.53 Table 2); in The Hound of The Baskervilles, 𝐺 is estimated to be 42.11 with Equation (13) with the values of Appendix B (the experimental average value is 46.16, Table 2). In the and 48.53 with the values of Appendix B (the experimental average value is 46.16, Table first case, the difference in the readability of the two novels is 19.71, and in the second case 2). In the first case, the difference in the readability of the two novels is 19.71, and in the it is 17.49, which implies an “error” of about 0.25 years of school (Figure 1). second case it is 17.49, which implies an “error” of about 0.25 years of school (Figure 1). It may be interesting to consider the most compact relationship between G and I , U P It may be interesting to consider the most compact relationship between 𝐺 and 𝐼 , given by the overall average values of the constants reported in Appendix B: given by the overall average values of the constants reported in Appendix B: (14) 𝐺 = −8.94𝐼 + 116 G = 8.94I + 116 (14) U P Figure 7 show this average relationship together with ±1 standard deviaton Figure 7 show this average relationship together with 1 standard deviaton bounds. bounds. These extremely compacted curves can synthetically represent how the capacity These extremely compacted curves can synthetically represent how the capacity of human of human short-term memory (modelled by 𝐼 ) is related to the difficulty of reading a text, short-term memory (modelled by I ) is related to the difficulty of reading a text, in any in any alphabetical language; therefore, Pit may be considered as a kind of “footprint” of alphabetical language; therefore, it may be considered as a kind of “footprint” of humans. humans. Figure 7. Average (blue line) and 1 standard deviation lines (cyan lines) of the universal readability index, G , versus the word interval, I , from Table A1. U P Analytics 2023, 2 309 6. Conclusions We have proposed a universal readability index, G , Equation (10). Compared to the current readability indices, this index considers also readers’ short-term memory processing capacity, here described by the word interval I , namely, the number of words between two interpunctions. The observation that differences give more insight than absolute values has justified, we think, the development of a universal readability formula which is useful for comparing texts written even in different languages and is applicable to alphabetical languages and related to cognitive psychology, the theory of communication, phonics and linguistics. Scholars have never considered including in the current readability formulae the word interval, I , but the scatterplots of I versus any readability index show that texts with the p p same readability index can have very different values of I . Now, it is unlikely that I has p P no impact on reading difficulty. By introducing I in the definition of a readability index, readability differences in texts are better “fine-tuned” for readers, e.g., to their school years as a reference. We have used the global readability index developed for Italian [11], after showing that Flesch’s index and ARI are connected to this index because they depend on the same variables. We have calculated an extremely compact formula, Equation (14), which can measure how the capacity of human short-term memory (modelled by I ) is likely related to the difficulty of reading a text, measured by the universal readability index G , here defined. We think that it synthetically models human reading difficulty, i.e., it might be considered a “footprint” of humans. However, there is an important aspect to be considered. Because, as far as we know, there are no direct experiments on the relationship between readability and short-term memory capacity, the universal index here proposed, Equation (10), should be considered a first step in researching this important relationship. Therefore, further work needs to be carried out by a multidisciplinary team of researchers to fully validate Equation (10). Funding: This research received no external funding. Data Availability Statement: Data are available, on request, by the author. Conflicts of Interest: The authors declare no conflict of interest. Appendix A Scatterplots between G and I , for selected languages. We show the scatterplots be- U P tween G and I (red circles), and between G and I , (blue circles) for some selected languages. P U P Analytics 2023, 2, FOR PEER REVIEW 15 Analytics 2023, 2 310 (a) (b) Figure A1. Scatter plot of the readability index G versus the word interval I (blue circles). The red Figure A1. Scatter plot of the readability index 𝐺 versus the word interval 𝐼 (blue circles). The U P red circles refer to the circles refer to scatterplots of the scatterplots 𝐺 of versus G versus 𝐼 . ( I a . )(:a Greek; ): Greek; (b() b : L ): a Latin. tin. M Miller iller’s ’s b bounds ounds a arre e given givenby by 𝐼 =7 ∓ I 2= . Th 7  e 2 valu . The esvalues of 𝑎 and of a and 𝑏 of b of Equation (12) Equation (12) a and nd the the corr correlat elation ion coe coefficient fficient between between G and 𝐺 I P U P and 𝐼 are repor are reported ted inin Table Table A1 A1 . . Analytics 2023, 2, FOR PEER REVIEW 16 Analytics 2023, 2 311 (a) (b) Figure A2. Scatter plot of the readability index G versus the word interval I (blue circles). The red Figure A2. Scatter plot of the readability index 𝐺 versus the word U interval 𝐼 P(blue circles). The red circles refe circles r to the scatter refer to the scatterplots plots of of of G 𝐺 ver versus sus I 𝐼 . (. ( a):aSpanish; ): Spanish; ( (b): Portuguese. b): Portugues Miller e. Mil ’s bounds ler’s bound are given s by I = 7  2. The values of a and b of Equation (12) and the correlation coefficient between G and are given by 𝐼 =7 ∓ 2. The values of 𝑎 and 𝑏 of Equation (12) and the correlation coefficient be- P U I are reported in Table A1. tween 𝐺 and 𝐼 are reported in Table A1. Analytics 2023, 2, FOR PEER REVIEW 17 Analytics 2023, 2 312 (a) (b) Figure A3. Scatter plot of the readability index G versus the word interval I (blue circles). The red Figure A3. Scatter plot of the readability index 𝐺 versus the word U interval 𝐼 P(blue circles). The circles refer to the scatterplots of G versus I . (a): French; (b): German. Miller ’s bounds are given by red circles refer to the scatterplots of 𝐺 versus 𝐼 P. (a): French; (b): German. Miller’s bounds are I = 7  2. The values of a and b of Equation (13) and the correlation coefficient between G and I given by 𝐼 =7 ∓ 2. The values of 𝑎 and 𝑏 of Equation (13) and the correlation coefficient between P U P are reported in Table A1. 𝐺 and 𝐼 are reported in Table A1. Analytics 2023, 2 313 Appendix B Table A1. Values of a and b of Equation (12), and correlation coefficient between G and I for the U P indicated languages [11]. Language a b Correlation Coefficient Greek 8.62 113.66 0.9477 Latin 10.59 120.82 0.8666 Esperanto 9.87 114.20 0.8803 French 7.46 107.51 0.9311 Italian 7.80 108.54 0.9065 Portuguese 8.34 112.33 0.8261 Romanian 8.08 111.11 0.8163 Spanish 8.46 112.60 0.9061 Danish 9.46 120.71 0.9182 English 7.88 110.23 0.9129 Finnish 10.06 118.22 0.8057 German 8.68 113.79 0.8563 Icelandic 8.68 114.98 0.8848 Norwegian 7.32 110.28 0.9426 Swedish 7.32 109.98 0.9546 Bulgarian 9.00 117.63 0.8697 Czech 10.41 125.50 0.8269 Croatian 9.86 122.33 0.8868 Polish 9.98 123.60 0.7160 Russian 10.70 118.04 0.7326 Serbian 8.71 117.24 0.8312 Slovak 10.03 124.83 0.8417 Ukrainian 8.34 113.42 0.7092 Estonian 9.97 120.11 0.8643 Hungarian 10.83 118.91 0.8034 Albanian 8.01 107.04 0.8776 Armenian 12.11 133.87 0.7805 Welsh 7.74 103.12 0.7828 Basque 9.99 117.48 0.8361 Hebrew 10.27 129.58 0.8163 Cebuano 6.97 107.50 0.9683 Tagalog 7.78 112.54 0.9188 Chichewa 8.40 118.76 0.9325 Luganda 8.69 118.42 0.8713 Somali 8.65 113.41 0.9492 Haitian 8.25 115.41 0.9132 Nahuatl 7.55 113.02 0.9420 Overall 8.94  1.22 116.00  6.49 0.8681  0.0661 References 1. Flesch, R. A New Readability Yardstick. J. Appl. Psychol. 