Letter frequency
The frequency of letters in text has often been studied for use in cryptography, and frequency analysis in particular. No exact letter frequency distribution underlies a given language, since all writers write slightly differently. Linotype machines sorted the letters' frequencies as etaoin shrdlu cmfwyp vbgkqj xz based on the experience and custom of manual compositors. Likewise, Modern International Morse code encodes the most frequent letters with the shortest symbols; arranging the Morse alphabet into groups of letters that require equal amounts of time to transmit, and then sorting these groups in increasing order, yields e it san hurdm wgvlfbk opjxcz yq. Similar ideas are used in modern data-compression techniques such as Huffman coding.
More recent analyses show that letter frequencies, like word frequencies, tend to vary, both by writer and by subject. One cannot write an essay about x-rays without using frequent Xs, and the essay will have an especially strange letter frequency if the essay is about the frequent use of x-rays to treat zebras in Qatar. Different authors have habits which can be reflected in their use of letters. Hemingway's writing style, for example, is visibly different from Faulkner's. Letter, bigram, trigram, word frequencies, word length, and sentence length can be calculated for specific authors, and used to prove or disprove authorship of texts, even for authors whose styles aren't so divergent.
Accurate average letter frequencies can only be gleaned by analyzing a large amount of representative text. With the availability of modern computing and collections of large text corpora, such calculations are easily made. This Deafandblind link details examples from a variety of sources, (press reporting, religious text, scientific text and general fiction) and there are differences especially for general fiction with the position of 'h' and 'i'. The example differs from the linotype 'etaoin shrdlu' to come out as 'etaoHn Isrdlu'. There is an unproven statement that conversation is similar in frequency to general fiction.
Herbert S. Zim, in his classic introductory cryptography text "Codes and Secret Writing", gives the English letter frequency sequence as "ETAON RISHD LFCMU GYPWB VKXJQ Z", the most common letter pairs as "TH HE AN RE ER IN ON AT ND ST ES EN OF TE ED OR TI HI AS TO", and the most common doubled letters as "LL EE SS OO TT FF RR NN PP CC".[1]
The 'top twelve' letters comprise about 80% of the total usage. The 'top eight" letters comprise about 65% of the total usage. A spy using the VIC cipher or some other cipher based on a straddling checkerboard typically uses a mnemonic such as "a sin to err" (dropping the second "r") to remember the top 8 characters.
The use of letter frequencies and frequency analysis plays a fundamental role in cryptograms and several word puzzle games, including Hangman, Scrabble, Bananagrams, and the television game show Wheel of Fortune. One of the earliest description in classical literature of applying the knowledge of English letter frequency to solving a cryptogram is found in E.A. Poe's famous story The Gold Bug, where the method is successfully applied to decipher a message instructing on the whereabouts of a treasure hidden by Captain Kidd [2].
Letter frequencies had a strong effect on the design of some keyboard layouts. The most-frequent letters are on the bottom row of the Blickensderfer typewriter. The most-frequent letters are on the home row of the Dvorak Simplified Keyboard.
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Relative frequencies of letters in the English language
The letter frequencies for English are listed below.[3] However, this table differs slightly from others, such as Cornell University Math Explorer's Project, which produced this table after measuring 40,000 words.
