letter frequency

File:CaliforniaJobCaseLayout.png was a compartmentalized box for printing in the 19th century, sizes corresponding to the commonality of letters]]

The frequency of letters in text has been studied for use in cryptanalysis, and frequency analysis in particular, dating back to the Arab mathematician al-Kindi (c. AD 801–873 ), who formally developed the method (the ciphers breakable by this technique go back at least to the Caesar cipher used by Julius Caesar,{{Citation needed|date=January 2024}} so this method could have been explored in classical times). Letter frequency analysis gained additional importance in Europe with the development of movable type in AD 1450, wherein one must estimate the amount of type required for each letterform, as evidenced by the variations in letter compartment size in typographer's type cases.

No exact letter frequency distribution underlies a given language, since all writers write slightly differently. However, most languages have a characteristic distribution which is strongly apparent in longer texts. Even language changes as extreme as from Old English to modern English (regarded as mutually unintelligible) show strong trends in related letter frequencies: over a small sample of Biblical passages, from most frequent to least frequent, {{not a typo|enaid sorhm tgþlwu æcfy ðbpxz}} of Old English compares to {{not a typo|eotha sinrd luymw fgcbp kvjqxz}} of modern English, with the most extreme differences concerning letterforms not shared.{{cite web |url=http://www.math.ucsd.edu/~crypto/Projects/MarshaMoreno/TimeComparisonFrequency.pdf |title=Frequency Analysis in Light of Language Innovation |last1=Moreno |first1=Marsha Lynn |date=Spring 2005 |department=Math |publisher=University of California – San Diego |access-date=19 February 2015}}

Linotype machines for the English language assumed the letter order, from most to least common, to be {{not a typo|etaoin shrdlu cmfwyp vbgkqj xz}} based on the experience and custom of manual compositors. The equivalent for the French language was {{not a typo|elaoin sdrétu cmfhyp vbgwqj xz}}.

Arranging the alphabet in Morse into groups of letters that require equal amounts of time to transmit, and then sorting these groups in increasing order, yields {{not a typo|e it san hurdm wgvlfbk opxcz jyq}}.American Morse code was developed in the 1830s by Alfred Vail, based on English-language letter frequencies, to encode the most frequent letters with the shortest symbols. Some efficiency was lost in the reformed version now used: the International Morse Code. Letter frequency was used by other telegraph systems, such as the Murray Code.

Similar ideas are used in modern data-compression techniques such as Huffman coding.

Letter frequencies, like word frequencies, tend to vary, both by writer and by subject. For instance, {{Vr|d}} occurs with greater frequency in fiction, as most fiction is written in past tense and thus most verbs will end in the inflectional suffix -ed / -d. One cannot write an essay about x-rays without using {{Vr|x}} frequently. 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 are not 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. Examples can be drawn from a variety of sources (press reporting, religious texts, scientific texts and general fiction) and there are differences especially for general fiction with the position of {{Vr|h}} and {{Vr|i}}, with {{Vr|h}} becoming more common.

Different dialects of a language will also affect a letter's frequency. For example, an author in the United States would produce something in which {{Vr|z}} is more common than an author in the United Kingdom writing on the same topic: words like "analyze", "apologize", and "recognize" contain the letter in American English, whereas the same words are spelled "analyse", "apologise", and "recognise" in British English. This would highly affect the frequency of the letter {{Vr|z}}, as it is rarely used by British writers in the English language.{{cite web |url=http://www.oxforddictionaries.com/words/british-and-american-spelling |archive-url=https://web.archive.org/web/20111228045450/http://oxforddictionaries.com/words/british-and-american-spelling |url-status=dead |archive-date=December 28, 2011 |title=British and American spelling - Oxford Dictionaries |website=Oxford Dictionaries - English |access-date=18 April 2018}}