1948, 32, 222–233. [CrossRef] [PubMed] 2. Flesch, R. The Art of Readable Writing; revised and enlarged edition; Harper & Row: New York, NY, USA, 1974. 3. Kincaid, J.P.; Fishburne, R.P.; Rogers, R.L.; Chissom, B.S. Derivation of New Readability Formulas (Automated Readability Index, Fog Count and Flesch Reading Ease Formula) For Navy Enlisted Personnel; Research Branch Report 8-75; Chief of Naval Technical Training, Naval Air Station: Memphis, TN, USA, 1975. 4. DuBay, W.H. The Principles of Readability; Impact Information: Costa Mesa, CA, USA, 2004. 5. Bailin, A.; Graftstein, A. The linguistic assumptions underlying readability formulae: A critique. Lang. Commun. 2001, 21, 285–301. [CrossRef] 6. DuBay, W.H. (Ed.) The Classic Readability Studies; Impact Information: Costa Mesa, CA, USA, 2006. 7. Zamanian, M.; Heydari, P. Readability of Texts: State of the Art. Theory Pract. Lang. Stud. 2012, 2, 43–53. [CrossRef] 8. Benjamin, R.G. 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Studi Sulle Redazioni de I Promessi Sposi; Edizioni Paoline: Milan, Ialy, 1968. 31. Giovanni Nencioni, N. La Lingua di Manzoni. Avviamento Alle Prose Manzoniane; Il Mulino: Bologna, Italy, 1993. 32. Guntert, G. Manzoni Romanziere: Dalla Scrittura Ideologica Alla Rappresentazione Poetica; Franco Cesati Editore: Firenze, Italy, 2000. 33. Frare, P. Leggere I Promessi Sposi; Il Mulino: Bologna, Italy, 2016. Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analytics Multidisciplinary Digital Publishing Institute

Readability Indices Do Not Say It All on a Text Readability

Matricciani, Emilio
Analytics , Volume 2 (2) – Mar 30, 2023
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Abstract

Article Emilio Matricciani Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB), Politecnico di Milano, 20133 Milan, Italy; emilio.matricciani@polimi.it Abstract: We propose a universal readability index, G , applicable to any alphabetical language and related to cognitive psychology, the theory of communication, phonics and linguistics. This index also considers readers’ short-term-memory processing capacity, here modeled by the word interval I , namely, the number of words between two interpunctions. Any current readability formula does not consider I , but scatterplots of I versus a readability index show that texts with the same p p readability index can have very different I , ranging from 4 to 9, practically Miller ’s range, which refers to 95% of readers. It is unlikely that I has no impact on reading difficulty. The examples shown are taken from Italian and English Literatures, and from the translations of The New Testament in Latin and in contemporary languages. We also propose an extremely compact formula, relating the capacity of human short-term memory to the difficulty of reading a text. It should synthetically model human reading difficulty, a kind of “footprint” of humans. However, further experimental and multidisciplinary work is necessary to confirm our conjecture about the dependence of a readability index on a reader ’s short-term-memory capacity. Keywords: alphabetical languages; ARI; English literature; Flesch Reading Ease Index; GULPEASE; human footprint; Italian literature; Miller ’s Law; short-term capacity; universal readability index; word interval 1. Introduction First developed in the United States [1–9], readability formulae are applicable to any alphabetical language. They are based on the length of words and sentences, and therefore they allow the comparison of different texts automatically and objectively to assess the Citation: Matricciani, E. Readability Indices Do Not Say It All on a Text difficulty that readers may find in reading them. From the point of view of the writer, a Readability. Analytics 2023, 2, readability formula allows the design of the best possible match between readers and texts. 296–314. https://doi.org/10.3390/ Many readability formulae have been proposed for English [6], and only some for very few analytics2020016 languages [10]. In Reference [11] we have defined a global readability formula applicable to any Academic Editor: Jong-Min Kim alphabetical language, based on a calque of the readability formula used in Italian [12], Received: 2 February 2023 both for providing it for languages that have none, and also for estimating, on common Revised: 2 March 2023 grounds, the readability of texts belonging to different languages/translations. Accepted: 21 March 2023 In fact, because an “absolute” readability formula—i.e., a formula that provides Published: 30 March 2023 numerical indices related to a universal origin, such as “zero”—might not exist at all, the readability formula proposed in Reference [11] can be used to compare different texts, because what counts, in this comparison, is the difference between numerical values. In other words, differences give more insight than absolute values for the purpose of Copyright: © 2023 by the author. comparing texts [11]. Licensee MDPI, Basel, Switzerland. As the title of this article claims, any current readability formula, however, does not say This article is an open access article everything about a text readability, because it neglects the response of readers’ short-term distributed under the terms and conditions of the Creative Commons memory to the partial stimuli contained in a sentence, i.e., to how the words of a sentence Attribution (CC BY) license (https:// are punctuated, a process described by the word interval I [13]. All readability formula creativecommons.org/licenses/by/ neglect, in fact, the empirical connection between the shortterm memory capacity of 4.0/). Analytics 2023, 2, 296–314. https://doi.org/10.3390/analytics2020016 https://www.mdpi.com/journal/analytics Analytics 2023, 2 297 readers (approximately described by Miller ’s 7  2 law [14]) and the word interval I , which appears, at least empirically, justified and natural [11,13,15–17]. The purpose of this article is to propose a universal readability formula, applicable to any alphabetical language, which includes the effect of short-term memory capacity. We base this formula on the global readability formula defined in Reference [11], which we will modify by including the word interval I . After this Introduction, Section 2 revisits the classical readability formula of Italian and its relationship with the Flesch Reading Ease Index and the Automated Readability Index, largely used in English texts; Section 3 summarizes the relationship between the word interval (number of words between two interpunctions, modeling the short-term memory capacity [13]) and the number of words per sentence; examples are drawn from Italian [13] and English literature [17]; Section 4 defines and discusses our proposal of a universal readability index; Section 5 proposes a synthetic readability index of humans, a kind of “footprint” that links human short-term memory to reading difficulty; finally Section 6 draws a