| Letter | Relative frequency in the English language | |
|---|---|---|
| a | 8.167% |
|
| b | 1.492% |
|
| c | 2.782% |
|
| d | 4.253% |
|
| e | 12.702% |
|
| f | 2.228% |
|
| g | 2.015% |
|
| h | 6.094% |
|
| i | 6.966% |
|
| j | 0.153% |
|
| k | 0.772% |
|
| l | 4.025% |
|
| m | 2.406% |
|
| n | 6.749% |
|
| o | 7.507% |
|
| p | 1.929% |
|
| q | 0.095% |
|
| r | 5.987% |
|
| s | 6.327% |
|
| t | 9.056% |
|
| u | 2.758% |
|
| v | 0.978% |
|
| w | 2.360% |
|
| x | 0.150% |
|
| y | 1.974% |
|
| z | 0.074% |
|
In English, the space is slightly more frequent than the top letter (7% more frequent than, or 107% as frequent as, e), and the non-alphabetic characters (digits, punctuation, etc.) occupy the fourth position, between t and a.[4]
Relative frequencies of the first letters of a word in the English language
First Letter of a word frequencies:[5]
| Letter | Relative frequency as the first letter of an English word | |
|---|---|---|
| a | 11.602% |
|
| b | 4.702% |
|
| c | 3.511% |
|
| d | 2.670% |
|
| e | 2.000% |
|
| f | 3.779% |
|
| g | 1.950% |
|
| h | 7.232% |
|
| i | 6.286% |
|
| j | 0.631% |
|
| k | 0.690% |
|
| l | 2.705% |
|
| m | 4.374% |
|
| n | 2.365% |
|
| o | 6.264% |
|
| p | 2.545% |
|
| q | 0.173% |
|
| r | 1.653% |
|
| s | 7.755% |
|
| t | 16.671% |
|
| u | 1.487% |
|
| v | 0.619% |
|
| w | 6.661% |
|
| x | 0.005% |
|
| y | 1.620% |
|
| z | 0.050% |
|
Relative frequencies of letters in other languages
| Letter | French [6] | German [7] | Spanish [8] | Portuguese [9] | Esperanto [10] | Italian[11] | Turkish | Swedish[12] | Polish[13] | Dutch [14] |
|---|---|---|---|---|---|---|---|---|---|---|
| a | 7.636% | 6.51% | 12.53% | 14.63% | 12.12% | 11.74% | 11.68% | 9.3% | 8.0% | 7.49% |
| b | 0.901% | 1.89% | 1.42% | 1.04% | 0.98% | 0.92% | 2.95% | 1.3% | 1.3% | 1.58% |
| c | 3.260% | 3.06% | 4.68% | 3.88% | 0.78% | 4.5% | 0.97% | 1.3% | 3.8% | 1.24% |
| d | 3.669% | 5.08% | 5.86% | 4.99% | 3.04% | 3.73% | 4.87% | 4.5% | 3.0% | 5.93% |
| e | 14.715% | 17.40% | 13.68% | 12.57% | 8.99% | 11.79% | 9.01% | 9.9% | 6.9% | 18.91% |
| f | 1.066% | 1.66% | 0.69% | 1.02% | 1.03% | 0.95% | 0.44% | 2.0% | 0.1% | 0.81% |
| g | 0.866% | 3.01% | 1.01% | 1.30% | 1.17% | 1.64% | 1.34% | 3.3% | 1.0% | 3.40% |
| h | 0.737% | 4.76% | 0.70% | 1.28% | 0.38% | 1.54% | 1.14% | 2.1% | 1.0% | 2.38% |
| i | 7.529% | 7.55% | 6.25% | 6.18% | 10.01% | 11.28% | 8.27%* | 5.1% | 7.0% | 6.50% |
| j | 0.545% | 0.27% | 0.44% | 0.40% | 3.50% | 0.00% | 0.01% | 0.7% | 1.9% | 1.46% |
| k | 0.049% | 1.21% | 0.01% | 0.02% | 4.16% | 0.00% | 4.71% | 3.2% | 2.7% | 2.25% |
| l | 5.456% | 3.44% | 4.97% | 2.78% | 6.14% | 6.51% | 5.75% | 5.2% | 3.1% | 3.57% |
| m | 2.968% | 2.53% | 3.15% | 4.74% | 2.99% | 2.51% | 3.74% | 3.5% | 2.4% | 2.21% |
| n | 7.095% | 9.78% | 6.71% | 5.05% | 7.96% | 6.88% | 7.23% | 8.8% | 4.7% | 10.03% |
| o | 5.378% | 2.51% | 8.68% | 10.73% | 8.78% | 9.83% | 2.45% | 4.1% | 7.1% | 6.06% |
| p | 3.021% | 0.79% | 2.51% | 2.52% | 2.74% | 3.05% | 0.79% | 1.7% | 2.4% | 1.57% |
| q | 1.362% | 0.02% | 0.88% | 1.20% | 0.00% | 0.51% | 0 | 0.007% | - | 0.009% |
| r | 6.553% | 7.00% | 6.87% | 6.53% | 5.91% | 6.37% | 6.95% | 8.3% | 3.5% | 6.41% |
| s | 7.948% | 7.27% | 7.98% | 7.81% | 6.09% | 4.98% | 2.95% | 6.3% | 3.8% | 3.73% |
| t | 7.244% | 6.15% | 4.63% | 4.74% | 5.27% | 5.62% | 3.09% | 8.7% | 2.4% | 6.79% |
| u | 6.311% | 4.35% | 3.93% | 4.63% | 3.18% | 3.01% | 3.43% | 1.8% | 1.8% | 1.99% |
| v | 1.628% | 0.67% | 0.90% | 1.67% | 1.90% | 2.10% | 0.98% | 2.4% | - | 2.85% |