The "top twelve" letters constitute about 80% of the total usage. The "top eight" letters constitute about 65% of the total usage. Letter frequency as a function of rank can be fitted well by several rank functions, with the two-parameter Cocho/Beta rank function being the best.{{cite journal |author1=Li, Wentian |author2=Miramontes, Pedro |title=Fitting ranked English and Spanish letter frequency distribution in US and Mexican presidential speeches |journal=Journal of Quantitative Linguistics |volume=18 |issue=4 |year=2011 |doi=10.1080/09296174.2011.608606 |pages=359 |arxiv=1103.2950|s2cid=1716455 }} Another rank function with no adjustable free parameter also fits the letter frequency distribution reasonably well{{cite journal |author=Gusein-Zade, S.M.|author-link=Sabir Gusein-Zade |title=Frequency distribution of letters in the Russian language |journal=Probl. Peredachi Inf. |volume=24 |issue=4 |pages=102–107 |year=1988}} (the same function has been used to fit the amino acid frequency in protein sequences.{{cite journal |author1=Gamow, George |author2=Ycas, Martynas |title=Statistical correlation of protein and ribonucleic acid composition |journal=Proc. Natl. Acad. Sci. |volume=41 |issue=12 |pages=1011–1019 |year=1955 |pmc=528190 |doi=10.1073/pnas.41.12.1011 |pmid=16589789|bibcode=1955PNAS...41.1011G|doi-access=free }}) 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"){{cite book |first=Friedrich L. |last=Bauer |url=https://books.google.com/books?id=hfWTDr_bvMwC |title=Decrypted Secrets: Methods and maxims of cryptology |year=2006 |page=57 |publisher=Springer |isbn=9783540481218 |via=Google Books}}{{cite book |first=Greg |last=Goebel |url=http://www.vectorsite.net/ttcode_03.html |archive-url=https://web.archive.org/web/20051205013154/http://www.vectorsite.net/ttcode_03.html |url-status=usurped |archive-date=December 5, 2005 |title=The Rise Of Field Ciphers: straddling checkerboard ciphers |year=2009}} or "at one sir"{{cite web |first=Dirk |last=Rijmenants |url=https://www.ciphermachinesandcryptology.com/en/onetimepad.htm |title=One-time Pad}} to remember the top eight characters.

File:Worn keyboard of English speaker.jpg

There are three ways to count letter frequency that result in very different charts for common letters. The first method, used in the chart below, is to count letter frequency in lemmas of a dictionary. The lemma is the word in its canonical form. The second method is to include all word variants when counting, such as "abstracts", "abstracted" and "abstracting" and not just the lemma of "abstract". This second method results in letters like {{Vr|s}} appearing much more frequently, such as when counting letters from lists of the most used English words on the Internet. {{Vr|s}} is especially common in inflected words (non-lemma forms) because it is added to form plurals and third person singular present tense verbs. A final method is to count letters based on their frequency of use in actual texts, resulting in certain letter combinations like {{Vr|th}} becoming more common due to the frequent use of common words like "the", "then", "both", "this", etc. Absolute usage frequency measures like this are used when creating keyboard layouts or letter frequencies in old fashioned printing presses.

An analysis of entries in the Concise Oxford dictionary, ignoring frequency of word use, gives an order of "EARIOTNSLCUDPMHGBFYWKVXZJQ".{{cite web |title=What is the frequency of the letters of the alphabet in English? |url=http://oxforddictionaries.com/words/what-is-the-frequency-of-the-letters-of-the-alphabet-in-english |archive-url=https://web.archive.org/web/20111224230632/http://oxforddictionaries.com/words/what-is-the-frequency-of-the-letters-of-the-alphabet-in-english |url-status=dead |archive-date=December 24, 2011 |work=Oxford Dictionary |publisher=Oxford University Press |access-date=29 December 2012}}

The letter-frequency table above is taken from Pavel Mička's website, which cites Robert Lewand's Cryptological Mathematics.{{cite web |last=Mička |first=Pavel |title=Letter frequency (English) |url=http://en.algoritmy.net/article/40379/Letter-frequency-English|publisher=Algoritmy.net}}