conclusion and suggests future work. 2. A Readability Formula for Alphabetical Languages The observation that differences are more important than absolute values in using readability formulae [13] justifies the development of a readability formula that can be used to compare texts, even those written in different languages [15]. For most languages, in fact, no readability formula has been defined, and only few adapt English formulae to their texts [10,18]. The proposed formula, of course, does not exclude using other readability formulae specifically devised for a language—e.g., the large choice for English—[4,6] but it allows the comparison, on the same ground, of the readability of texts written in any language and in translation. For this purpose, we have proposed in Reference [11] to adopt, as a reference, the readability formula developed for Italian, known by the acronym GULPEASE [12]: G = 89 10C + 300/P (1) P F In Equation (1) C is the number of characters per word, and P , is the number P F of words per sentence. Notice that, like all readability formulae, Equation (1) does not contain any reference to interpunctions (besides, of course, full stops, question marks and exclamation marks, which determine the length of sentences), and therefore it does not consider the parameter very likely linked to the short–term memory capacity, namely the word interval I [13]. G can be interpreted as a readability index by considering the number of years of school attended in Italy’s school system (see Reference [12]), as shown in Figure 1. The larger G, the more readable the text for any number of school years. The continuous lines shown in Figure 1 divide the quadrant into areas of the same performance of texts, such as “almost unintelligible”, “very difficult”, etc. For example, the area labelled “easy” indicates all combinations of values of G and school years required to declare a text “easy” to read. In all cases, it is shown that, as the number of school years of the reader increases, the readability index he/she can tolerate decreases. In Reference [11] we have shown, for Italian literature, that the term 10C varies very little from text to text and across seven centuries, while the term 300/P varies very much and, in practice, determines the value of the readability index. Equation (1) says that a text is more difficult to read if P is large, i.e., if sentences are long, and if C is large, i.e., if words are long. In other words, a text is easier to read if it contains short words and short sentences, a result that is predicted by any known readability formula and should be true, of course, in any language. Analytics 2023, 2, FOR PEER REVIEW 3 Analytics 2023, 2 298 Figure 1. Readability index, G, of Italian (GULPEASE, see Reference [12]), as a function of the number Figure 1. Readability index, 𝐺 , of Italian (GULPEASE, see Reference [12]), as a function of the of school years attended in Italy. The continuous lines divide the quadrant into areas of the same number of school years attended in Italy. The continuous lines divide the quadrant into areas of the performance of texts. Elementary school lasts 5 years, junior high school lasts 3 years, and high school same performance of texts. Elementary school lasts 5 years, junior high school lasts 3 years, and high lasts 5 years. Children stay at school till they are 19 years old. For comparison, the green vertical axis school lasts 5 years. Children stay at school till they are 19 years old. For comparison, the green on the right refers to the Flesh Reading Ease index. vertical axis on the right refers to the Flesh Reading Ease index. In Reference [11], we have proposed the adoption of Equation (1) also for the other Equation (1) says that a text is more difficult to read if 𝑃 is large, i.e., if sentences languages, such as those listed in Table 1, by scaling the constant 10 according to the ratio are long, and if 𝐶 is large, i.e., if words are long. In other words, a text is easier to read if between the average number of characters per word in Italian, < C > = 4.48 and the p,I TA it contains short words and short sentences, a result that is predicted by any known average number of characters per word in another language, e.g., < C > = 4.24 for p,EN G readability formula and should be true, of course, in any language. English. The rationale for this choice is that C is a parameter typical of a language which, In Reference [11], we have proposed the adoption of Equation (1) also for the other if not scaled, would bias G without really quantifying the change in reading difficulty of languages, such as those listed in Table 1, by scaling the constant 10 according to the ratio readers, who are surely accustomed to reading, in their language, shorter or longer words, between the average number of characters per word in Italian, <𝐶 > = 4.48 and the on average, than those found in Italian. This scaling, therefore, avoids changing G for the average number of characters per word in another language, e.g., <𝐶 > = 4.24 for only reason that a language has, on average, words shorter or longer than , Italian. In any English. The rationale for this choice is that 𝐶 is a parameter typical of a language which, case, as recalled above, C affects a readability formula much less than P [13]. if not scaled, would bias 𝐺 without really quantifying the change in reading difficulty of readers, who are surely accustomed to reading, in their language, shorter or longer words, Table 1. Values of C and k of Equations (2) and (3) in the New Testament texts in the indicated on average, than those found in Italian. This scaling, therefore, avoids changing 𝐺 for the languages. Languages are listed according to their language family (see Reference [11]). only reason that a language has, on average, words shorter or longer than Italian. In any Language Language Family C k case, as recalled above, 𝐶 affects a readability formula much less than 𝑃 [13]. Greek Hellenic 4.86 0.92 Latin Italic 5.16 0.87 Table 1. Values of 𝐶 and 𝑘 of Equations (2) and (3) in the New Testament texts in the indicated Esperanto Constructed 4.43 1.01 languages. Languages are listed according to their language family (see Reference [11]). French Romance 4.20 1.07 Language Language Family 𝑪 𝒌 Italian Romance 4.48 1.00 Portuguese Romance 4.43 1.01 Greek Hellenic 4.86 0.92 Romanian Romance 4.34 1.03 Latin Italic 5.16 0.87 Spanish Romance 4.30 1.04 Esperanto Constructed 4.43 1.01 Danish Germanic 4.14 1.08 French Romance 4.20 1.07 English Germanic 4.24 1.06 Finnish Germanic 5.90 0.76 Italian Romance 4.48 1.00 German Germanic 4.68 0.96 Portuguese Romance 4.43 1.01 Analytics 2023, 2 299 Table 1. Cont. Language Language Family C k Icelandic Germanic 4.34 1.03 Norwegian Germanic 4.08 1.10 Swedish Germanic 4.23 1.06 Bulgarian BaltoSlavic 4.41 1.02 Czech BaltoSlavic 4.51 0.99 Croatian BaltoSlavic 4.39 1.02 Polish BaltoSlavic 5.10 0.88 Russian BaltoSlavic 4.67 0.96 Serbian BaltoSlavic 4.24 1.06 Slovak BaltoSlavic 4.65 0.96 Ukrainian BaltoSlavic 4.56 0.98 Estonian Uralic 4.89 0.92 Hungarian Uralic 5.31 0.84 Albanian Albanian 4.07 1.10 Armenian Armenian 4.75 0.94 Welsh Celtic 4.04 1.11 Basque Isolate 6.22 0.72 Hebrew Semitic 4.22 1.06 Cebuano Austronesian 4.65 0.96 Tagalog Austronesian 4.83 0.93 Chichewa NigerCongo 6.08 0.74 Luganda NigerCongo 6.23 0.72 Somali AfroAsiatic 5.32 0.84 Haitian French Creole 3.37 1.33 Nahuatl UtoAztecan 6.71 0.67 On the other hand, we have