| w | 0.114% | 1.89% | 0.02% | 0.01% | 0.00% | 0.00% | 0 | 0.03% | 3.6% | 1.52% |
| x | 0.387% | 0.03% | 0.22% | 0.21% | 0.00% | 0.00% | 0 | 0.1% | - | 0.04% |
| y | 0.308% | 0.04% | 0.90% | 0.01% | 0.00% | 0.00% | 3.37% | 0.6% | 3.2% | 0.035% |
| z | 0.136% | 1.13% | 0.52% | 0.47% | 0.50% | 0.49% | 1.50% | 0.02% | 5.1% | 1.39% |
| à | 0.486% | 0 | 0 | see a | 0 | see a | 0 | 0.0% | 0 | see a |
| å | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1.6% | 0 | - |
| ä | 0 | - | 0 | 0 | 0 | 0 | 0 | 2.1% | 0 | see a |
| ą | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | see a | - |
| œ | 0.018% | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | - |
| ç | 0.085% | 0 | 0 | see c | 0 | 0 | 1.26% | 0 | 0 | - |
| ĉ | 0 | 0 | 0 | 0 | 0.66% | 0 | 0 | 0 | 0 | - |
| ć | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | see c | - |
| è | 0.271% | 0 | 0 | 0 | 0 | see e | 0 | 0.0% | 0 | see e |
| é | 1.904% | 0 | 0 | see e | 0 | see e | 0 | 0.0% | 0 | see e |
| ê | 0.225% | 0 | 0 | see e | 0 | 0 | 0 | 0 | 0 | - |
| ë | 0.001% | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | see e |
| ę | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | see e | - |
| ĝ | 0 | 0 | 0 | 0 | 0.69% | 0 | 0 | 0 | 0 | - |
| ğ | 0 | 0 | 0 | 0 | 0 | 0 | 1.13% | 0 | 0 | - |
| ĥ | 0 | 0 | 0 | 0 | 0.02% | 0 | 0 | 0 | 0 | - |
| î | 0.045% | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | - |
| ì | 0 | 0 | 0 | 0 | 0 | see i | 0 | 0 | 0 | see i |
| ï | 0.005% | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | see i |
| ı | 0 | 0 | 0 | 0 | 0 | 0 | 5.20%* | 0 | 0 | - |
| ĵ | 0 | 0 | 0 | 0 | 0.12% | 0 | 0 | 0 | 0 | - |
| ł | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | see l | - |
| ñ | 0 | 0 | 0.31% | 0 | 0 | 0 | 0 | 0 | 0 | - |
| ń | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | see n | - |
| ò | 0 | 0 | 0 | 0 | 0 | see o | 0 | 0 | 0 | see o |
| ö | 0 | - | 0 | 0 | 0 | 0 | 0.87% | 1.5% | 0 | see o |
| ó | 0 | - | 0 | see o | 0 | 0 | 0 | 0 | see o | see o |
| ŝ | 0 | 0 | 0 | 0 | 0.38% | 0 | 0 | 0 | 0 | - |
| ş | 0 | 0 | 0 | 0 | 0 | 0 | 1.94% | 0 | 0 | - |
| ś | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | see s | - |
| ß | 0 | 0.31% | 0 | 0 | 0 | 0 | 0 | 0 | 0 | - |
| ù | 0.058% | 0 | 0 | 0 | 0 | see u | 0 | 0 | 0 | see u |
| ŭ | 0 | 0 | 0 | 0 | 0.52% | 0 | 0 | 0 | 0 | - |
| ü | 0 | - | 0 | 0 | 0 | 0 | 1.99% | 0 | 0 | see u |
| ź | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | see z | - |
| ż | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0.7% | - |
*See Turkish dotted and dotless I
The figure below illustrates the frequency distributions of the 26 most common Latin letters across some languages. 
Based on these tables, the 'etaoin shrdlu'-equivalent results for each language is as follows:
- French: 'esait nrulo'; (Indo-European: Romance; traditionally, 'esartinulop' is used, in part for its ease of pronunciation[15])
- Spanish: 'eaosr nidlc'; (Indo-European: Romance)
- Portuguese: 'aeosr indmt' (Indo-European: Romance)
- Italian: 'eaion lrtsc'; (Indo-European: Romance)
- Esperanto: 'aieon lsrtk' (artificial language – influenced by Indo-European languages, Romance, Germanic mostly)
- German: 'enisr atdhu'; (Indo-European: Germanic)
- Swedish: 'eantr slido'; (Indo-European: Germanic)
- Turkish: 'aeinr ldkmu'; (Turkic: a non Indo-European language)
- Dutch: 'enati rodsl'; (Indo-European: Germanic)[16]
- Polish: 'aoiez nscwr'; (Indo-European: Slavic)
All these languages use a basically similar 25+ character alphabet.