According to Lewand, arranged from most to least common in appearance, the letters are: etaoinshrdlcumwfgypbvkjxqz. Lewand's ordering differs slightly from others, such as Cornell University Math Explorer's Project, which produced a table after measuring 40,000 words.{{cite web |last= |first= |date= |title=English Letter Frequency (based on a sample of 40,000 words) |url=http://pi.math.cornell.edu/~mec/2003-2004/cryptography/subs/frequencies.html |access-date=2021-01-24 |work=cornell.edu}}

In English, the space character occurs almost twice as frequently as the top letter ({{Vr|e}}){{cite web |url=http://www.data-compression.com/english.html |title=Statistical Distributions of English Text |website=data-compression.com |archive-url=https://web.archive.org/web/20170918020907/http://www.data-compression.com/english.html |archive-date=2017-09-18 |url-status=dead}} and the non-alphabetic characters (digits, punctuation, etc.) collectively occupy the fourth position (having already included the space) between {{Vr|t}} and {{Vr|a}}.{{cite web |last=Lee |first=E. Stewart |title=Essays about Computer Security |publisher=University of Cambridge Computer Laboratory |url=http://www.cl.cam.ac.uk/~mgk25/lee-essays.pdf |page=181}}

The frequency of the first letters of words or names is helpful in pre-assigning space in physical files and indexes.{{cite book |author-link=Herbert Marvin Ohlman |author=Ohlman, Herbert Marvin |title=Subject-Word Letter Frequencies with Applications to Superimposed Coding |url=http://books.nap.edu/openbook.php?record_id=10866&page=903 |publisher=Proceedings of the International Conference on Scientific Information |year=1959|doi=10.17226/10866 |isbn=978-0-309-57421-1 }} Given 26 filing cabinet drawers, rather than a 1:1 assignment of one drawer to one letter of the alphabet, it is often useful to use a more equal-frequency-letter code by assigning several low-frequency letters to the same drawer (often one drawer is labeled VWXYZ), and to split up the most-frequent initial letters ({{Vr|s, a, c}}) into several drawers (often 6 drawers Aa-An, Ao-Az, Ca-Cj, Ck-Cz, Sa-Si, Sj-Sz). The same system is used in some multi-volume works such as some encyclopedias. Cutter numbers, another mapping of names to a more equal-frequency code, are used in some libraries.

Both the overall letter distribution and the word-initial letter distribution approximately match the Zipf distribution and even more closely match the Yule distribution.{{cite journal |author1=Pande, Hemlata |author2=Dhami, H.S. |url=http://www.skase.sk/Volumes/JTL16/pdf_doc/02.pdf |title=Mathematical Modelling of Occurrence of Letters and Word's Initials in Texts of Hindi Language |journal=JTL |volume=16}}

Often the frequency distribution of the first digit in each datum is significantly different from the overall frequency of all the digits in a set of numeric data, an observation known as Benford's law.

An analysis by Peter Norvig on words that appear 100,000 times or more in Google Books data transcribed using optical character recognition (OCR) determined the frequency of first letters of English words, among other things.{{cite web |url=http://norvig.com/mayzner.html |title=English Letter Frequency Counts: Mayzner revisited or ETAOIN SRHLDCU |website=norvig.com |access-date=18 April 2018}}{{clear}}

{{Cite check|section|date=July 2014}}

*See İ and dotless I.

The figure below illustrates the frequency distributions of the 26 most common Latin letters across some languages. All of these languages use a similar 25+ character alphabet.

