## 2 Readability Formulae  
A readability index is a measure to express the complexity of written text. Quite often they are based on simple features, such as sentence and word length, and indicate how easy it is to read and comprehend a text. While there is a wide variety of readability indices covered in the related literature, we focus here on three well established methods: Flesch Reading Ease, the SMOG grading index and the Gunning fog index. The _Flesch Reading Ease (FRE)_ index [8] is a long established index in this context. The score of FRE typically ranges between 0 and 100 [9]. A higher score indicates a text that is easier to read and comprehend. For instance, a text with a score between 100 and 90 should be understandable for 11-year old students, while a score lower than 30 requires the reader to be at the level of a college graduate. Let us denote the total number of syllables in a text with \(y\), the number of words with \(w\) and let \(s\) be the number of sentences, then the FRE index \(r_{\text{FRE}}\) is defined by:  
\[r_{\text{FRE}}=206.835-84.6\cdot\frac{y}{w}-1.015\cdot\frac{w}{s} \tag{1}\]  
[PERSON] [10] introduced a different parameter in his readability formula: the number of polysyllables. A polysyllable is a word made of three or more syllables. If we denote the number of polysyllables in a text with \(p\), and use \(s\) again for the number of sentences, then the _SMOG grading_ index \(r_{\text{SMOG}}\) is defined as:  
\[r_{\text{SMOG}}=1.043\cdot\sqrt{\frac{p}{s}}+3.1291 \tag{2}\]  
The SMOG grading index indicates the educational level required to comprehend a text, _i.e._, the years of school education, required to understand it. For instance, a SMOG reading index value of 5 indicates that a text can be understood after five years of school education. To calculate the index of a large text, [PERSON] stated that it is sufficient to use three text samples of 10 sentences each. Another metric with a similar intention is the _Gunning fog_ index [11]. Comparable to SMOG, also this index estimates the years of education required to understand a given text. To calculate the Gunning fog index, a passage of around 100 words needs to be analysed. Polysyllables are considered in the Gunning fog index, too, but only those which are not proper nouns, compound words, _etc_. The Gunning fog index \(r_{\text{FOG}}\) is defined as:  
\[r_{\text{FOG}}=0.4\cdot\left(\frac{w}{s}+100\cdot\frac{p}{w}\right) \tag{3}\]  
The original Gunning fog formula was based on clauses and not on the number of sentences \(s\). This rendered the Gunning fog too difficult to be calculated automatically. Thus, by now, that the formulation presented in Equation (3) is generally recommended. Both, the Gunning fog and the SMOG grading index use only small samples of a text to calculate the a readability score. This was originally simply motivated by practical reasons. Both formulae are relatively old. The Gunning fog index was published in 1952, the SMOG grading index in 1969. At this time the formulae needed to be calculated by hand. With the rise of electronic means to process texts both formulae have been implemented for automatic computation. Hence, their computation can easily be extended to a full text even for entire books. One question of interest is, whether there is a difference when considering the full text rather than just some parts and samples of a text. The chart in Figure 1 shows the development of the SMOG grading index across sub-samples of 30 sentences over a full text (in this case a novel for children). The average of the SMOG grading index over all sub-samples is 4.00 and the SMOG grading index calculated with the whole text is 4.01. This difference can be neglected. The minimal index is 3.73 and the maximal index is 4.35. If one of these parts is selected randomly the readability might be miscalculated to the extent of half a school year. The SMOG grading index and the Gunning fog index both promise to calculate the years of school education.