English Letter Frequency Counts:
Mayzner Revisited


On December 17th 2012, I got a nice letter from Mark Mayzner, a retired 85-year-old researcher who studied the frequency of letter combinations in English words in the early 1960s. His 1965 publication has been cited in hundreds of articles. Mayzner describes his work:
I culled a corpus of 20,000 words from a variety of sources, e.g., newspapers, magazines, books, etc. For each source selected, a starting place was chosen at random. In proceeding forward from this point, all three, four, five, six, and seven-letter words were recorded until a total of 200 words had been selected. This procedure was duplicated 100 times, each time with a different source, thus yielding a grand total of 20,000 words. This sample broke down as follows: three-letter words, 6,807 tokens, 187 types; four-letter words, 5,456 tokens, 641 types; five-letter words, 3,422 tokens, 856 types; six-letter words, 2,264 tokens, 868 types; seven-letter words, 2,051 tokens, 924 types. I then proceeded to construct tables that showed the frequency counts for three, four, five, six, and seven-letter words, but most importantly, broken down by word length and letter position, which had never been done before to my knowledge.
and he wonders if:
perhaps your group at Google might be interested in using the computing power that is now available to significantly expand and produce such tables as I constructed some 50 years ago, but now using the Google Corpus Data, not the tiny 20,000 word sample that I used.
The answer is: yes indeed, I am interested! And it will be a lot easier for me than it was for Mayzner. Working 60s-style, Mayzner had to gather his collection of text sources, then go through them and select individual words, punch them on Hollerith cards, and use a card-sorting machine.

Here's what we can do with today's computing power (using publicly available data and the processing power of my own personal computer; I'm not relying on access to corporate computing power):

  1. I consulted the Google books Ngrams raw data set, which gives word counts of the number of times each word is mentioned (broken down by year of publication) in the books that have been scanned by Google.
  2. I downloaded the English Version 20120701 "1-grams" (that is, word counts) from that data set given as the files "a" to "z" (that is, http://storage.googleapis.com/books/ngrams/books/googlebooks-eng-all-1gram-20120701-a.gz to http://storage.googleapis.com/books/ngrams/books/googlebooks-eng-all-1gram-20120701-z.gz). I unzipped each file; the result is 23 GB of text (so don't try to download them on your phone).
  3. I then condensed these entries, combining the counts for all years, and for different capitalizations: "word", "Word" and "WORD" were all recorded under "WORD". I discarded any entry that used a character other than the 26 letters A-Z. I also discarded any word with fewer than 100,000 mentions. (If you want you can download the word count file; note that it is 1.5 MB.)
  4. I generated tables of counts, first for words, then for letters and letter sequences, keyed off of the positions and word lengths.

Word Counts

My distillation of the Google books data gives us 97,565 distinct words, which were mentioned 743,842,922,321 times (37 million times more than in Mayzner's 20,000-mention collection). Each distinct word is called a "type" and each mention is called a "token." To no surprise, the most common word is "the". Here are the top 50 words, with their counts (in billions of mentions) and their overall percentage (looking like a Zipf distribution):
WORD    COUNT  PERCENT bar graph
the     53.10 B  7.14%  the
of      30.97 B  4.16%  of
and     22.63 B  3.04%  and
to      19.35 B  2.60%  to
in      16.89 B  2.27%  in
a       15.31 B  2.06%  a
is       8.38 B  1.13%  is
that     8.00 B  1.08%  that
for      6.55 B  0.88%  for
it       5.74 B  0.77%  it
as       5.70 B  0.77%  as
was      5.50 B  0.74%  was
with     5.18 B  0.70%  with
be       4.82 B  0.65%  be
by       4.70 B  0.63%  by
on       4.59 B  0.62%  on
not      4.52 B  0.61%  not
he       4.11 B  0.55%  he
i        3.88 B  0.52%  i
this     3.83 B  0.51%  this
are      3.70 B  0.50%  are
or       3.67 B  0.49%  or
his      3.61 B  0.49%  his
from     3.47 B  0.47%  from
at       3.41 B  0.46%  at
which    3.14 B  0.42%  which
but      2.79 B  0.38%  but
have     2.78 B  0.37%  have
an       2.73 B  0.37%  an
had      2.62 B  0.35%  had
they     2.46 B  0.33%  they
you      2.34 B  0.31%  you
were     2.27 B  0.31%  were
their    2.15 B  0.29%  their
one      2.15 B  0.29%  one
all      2.06 B  0.28%  all
we       2.06 B  0.28%  we
can      1.67 B  0.22%  can
her      1.63 B  0.22%  her
has      1.63 B  0.22%  has
there    1.62 B  0.22%  there
been     1.62 B  0.22%  been
if       1.56 B  0.21%  if
more     1.55 B  0.21%  more
when     1.52 B  0.20%  when
will     1.49 B  0.20%  will
would    1.47 B  0.20%  would
who      1.46 B  0.20%  who
so       1.45 B  0.19%  so
no       1.40 B  0.19%  no