{{Letter frequencies in 11 languages}}

Based on these tables, the 'etaoin shrdlu' equivalent for each language is as follows:

• French: '{{not a typo|esaitn ruoldc}}'; (Indo-European: Romance; traditionally, 'esartinulop' is used, in part for its ease of pronunciationPerec, Georges; Alphabets; Éditions Galilée, 1976)
• Spanish: '{{not a typo|eaosrn idltcm}}'; (Indo-European: Romance)
• Portuguese: '{{not a typo|aeosri dmntcu}}' (Indo-European: Romance)
• Italian: '{{not a typo|eaionl rtscdu}}'; (Indo-European: Romance)
• German: '{{not a typo|ensria tdhulg}}'; (Indo-European: Germanic)
• Swedish: '{{not a typo|eanrts ildomk}}'; (Indo-European: Germanic)
• Turkish: '{{not a typo|aeinrl ıdkmyt}}'; (Turkic: Oghuz)
• Dutch: '{{not a typo|enatir odslgv}}'; (Indo-European: Germanic)
• Polish: '{{not a typo|aioezn rwstcy}}'; (Indo-European: Slavic)
• Danish: '{{not a typo|erntai dslogk}}'; (Indo-European: Germanic)
• Icelandic: '{{not a typo|arnies tulðgm}}'; (Indo-European: Germanic)
• Finnish: '{{not a typo|aintes loukäm}}'; (Uralic: Finnic)
• Czech: '{{not a typo|aeonit vsrldk}}'; (Indo-European: Slavic)
• Hungarian: '{{not a typo|eatlsn kizroá}}'; (Uralic: Ugric)
• Welsh: '{{not a typo|ayneir odwgldd}}; (Indo-European: Celtic)

• Arabic letter frequency
• English word frequency
• Letter frequency effect
• Lipogram
• RSTLNE (Wheel of Fortune)

{{notelist}}

{{Reflist}}

• [https://letterfrequency.org/ Letter Frequencies]
• {{cite web |url=http://pages.central.edu/emp/LintonT/ |title=Cryptographical Mathematics |author=Lewand, Robert Edward |publisher=pages.central.edu|archive-url=https://web.archive.org/web/20070402181401/http://pages.central.edu/emp/LintonT/ |archive-date=2007-04-02 }}
• {{cite web |url=https://www.bckelk.org.uk/words/etaoin.html |title=Some examples of letter frequency rankings in some common languages |publisher=www.bckelk.org.uk}}
• {{cite web |url=http://www.patrick-wied.at/projects/heatmap-keyboard/ |title=JavaScript Heatmap Visualization showing letter frequencies of texts on different keyboard layouts |publisher=www.patrick-wied.at}}
• {{cite web |url=http://norvig.com/mayzner.html |title=An updated version of Mayzner's work using Google books Ngrams data set |author=Norvig, Peter |publisher=norvig.com}}
• [http://simia.net/letters/ Letter frequency]—simia.net

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:

• {{cite journal |last1=Mayzner |first1=M.S. |last2=Tresselt |first2=M.E. |last3=Wolin |first3=B.R. |title=Tables of single-letter and digram frequency counts for various word-length and letter-position combinations |journal=Psychonomic Monograph Supplements |volume=1 |issue=2 |pages=13–32 |year=1965 |oclc=639975358}}
• {{cite journal |last1=Mayzner |first1=M.S. |last2=Tresselt |first2=M.E. |last3=Wolin |first3=B.R. |title=Tables of trigram frequency counts for various word-length and letter-position combinations |journal=Psychonomic Monograph Supplements |volume=1 |issue=3 |pages=33–78 |year=1965 }}
• {{cite journal |last1=Mayzner |first1=M.S. |last2=Tresselt |first2=M.E. |last3=Wolin |first3=B.R. |title=Tables of tetragram frequency counts for various word-length and letter-position combinations |journal=Psychonomic Monograph Supplements |volume=1 |issue=4 |pages=79–143 |year=1965 }}
• {{cite journal |last1=Mayzner |first1=M.S. |last2=Tresselt |first2=M.E. |last3=Wolin |first3=B.R. |title=Tables of pentagram frequency counts for various word-length and letter-position combinations |journal=Psychonomic Monograph Supplements |volume=1 |issue=5 |pages=144–190 |year=1965 }}

Category:Cryptography

Category:Frequency distribution

Category:Grammatology

Category:Phonology

Category:Quantitative linguistics