See also
- Corpus linguistics
- ETAOIN SHRDLU
- RSTLNE (Wheel of Fortune)
- Frequency analysis (cryptanalysis)
- Linotype machine
- Most common words in English
- Scrabble
References
- ^ Zim, Herbert Spencer. Codes and secret writing (abridged edition). Scholastic Book Services, fourth printing, 1962. Copyright 1948 Herbert S. Zim. Originally published by William Morrow.
- ^ http://www.gutenberg.org/catalog/world/readfile?fk_files=1977099 THE WORKS OF EDGAR ALLAN POE IN FIVE VOLUMES at the Project Gutenberg
- ^ Beker, Henry; Piper, Fred (1982). Cipher Systems: The Protection of Communications. Wiley-Interscience. p. 397. Table also available from Lewand, Robert (2000). Cryptological Mathematics. The Mathematical Association of America. p. 36. ISBN 978-0883857199. http://books.google.com/books?id=CyCcRAm7eQMC&pg=PA36. and [1]
- ^ Lee, E. Stewart; Essays about Computer Security; University of Cambridge Computer Laboratory, p. 181
- ^ Calculated from "Project Gutenberg Selections" available from the NLTK Corpora
- ^ "CorpusDeThomasTempé". http://gpl.insa-lyon.fr/Dvorak-Fr/CorpusDeThomasTemp%C3%A9. Retrieved 2007-06-15.
- ^ Albrecht Beutelspacher, Kryptologie, 7. Aufl., Wiesbaden: Vieweg Verlagsgesellschaft, 2005, ISBN 3-8348-0014-7, p.10
- ^ Fletcher Pratt, Secret and Urgent: the Story of Codes and Ciphers Blue Ribbon Books, 1939, pp. 254-255.
- ^ "Frequência da ocorrência de letras no Português". http://www.numaboa.com/criptografia/criptoanalise/310-Frequencia-no-Portugues. Retrieved 2009-06-16.
- ^ "La Oftecoj de la Esperantaj Literoj". http://lingvakritiko.com/2007/09/13/literoftecoj-kaj-tabelvortoftecoj/. Retrieved 2007-09-14.
- ^ Simon Singh, Codici e Segreti, 1999, RCS, ISBN 88-17-12539-3
- ^ Simon Singh, Kodboken, 1999, Norstedts, ISBN 91-1-1300708-4
- ^ Wstęp do kryptologii, counting [space] 17.2%, [dot point] 0.9%, [comma] 0.9% and [semicolon] 0.5%
- ^ "Letterfrequenties". Genootschap OnzeTaal. http://www.onzetaal.nl/advies/letterfreq.php. Retrieved 2009-05-17.
- ^ Perec, Georges; Alphabets; Éditions Galilée, 1976
- ^ "Letterfrequenties". Genootschap OnzeTaal. http://www.onzetaal.nl/advies/letterfreq.php. Retrieved 2008-12-26.
- Notes
Some useful tables for single letter, digram, trigram, tetragram, and pentagram frequencies based on 20,000 words that take into account word-length and letter-position combinations for words 3 to 7 letters in length. The references are as follows:
- 1. Mayzner, M.S. & Tresselt, M.E. Tables of single-letter and digram frequency counts for various word-length and letter-position combinations. Psychonomic Monograph Supplements, 1965, 1, #2, 13-32.
- 2. Mayzner, M.S., Tresselt, M.E. & Wolin, B.R. Tables of trigram frequency counts for various word-length and letter-position combinations. Psychonomic Monograph Supplements, 1965, 1, #3, 33-78.
- 3. Mayzner, M.S., Tresselt, M.E. & Wolin, B.R. Tables of tetragram frequency counts for various word-length and letter-position combinations. Psychonomic Monograph Supplements, 1965, 1, #4, 79-143.
- 4. Mayzner, M.S., Tresselt, M.E. & Wolin, B.R. Tables of pentagram frequency counts for various word-length and letter-position combinations. Psychonomic Monograph Supplements, 1965, 1, #5, 144-190.
External links
- A site with content of Cryptographical Mathematics by Robert Edward Lewand
- Some examples of letter frequency rankings in some common languages
- Java-Application for building letter frequencies out of a text file
- JavaScript Heatmap Visualization showing letter frequencies of texts on different keyboard layouts