maintained the constant 300 because P depends signifi- cantly on author ’s style [13,15], not on language. Finally, notice that the constant 89 sets just the absolute ordinate scale, and therefore it has no impact on comparisons [13]. In conclusion, in Reference [11] we have defined a global readability index applicable to texts written in a language as: G = 89 10kC + 300/P (2) P F with k = < C >/< C > (3) P,I TA P By using Equations (2) and (3), we force the average value of 10  C of any language to be equal to that found in Italian, namely 10  4.48. Table 1 reports for Greek, Latin and 35 contemporary languages, the average values of C [11] and the calculated values of the constant k of Equation (3). For example, for English texts, C of a sample text is multiplied by 10.6, instead of 10; for Nahuatl (longer words), C is multiplied by 6.7, and for Haitian (shorter words) by 13.3. Notice that k seems to be a stable factor. For example, in the sample of the English literature studied in Reference [17], we have found < C > = 4.23 (instead of the 4.24 P,EN G of Table 1). Now, because the value found in the Italian literature [13] is < C > = 4.67, P,I TA therefore k = 4.67/4.23 = 1.10, instead of the k = 4.48/4.24 = 1.06 of Table 1. As recalled above, all readability formulae substantially tell the same story, and therefore they should be very similar and it is very likely that any one of them can be obtained from another. We illustrate this fact with an example. Because English is the language that has more readability formulae than any other language, let us compare G to the most classical English readability formula proposed and amply discussed by Flesch [1,2], known as the Flesch Reading Ease (RE) formula: RE = 206.8 1.015w 84.6s (4) Analytics 2023, 2, FOR PEER REVIEW 5 As recalled above, all readability formulae substantially tell the same story, and therefore they should be very similar and it is very likely that any one of them can be obtained from another. We illustrate this fact with an example. Because English is the language that has more readability formulae than any other language, let us compare 𝐺 to the most classical English readability formula proposed and amply discussed by Flesch [1,2], known as the Flesch Reading Ease ( ) formula: Analytics 2023, 2 300 = 206.8 − 1.015𝑤 − 84.6𝑠 (4) In Equation (4), 𝑤 is the average number of words per sentence, and 𝑠 is the average number of syllables per word. Because the number of characters per word is, on In Equation (4), w is the average number of words per sentence, and s is the average average, proportional to the number of syllables per word, the parameter 𝑠 paralles 𝐶 number of syllables per word. Because the number of characters per word is, on average, and, of course, 𝑤= 𝑃 . proportional to the number of syllables per word, the parameter s paralles C and, of How Equation (4) quantifies the degree of difficulty was defined by Flesch himself course, w = P . [1,2], and its values are reported in the vertical scale of Figure 1 (right ordinate scale), for How Equation (4) quantifies the degree of difficulty was defined by Flesch himself [1,2], comparison with 𝐺 (left ordinate scale). Figure 2 shows the scatterplot between the and its values are reported in the vertical scale of Figure 1 (right ordinate scale), for values calculated with the global readability index 𝐺 , Equation (2), versus those comparison with G (left ordinate scale). Figure 2 shows the scatterplot between the values calculated with , Equation (4), according to WinWord, in novels from English literature calculated with the global readability index G, Equation (2), versus those calculated with [17], Table 2. RE, Equation (4), according to WinWord, in novels from English literature [17], Table 2. Figure 2. Flesch Reading Ease (RE) index, Equation (4), versus the global index G, Equation (2), for Figure 2. Flesch Reading Ease (RE) index, Equation (4), versus the global index 𝐺 , Equation (2), for the novels of the English Literature listed in Table 2. Robinson Crusoe, cyan “o”; Pride and Prejudice, the novels of the English Literature listed in Table 2. Robinson Crusoe, cyan “o”; Pride and Prejudice, black “o”; Vanity Fair, blue “o”; Alice’s Adventures in Wonderland, magenta “o”; Treasure Island, green black “o”; Vanity Fair, blue “o”; Alice’s Adventures in Wonderland, magenta “o”; Treasure Island, green “o”; Adventures of Huckleberry Finn, red “+”; Peter Pan, blue “+”; The Sun Also Rises, green “+”; A “o”; Adventures of Huckleberry Finn, red “+”; Peter Pan, blue “+”; The Sun Also Rises, green “+”; A Farewell to Arms, black “+”. Farewell to Arms, black “+”. Table 2. Novels from English literature. Deep-language parameters C , P , I , G and universal P F P Table 2. Novels from English literature. Deep-language parameters 𝐶 , 𝑃 , 𝐼 , 𝐺 and universal readability index G , the latter discussed in Section 4. Novels are listed according to the year readability index 𝑮 , the latter discussed in Section 4. Novels are listed according to the year of of publication. publication. Literary Work C P I G G p F P U Literary Work 𝑷 𝑰 𝑮 𝑮 𝑭 𝑷 𝑼 Matthew King James translation (1611) 4.27 23.51 5.91 55.14 55.86 Matthew King James translation (1611) 4.27 23.51 5.91 55.14 55.86 Robinson Crusoe (D. Defoe, 1719) 3.94 57.75 7.12 50.84 42.22 Robinson Crusoe (D. Defoe, 1719) 3.94 57.75 7.12 50.84 42.22 Pride and Prejudice (J. Austen, 1813) 4.40 24.86 7.16 52.79 43.89 Wuthering Heights (E. Brontë, 1845–1846) 4.27 25.82 5.97 53.65 53.89 Vanity Fair (W. Thackeray, 1847–1848) 4.63 25.74 6.73 49.75 44.10 David Copperfield (C. Dickens, 1849–1850) 4.04 24.40 5.61 56.68 59.66 Moby Dick (H. Melville, 1851) 4.52 31.18 6.45 49.11 45.66 The Mill on The Floss (G. Eliot, 1860) 4.29 28.03 7.09 52.70 44.32 Alice’s Adventures in Wonderland (L. Carroll, 1865) 3.96 30.92 5.79 56.14 57.76 𝑅𝐸 𝑅𝐸 𝑅𝐸 Analytics 2023, 2 301 Table 2. Cont. Literary Work C P I G G p F P U Little Women (L.M. Alcott, 1868–1869) 4.18 21.08 6.30 57.31 54.99 Treasure Island (R. L. Stevenson, 1881–1882) 4.02 21.89 6.05 58.78 58.39 Adventures of Huckleberry Finn (M. Twain, 1884) 3.85 24.89 6.63 59.01 54.14 Three Men in a Boat (J.K. Jerome, 1889) 4.25 13.71 6.14 64.19 63.13 The Picture of Dorian Gray (O. Wilde, 1890) 4.19 16.56 6.29 62.83 60.58 The Jungle Book (R. Kipling, 1894) 4.11 21.52 7.15 57.95 49.14 The War of the Worlds (H.G. Wells, 1897) 4.38 20.85 7.67 55.31 42.48 The Wonderful Wizard of Oz (L.F. Baum, 1900) 4.02 20.55 7.63 59.38 46.85 The Hound of The Baskervilles (A.C. Doyle, 1901–1902) 4.15 17.79 7.83 60.27 46.16 Peter Pan (J.M. Barrie, 1902) 4.12 18.20 6.35 60.53 57.85 A Little Princess (F.H. Burnett, 1902–1905) 4.18 16.38 6.80 61.57 55.45 Martin Eden (J. London, 1908–1909) 4.32 16.94 6.76 59.38 53.50 Women in love (D.H. Lawrence, 1920) 4.26 13.71 5.22 63.98 70.02 The Secret Adversary (A. Christie, 1922) 4.28 11.02 5.52 69.08 72.76 The Sun Also Rises (E. Hemingway, 1926) 3.92 10.70 6.02 72.58 72.45 A Farewell to Arms (H. Hemingway,1929) 3.94 10.12 6.80 73.17 66.99 Of Mice and Men (J. Steinbeck, 1937) 4.02 9.67 5.61 74.20 77.24 We can notice a fair agreement between the two indices, with a correlation coefficient of 0.850. The bias could be compensated by downscaling RE. The attribution of the grade level G L in the USA school system was defined by Kincaid et al. [3], by using the same parameters w and s. The grade level is similar to that attributed to G. Another readability formula, the Automated Readability Index (ARI), was also defined by Kincaid et al. ii for specific military documents [3]. It is fully related to G because it depends on the same parameters, C and P : P F AR I = 4.71C + 0.5P 21.43 (5) p F As AR I increases, the age of required readers increases too. Figure 3 shows the scatterplot between the global G, Equation (2), and AR I, for the the same English novels considered in Figure 2. We can see a very tight relationship for fixed C . In conclusion, the global readability formula, Equation (2), provides a readability index that can be directly scaled to AR I and approximately also to RE. For this reason, we continue studying G, which we will modify by introducing the word interval I to obtain the universal readability formula/index mentioned above. To do so we need to recall, in the next section, some fundamental knowledge on I . P Analytics 2023, 2, FOR PEER REVIEW 7 Analytics 2023, 2 302 Figure 3. Automated Readability Index (ARI), Equation (5), versus the global index G, Equation (3), Figure 3. Automated Readability Index (ARI), Equation (5), versus the global index 𝐺 , Equation (3), for the novels of English literature listed in Table 2. The continuous lines assume constant values of for the novels of English literature listed in Table 2. The continuous lines assume constant values of C . Robinson Crusoe, cyan “o”; Pride and Prejudice, black “o”; Vanity Fair, blue “o”; Alice’s Adventures 𝐶 . Robinson Crusoe, cyan “o”; Pride and Prejudice, black “o”; Vanity Fair, blue “o”; Alice’s Adventures in Wonderland, magenta “o”; Treasure Island, green “o”; Adventures of Huckleberry Finn, red “+”; Peter in Wonderland, magenta “o”; Treasure Island, green “o”; Adventures of Huckleberry Finn, red “+”; Peter Pan, blue “+”; The Sun Also Rises, green “+”; A Farewell to Arms, black “+”. Pan, blue “+”; The Sun Also Rises, green “+”; A Farewell to Arms, black “+”. 3. Word Interval and Short-Term Memory In conclusion, the global readability formula, Equation (2), provides a readability As we have discussed in References [11,13,15], the word interval I ¯namely the number index that can be directly scaled to and approximately also to . For this reason, of words per interpunctions—varies in the same range of the short-term memory capacity- we continue studying 𝐺 , which we will modify by introducing the word interval 𝐼 to given by Miller ’s 7  2 law [14], a range that includes 95% of all cases, and very likely obtain the universal readability formula/index mentioned above. To do so we need to the two ranges are deeply related because interpunctions organize small portions of more recall, in the next section, some fundamental knowledge on 𝐼 . complex arguments (which make a sentence) in short chunks of text, which are the natural input to short-term memory [19–27]. Moreover, I , drawn against the number of words per 3. Word Interval and Short-Term Memory sentence, P , tends to approach a horizontal asymptote as P increases, and this occurs both F F in ancient As we have classical disc languages ussed in (Gr References eek and Latin) [11,13,15], th and in contemporary e word interval languages, 𝐼 —namely the as shown in References [11,13] by studying translations of the New Testament books from Greek. In number of words per interpunctions—varies in the same range of the short-term memory other words, even if sentences get longer, I cannot get larger than about the upper limit capacity-given by Miller’s 7±2 law [14], a ra pnge that includes 95% of all cases, and very of Millers’ law (namely 9), because of the constraints imposed by the short-term memory likely the two ranges are deeply related because interpunctions organize small portions capacity of readers and writers, as well. of more complex arguments (which make a sentence) in short chunks of text, which are The average value of I can be empirically related to the average value of P according the natural input to short-term p memory [19–27]. Moreover, 𝐼 , drawn against the F number to the non-linear relationship [13]: of words per sentence, 𝑃 , tends to approach a horizontal asymptote as 𝑃 increases, and this occurs both in ancient classical languages (Greek and Latin) and in contemporary (<P >1) (P 1) languages, as shown in References [11,13] by studying translations of the New Testament Fo < I >= ( I 1)  1 e + 1 (6) P P¥ books from Greek. In other words, even if sentences get longer, 𝐼 cannot get larger than about the upper limit of Millers’ law (namely 9), because of the constraints imposed by where I gives the horizontal asymptote, and P gives the value of < P > at which the P¥ Fo F the short-term memory capacity of readers and writers, as well. exponential falls at 1/e of its maximum value. The average value of 𝐼 can be empirically related to the average value of 𝑃 Equation (6) is a good average mathematical model for Italian literature [13] and also according to the non-linear relationship [13]: for Greek, Latin and contemporary languages [11,15]. Reference [11] reports the values of I and P for each language considered. P¥ Fo <𝐼 >= 𝐼 −1 ×1−𝑒 +1 (6) Presently, we have carried out the same analysis as for the large corpus of Italian literature [13] for a smaller but useful corpus of the English literature recently studied in 𝑅𝐸 𝐴𝑅𝐼 Analytics 2023, 2, FOR PEER REVIEW 8 where 𝐼 gives the horizontal asymptote, and 𝑃 gives the value of <𝑃 > at which the exponential falls at 1/𝑒 of its maximum value. Equation (6) is a good average mathematical model for Italian literature [13] and also for Greek, Latin and contemporary languages [11,15]. Reference [11] reports the values of Analytics 2023, 2 303 𝐼 and 𝑃 for each language considered. Presently, we have carried out the same analysis as for the large corpus of Italian literature [13] for a smaller but useful corpus of the English literature recently studied in Refer Reference [17], and ence [17], and have havecalculated calculated the the best-fit best-fitvalues values of of Equation (6 Equation (6).). Figur Figuree4 4 shows shows the the scatter scatter plo plottof of I 𝐼 versus versus P𝑃 (values (values ca calculated lculated for for each ch each chapter) apter) and and the the b best-fit est-fit c curve, urve,with with I𝐼 = =6 6.70 .70and and P 𝑃 ==6 6.78.7 ,8 to, be to b compar e comed pare with d with I = 𝐼 7.37 =7 and .37 P and = 𝑃10.22 =1 of0the .22 of Italian the P¥ Fo P¥ Fo literature, whose curve is also drawn. Italian literature, whose curve is also drawn. Figure 4. Scatter plot of the word interval, I , and the number of words per sentence, P , for all P F Figure 4. Scatter plot of the word interval, 𝐼 , and the number of words per sentence, 𝑃 , for all samples of the novels considered in the English literature, Table 2. The continuous black line refers to samples of the novels considered in the English literature, Table 2. The continuous black line refers the best fit given by Equation (6). The green line refers to the Italian literature [13]. Miller ’s bounds to the best fit given by Equation (6). The green line refers to the Italian literature [13]. Miller’s bounds are given by I = 7  2. are given by 𝐼 =7 ∓ 2. Notice that the constants of the English literature differ from those reported in Notice that the constants of the English literature differ from those reported in Reference [17] ( I = 6.57, P = 4.16) for the same literary corpus, because the latter P¥ Fo Reference [17] (𝐼 =6.57, 𝑃 =4.16) for the same literary corpus, because the latter were the results of fitting Equation (6) to the average values of I and P , not to the values P F were the results of fitting Equation (6) to the average values of 𝐼 and 𝑃 , not to the values of I and P obtained by considering the samples (a sample for each chapter), which give P F of 𝐼 and 𝑃 obtained by considering the samples (a sample for each chapter), which give the scatterplot drawn in Figure 4. The different values are due, of course, to the non-linear the scatterplot drawn in Figure 4. The different values are due, of course, to the non-linear best fit. best fit. Now, as we have recalled in Section 2, any readability index is practically a function only Now, as we have recalled in Section 2, any readability index is practically a function of P . Readability formulae do not consider I , but the scatterplots of I versus G show an p p only of 𝑃 . Readability formulae do not consider 𝐼 , but the scatterplots of 𝐼 versus 𝐺 interesting story: texts with the same G do not show the same I . In other words, according show an interesting story: texts with the same 𝐺 do not show the same 𝐼 . In other words, to the theory of readability formulae, a text with a given index should be readable with the according to the theory of readability formulae, a text with a given index should be same effort both by readers who display a powerful short-term memory processing capacity readable with the same effort both by readers who display a powerful short-term memory (large I ) and by readers who do not (small I ). For example, for G = 60 (“easy/standard” p p processing capacity (large 𝐼 ) and by readers who do not (small 𝐼 ). For example, for 𝐺= texts for readers with 8 years of school, Figure 1), Figure 5 shows th at I can vary from 4 to 60 (“easy/standard” texts for readers with 8 years of school, Figure 1), Figure 5 shows that 9. This is practically Miller ’s range, which refers to 95% of readers [14]. We think that these readers should be distinguished, and therefore, our aim is to propose, in the next section, a possible “universal” readability index, G , based on G, which includes I . U p Analytics 2023, 2, FOR PEER REVIEW 9 𝐼 can vary from 4 to 9. This is practically Miller’s range, which refers to 95% of readers [14]. We think that these readers should be distinguished, and therefore, our aim is to Analytics 2023, 2 304 propose, in the next section, a possible “universal” readability index, 𝐺 , based on 𝐺 , which includes 𝐼 . (a) (b) Figure 5. Scatterplots of the readability index, 𝐺 , versus the word interval, 𝐼 . (a): Italian Literature Figure 5. Scatterplots of the readability index, G, versus the wor d interval, I . (a): Italian Litera- [11]; (b): English Literature, Table 2. Miller’s bounds are given by 𝐼 =7 ∓ 2. ture [11]; (b): English Literature, Table 2. Miller ’s bounds are given by I = 7  2. 4. A Universal Readability Formula 4. A Universal Readability Formula We suppose that the global readability index G should be modified by introducing We suppose that the global readability index 𝐺 should be modified by introducing a function that depends linearly on I . Our hypothesis is based on Miller ’s law, which a function that depends linearly on 𝐼 . Our hypot P hesis is based on Miller’s law, which quantifies linearly the processing capacity of the short-term memory. Moreover, the function quantifies linearly the processing capacity of the short-term memory. Moreover, the should not change the global value for a reader with an “average” processing short-term memory capacity. For words, this average is not 7, but about 6 [1,28]; therefore, in the following we assume this latter value. Notice that 6.03 is the average value of I (standard deviation 1.11) of the data listed in Table 2 of Reference [11], a further indication of its barycentric value. Analytics 2023, 2 305 We write our proposed universal readability formula as: DG G = G I 6 (7) ( ) U P D I where G is given by Equation (2). DG We assume that the numerical value of the discrete derivative is given by: D I DG G G max min = (8) D I I I P P,max P,min In Equation (8), the numerical values are the maximum and minimum averages found in the Italian literature—see Reference [13], whose oldest texts (seven centuries old, e.g., Boccaccio’s Decameron) are still read today in Italian high schools with a reasonable effort, a possibility not available in other Western languages. From [13], we calculate: DG 69.84 49.54 = = 6.15  6.00 (9) D I 8.24 4.94 Therefore, the proposed universal readability formula is given by G = G 6( I 6) (10) U P Equation (10) sets G = G for I = 6; G < G for I > 6 and G > G for I < 6. In U P U P U P other words, if a text with a given G, has a small word interval I , then it should be read more easily than a text with the same G, but larger I . For example, texts with G = 60 would be transformed in Miller ’s range of 5 to 9 to G = 66 for I = 5 and in G = 42 U P U for I = 9, and therefore, in the first case, the text considered “easy” after 8 years of school (Figure 1), is considered “easy” to read but only after 7.2 years of school; in the second case, the text would be considered “easy”, but only after about 13.2 years of school. The meaningful difference between the two indices is therefore very large: 66 42 = 24, corresponding to 13.2 7.2 = 5 years of school. This significant difference would be lost in the original formula of Equation (2), or in any other readability formula. Figure 6 shows the scatterplots between G and I (blue circles) for the samples U P concerning the literary texts considered in Italian [13] and in English Literatures, in Table 2. Compared to the scatterplots of Figure 5 (redrawn in Figure 6 with red circles), the difference between G and G is evident: the linear dependence of G on I , according to Equation (10), U U P spreads the values around a line and introduces significant correlation coefficients, 0.9016 for Italian, and 0.7730 for English. The regression line: G = a I + b (11) U P is very similar in the two languages: G = 9.47I + 115.71 (12) U,I TA P G = 8.88I + 111.64 (13) U,EN G P This result indicates that Equation (11) might be “universal”. Finally, some specific examples concerning novels taken from Italian and English literatures will further illustrate the relationship between G and G . U Analytics 2023, 2, FOR PEER REVIEW 11 Analytics 2023, 2 306 (a) (b) Figure 6. Scatt Figure er plot 6. Scatter of theplot readability in of the readability dex, 𝐺 index, , versus the G , versus word interval, the word interval, 𝐼 (blue Icirc (blue les).cir The cles). The red U P red circles refer to the scatterplots of Figure 5. (a): Italian Literature [13]; (b): English Literature, circles refer to the scatterplots of Figure 5. (a): Italian Literature [13]; (b): English Literature, Table 1. Table 1. Miller’s bounds are given by 𝐼 =7 ∓ 2. Miller ’s bounds are given by I = 7  2. Finally, some specific examples concerning novels taken from Italian and English Table 3 shows how the readability index is modified from G to G for some Italian literatures will further illustrate the relationship between 𝐺 and 𝐺 . novels written from the XIV to the XX century [13]. For example, it is interesting to notice Table how 3 sho Gw is s how th transformed e readability into G inde for the x is mod two novels ified from written 𝐺 to by 𝐺 Alessandr for some It o Manzoni. alian novels written from the XIV to the XX century [13]. For example, it is interesting to notice how 𝐺 is transformed into 𝐺 for the two novels written by Alessandro Manzoni. Analytics 2023, 2 307 Table 3. Novels from Italian Literature [13]. Average deep-language parameters C , P , I , and G P F P and corresponding universal readability index, G . Novels are listed according to the alphabetical order of the author‘s name. Novel C P I G G P F P U Anonymous (I Fioretti di San Francesco, XIV Century) 4.65 37.70 8.24 50.70 37.26 Boccaccio Giovanni (Decameron, XIV) 4.48 44.27 7.79 51.18 40.44 Buzzati Dino (Il deserto dei tartari, XX) 5.10 17.75 6.63 55.27 51.49 Calvino Italo (Marcovaldo, XX) 4.74 17.60 6.59 59.19 55.65 Cassola Carlo (La ragazza di Bube, XX) 4.48 11.93 5.64 69.84 72.00 Collodi Carlo (Pinocchio, XIX) 4.60 16.92 6.19 61.57 60.43 Deledda Grazia (Canne al vento, XX) 4.51 15.08 6.06 64.39 64.03 D’Annunzio Gabriele (Le novelle delle Pescara, XX) 4.91 17.99 6.38 58.16 55.88 Eco Umberto (Il nome della rosa, XX) 4.81 21.08 7.46 55.78 47.02 Fogazzaro (Piccolo mondo antico, XIX-XX) 4.79 16.08 6.10 61.46 60.86 Gadda (Quer pasticciaccio brutto . . . XX) 4.76 18.43 4.98 58.24 64.36 Machiavelli Niccolò (Il principe, XV-XVI) 4.71 40.17 6.45 49.54 46.84 Manzoni Alessandro (Fermo e Lucia, XIX) 4.75 30.98 7.17 51.72 44.70 Manzoni Alessandro (I promessi sposi, XIX) 4.60 24.83 5.30 56.00 60.20 Moravia Alberto (La ciociara, XX) 4.56 29.93 7.28 53.52 45.84 Pavese Cesare (La luna e i falò, XX) 4.47 17.83 6.83 61.90 56.92 Pirandello Luigi (Il fu Mattia Pascal) 4.63 14.57 4.94 63.94 70.30 Svevo Italo (Senilità, XX) 4.86 16.04 7.75 59.39 48.89 Tomasi di Lampedusa (Il gattopardo, XX) 4.99 26.42 7.90 50.72 39.32 Verga (I Malavoglia, XIX-XX) 4.46 20.45 6.82 59.34 54.42 Alessandro Manzoni (Milan 1785, Milan 1873), one of the most studied Italian novel- ist in Italian high schools (Licei) and universities, in 1827 published Fermo e Lucia (Fermo and Lucia), a text that scholars of Italian Literature—and Manzoni himself—consider the “first” version of his masterpiece I Promessi Sposi (The Betrothed, available in a new English translation [29]) published in the years 1840–1842. According to scholars of Italian litera- ture [30–33], the two versions differ very much, both in story structure and characters and, as far as we are here concerned, also in style and language; therefore, it is interesting to see how much the author transformed (mathematically) Fermo e Lucia into I Promessi Sposi, a study partially carried out in References [13,15]. As far as readability is concerned, from Table 3 we notice a large improvement in I Promessi Sposi, compared to Fermo e Lucia, if differences are considered. In fact, G = 51.72 in Fermo e Lucia and G = 56.00 in I Promessi Sposi, a difference of only 4.28 units, leading to a decrease in school years (for “easy” reading, Figure 1) of only about 0.8 years. This difference does not justify the reading difficulty of the two texts discussed by scholars of Italian literature [30–33]. However, if we consider G , then the difference is quite large, very likely measuring the relative reading difficulty, because G ranges from 44.70 to 60.20, a difference of 15.5 units leading to a decrease in school years (for “easy” reading, Figure 1) from 11.8 (Fermo e Lucia) to only 8 (I Promessi Sposi), well justified by scholars of Italian literature [30–33]. In conclusion, G is a better estimate than G in assessing the difference in reading difficulty between these two very studied novels. Table 2 shows also how the readability index is modified from G to G in some English novels. As we can read from Table 2, in Robinson Crusoe the readability index decreases from 50.84 to 42.22, therefore passing from about 10.3 to 12.4 years of school for “easy” reading Analytics 2023, 2 308 Analytics 2023, 2, FOR PEER REVIEW(Figur e 1). For Hemingway’s novels, The Sun Also Rises is more readable (72.45) 13 than A Farewell to Arms (66.99); the order given by G, i.e., 72.58 and 73.17, respectively, is reversed, therefore reducing the number of years of school required for “easy” reading by 1 (Figure 1). The Hound of The Baskervilles changes its readability index from 60.27 to 46.16, therefore reversed, therefore reducing the number of years of school required for “easy” reading by 1 (Figure 1). The Hound of The Baskervilles changes its readability index from 60.27 to 46.16, passing from 8 to 11.5 years of school for “easy” reading (Figure 1). therefore passing from 8 to 11.5 years of school for “easy” reading (Figure 1). In conclusion, by introducing the word interval I in the definition of a readability In conclusion, by introducing the word interval 𝐼 in the definition of a readability index, as in Equation (10), readability differences in texts are more “fine-tuned” for readers. index, as in Equation (10), readability differences in texts are more “fine-tuned” for readers. 5. A “Footprint” of Humans As already recalled, in Reference [11] we have studied the translation of the New 5. A “Footprint” of Humans Testament from Greek to Latin and to contemporary languages. For all these translations, As already recalled, in Reference [11] we have studied the translation of the New we have recently calculated the scatterplots between G and I , and between G and P U Testament from Greek to Latin and to contemporary languages. For all these translations, I , with results very similar to those shown in Figure 6. Some specific examples are we have recently calculated the scatterplots between 𝐺 and 𝐼 , and between 𝐺 and 𝐼 , reported in Appendix A. Similarly, we have calculated the linear best fit between G with results very similar to those shown in Figure 6. Some specific examples are reported and I . Appendix B lists the values of the constants a and b of Equation (11) for each in Appendix A. Similarly, we have calculated the linear best fit between 𝐺 and 𝐼 . translation/language. Appendix B lists the values of the constants 𝑎 and 𝑏 of Equation (11) for each translThis ation/ set langua of values ge. are useful because they could be used to compare texts written in any language. This seFor t of v example, alues are useful bec in David Copperfield ause they could be , G is estimated used to compare tex to be 61.82 ts written in with Equation (13) any language. For example, in David Copperfield, 𝐺 is estimated to be 61.82 with Equation and 66.02 with the values of Appendix B (the experimental average value is 59.66, Table 2); (13) and 66.02 with the values of Appendix B (the experimental average value is 59.66, in The Hound of The Baskervilles, G is estimated to be 42.11 with Equation (13) and 48.53 Table 2); in The Hound of The Baskervilles, 𝐺 is estimated to be 42.11 with Equation (13) with the values of Appendix B (the experimental average value is 46.16, Table 2). In the and 48.53 with the values of Appendix B (the experimental average value is 46.16, Table first case, the difference in the readability of the two novels is 19.71, and in the second case 2). In the first case, the difference in the readability of the two novels is 19.71, and in the it is 17.49, which implies an “error” of about 0.25 years of school (Figure 1). second case it is 17.49, which implies an “error” of about 0.25 years of school (Figure 1). It may be interesting to consider the most compact relationship between G and I , U P It may be interesting to consider the most compact relationship between 𝐺 and 𝐼 , given by the overall average values of the constants reported in Appendix B: given by the overall average values of the constants reported in Appendix B: (14) 𝐺 = −8.94𝐼 + 116 G = 8.94I + 116 (14) U P Figure 7 show this average relationship together with ±1 standard deviaton Figure 7 show this average relationship together with 1 standard deviaton bounds. bounds. These extremely compacted curves can synthetically represent how the capacity These extremely compacted curves can synthetically represent how the capacity of human of human short-term memory (modelled by 𝐼 ) is related to the difficulty of reading a text, short-term memory (modelled by I ) is related to the difficulty of reading a text, in any in any alphabetical language; therefore, Pit may be considered as a kind of “footprint” of alphabetical language; therefore, it may be considered as a kind of “footprint” of humans. humans. Figure 7. Average (blue line) and 1 standard deviation lines (cyan lines) of the universal readability index, G , versus the word interval, I , from Table A1. U P Analytics 2023, 2 309 6. Conclusions We have proposed a universal readability index, G , Equation (10). Compared to the current readability indices, this index considers also readers’ short-term memory processing capacity, here described by the word interval I , namely, the number of words between two interpunctions. The observation that differences give more insight than absolute values has justified, we think, the development of a universal readability formula which is useful for comparing texts written even in different languages and is applicable to alphabetical languages and related to cognitive psychology, the theory of communication, phonics and linguistics. Scholars have never considered including in the current readability formulae the word interval, I , but the scatterplots of I versus any readability index show that texts with the p p same readability index can have very different values of I . Now, it is unlikely that I has p P no impact on reading difficulty. By introducing I in the definition of a readability index, readability differences in texts are better “fine-tuned” for readers, e.g., to their school years as a reference. We have used the global readability index developed for Italian [11], after showing that Flesch’s index and ARI are connected to this index because they depend on the same variables. We have calculated an extremely compact formula, Equation (14), which can measure how the capacity of human short-term memory (modelled by I ) is likely related to the difficulty of reading a text, measured by the universal readability index G , here defined. We think that it synthetically models human reading difficulty, i.e., it might be considered a “footprint” of humans. However, there is an important aspect to be considered. Because, as far as we know, there are no direct experiments on the relationship between readability and short-term memory capacity, the universal index here proposed, Equation (10), should be considered a first step in researching this important relationship. Therefore, further work needs to be carried out by a multidisciplinary team of researchers to fully validate Equation (10). Funding: This research received no external funding. Data Availability Statement: Data are available, on request, by the author. Conflicts of Interest: The authors declare no conflict of interest. Appendix A Scatterplots between G and I , for selected languages. We show the scatterplots be- U P tween G and I (red circles), and between G and I , (blue circles) for some selected languages. P U P Analytics 2023, 2, FOR PEER REVIEW 15 Analytics 2023, 2 310 (a) (b) Figure A1. Scatter plot of the readability index G versus the word interval I (blue circles). The red Figure A1. Scatter plot of the readability index 𝐺 versus the word interval 𝐼 (blue circles). The U P red circles refer to the circles refer to scatterplots of the scatterplots 𝐺 of versus G versus 𝐼 . ( I a . )(:a Greek; ): Greek; (b() b : L ): a Latin. tin. M Miller iller’s ’s b bounds ounds a arre e given givenby by 𝐼 =7 ∓ I 2= . Th 7  e 2 valu . The esvalues of 𝑎 and of a and 𝑏 of b of Equation (12) Equation (12) a and nd the the corr correlat elation ion coe coefficient fficient between between G and 𝐺 I P U P and 𝐼 are repor are reported ted inin Table Table A1 A1 . . Analytics 2023, 2, FOR PEER REVIEW 16 Analytics 2023, 2 311 (a) (b) Figure A2. Scatter plot of the readability index G versus the word interval I (blue circles). The red Figure A2. Scatter plot of the readability index 𝐺 versus the word U interval 𝐼 P(blue circles). The red circles refe circles r to the scatter refer to the scatterplots plots of of of G 𝐺 ver versus sus I 𝐼 . (. ( a):aSpanish; ): Spanish; ( (b): Portuguese. b): Portugues Miller e. Mil ’s bounds ler’s bound are given s by I = 7  2. The values of a and b of Equation (12) and the correlation coefficient between G and are given by 𝐼 =7 ∓ 2. The values of 𝑎 and 𝑏 of Equation (12) and the correlation coefficient be- P U I are reported in Table A1. tween 𝐺 and 𝐼 are reported in Table A1. Analytics 2023, 2, FOR PEER REVIEW 17 Analytics 2023, 2 312 (a) (b) Figure A3. Scatter plot of the readability index G versus the word interval I (blue circles). The red Figure A3. Scatter plot of the readability index 𝐺 versus the word U interval 𝐼 P(blue circles). The circles refer to the scatterplots of G versus I . (a): French; (b): German. Miller ’s bounds are given by red circles refer to the scatterplots of 𝐺 versus 𝐼 P. (a): French; (b): German. Miller’s bounds are I = 7  2. The values of a and b of Equation (13) and the correlation coefficient between G and I given by 𝐼 =7 ∓ 2. The values of 𝑎 and 𝑏 of Equation (13) and the correlation coefficient between P U P are reported in Table A1. 𝐺 and 𝐼 are reported in Table A1. Analytics 2023, 2 313 Appendix B Table A1. Values of a and b of Equation (12), and correlation coefficient between G and I for the U P indicated languages [11]. Language a b Correlation Coefficient Greek 8.62 113.66 0.9477 Latin 10.59 120.82 0.8666 Esperanto 9.87 114.20 0.8803 French 7.46 107.51 0.9311 Italian 7.80 108.54 0.9065 Portuguese 8.34 112.33 0.8261 Romanian 8.08 111.11 0.8163 Spanish 8.46 112.60 0.9061 Danish 9.46 120.71 0.9182 English 7.88 110.23 0.9129 Finnish 10.06 118.22 0.8057 German 8.68 113.79 0.8563 Icelandic 8.68 114.98 0.8848 Norwegian 7.32 110.28 0.9426 Swedish 7.32 109.98 0.9546 Bulgarian 9.00 117.63 0.8697 Czech 10.41 125.50 0.8269 Croatian 9.86 122.33 0.8868 Polish 9.98 123.60 0.7160 Russian 10.70 118.04 0.7326 Serbian 8.71 117.24 0.8312 Slovak 10.03 124.83 0.8417 Ukrainian 8.34 113.42 0.7092 Estonian 9.97 120.11 0.8643 Hungarian 10.83 118.91 0.8034 Albanian 8.01 107.04 0.8776 Armenian 12.11 133.87 0.7805 Welsh 7.74 103.12 0.7828 Basque 9.99 117.48 0.8361 Hebrew 10.27 129.58 0.8163 Cebuano 6.97 107.50 0.9683 Tagalog 7.78 112.54 0.9188 Chichewa 8.40 118.76 0.9325 Luganda 8.69 118.42 0.8713 Somali 8.65 113.41 0.9492 Haitian 8.25 115.41 0.9132 Nahuatl 7.55 113.02 0.9420 Overall 8.94  1.22 116.00  6.49 0.8681  0.0661 References 1. 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Journal

Analytics – Multidisciplinary Digital Publishing Institute

Published: Mar 30, 2023

Keywords: alphabetical languages; ARI; English literature; Flesch Reading Ease Index; GULPEASE; human footprint; Italian literature; Miller’s Law; short-term capacity; universal readability index; word interval

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