Word Lengths

And here is the breakdown of mentions (in millions) by word length (looking like a Poisson distribution). The average is 4.79 letters per word, and 80% are between 2 and 7 letters long:
LEN    COUNT   PERCENT bar graph
 1  22301.22 M  2.998%  1
 2 131293.85 M 17.651%  2
 3 152568.38 M 20.511%  3
 4 109988.33 M 14.787%  4
 5  79589.32 M 10.700%  5
 6  62391.21 M  8.388%  6
 7  59052.66 M  7.939%  7
 8  44207.29 M  5.943%  8
 9  33006.93 M  4.437%  9
10  22883.84 M  3.076%  10
11  13098.06 M  1.761%  11
12   7124.15 M  0.958%  12
13   3850.58 M  0.518%  13
14   1653.08 M  0.222%  14
15    565.24 M  0.076%  15
16    151.22 M  0.020%  16
17     72.81 M  0.010%  17
18     28.62 M  0.004%  18
19      8.51 M  0.001%  19
20      6.35 M  0.001%  20
21      0.13 M  0.000%  21
22      0.81 M  0.000%  22
23      0.32 M  0.000%  23
Here is the distribution for distinct words (that is, counting each word only once regardless of how many times it is mentioned). Now the average is 7.60 letters long, and 80% are between 4 and 10 letters long:
 1      26  0.027%  1
 2     662  0.679%  2
 3   4,615  4.730%  3
 4   6,977  7.151%  4
 5  10,541 10.804%  5
 6  13,341 13.674%  6
 7  14,392 14.751%  7
 8  13,284 13.616%  8
 9  11,079 11.356%  9
10   8,468  8.679%  10
11   5,769  5.913%  11
12   3,700  3.792%  12
13   2,272  2.329%  13
14   1,202  1.232%  14
15     668  0.685%  15
16     283  0.290%  16
17     158  0.162%  17
18      64  0.066%  18
19      40  0.041%  19
20      16  0.016%  20
21       1  0.001%  21
22       5  0.005%  22
23       2  0.002%  23
Here are the 24 words with length of 20 or more (that are mentioned at least 100,000 times each in the book corpus):
electroencephalographic          radiopharmaceuticals
polytetrafluoroethylene		 electroencephalogram
forschungsgemeinschaft		 keratoconjunctivitis
deinstitutionalization		 counterrevolutionary
counterrevolutionaries		 immunohistochemistry
dehydroepiandrosterone		 internationalisation
electroencephalography		 hypercholesterolemia
immunoelectrophoresis		 phosphatidylinositol
institutionalisation		 compartmentalization
acetylcholinesterase		 electrophysiological
internationalization		 electrocardiographic
institutionalization		 uncharacteristically

Letter Counts

Enough of words; let's get back to Mayzner's request and look at letter counts. There were 3,563,505,777,820 letters mentioned. Here they are in frequency order:
E 445.2 B  12.49%  E
T 330.5 B   9.28%  T
A 286.5 B   8.04%  A
O 272.3 B   7.64%  O
I 269.7 B   7.57%  I
N 257.8 B   7.23%  N
S 232.1 B   6.51%  S
R 223.8 B   6.28%  R
H 180.1 B   5.05%  H
L 145.0 B   4.07%  L
D 136.0 B   3.82%  D
C 119.2 B   3.34%  C
U  97.3 B   2.73%  U
M  89.5 B   2.51%  M
F  85.6 B   2.40%  F
P  76.1 B   2.14%  P
G  66.6 B   1.87%  G
W  59.7 B   1.68%  W
Y  59.3 B   1.66%  Y
B  52.9 B   1.48%  B
V  37.5 B   1.05%  V
K  19.3 B   0.54%  K
X   8.4 B   0.23%  X
J   5.7 B   0.16%  J
Q   4.3 B   0.12%  Q
Z   3.2 B   0.09%  Z
Note there is a standard order of frequency used by typesetters, ETAOIN SHRDLU, that is slightly violated here: L, R, and C have all moved up one rank, giving us the less mnemonic ETAOIN SRHLDCU.

In the colored-bar chart below (inspired by the Wikipedia article on Letter Frequency), the frequency of each letter is proportional to the length of the color bar. If you hover the mouse over each color bar, you can see the exact percentages and counts. (This is the same information as in the table above, presented in a different way.)


Letter Counts by Position Within Word

Now we show the letter frequencies by position within word. That is, the frequencies for just the first letter in each word, just the second letter, and so on. We also show frequencies for positions relative to the end of the word: "-1" means the last letter, "-2" means the second to last, and so on. We can see that the frequencies vary quite a bit; for example, "e" is uncommon as the first letter (4 times less frequent than elsewhere); similarly "n" is 3 times less common as the first letter than it is overall. The letter "e" makes a comeback as the most common last letter (and also very common at 3rd and 5th letter places). The most common first letter is "t" and the most common second letter is "o".

 1 etaoinsrhldcumfpgwybvkxjqz
 2 etaoinsrhldcumfpgwybvkxjqz
 3 etaoinsrhldcumfpgwybvkxjqz
 4 etaoinsrhldcumfpgwybvkxjqz
 5 etaoinsrhldcumfpgwybvkxjqz
 6 etaoinsrhldcumfpgwybvkxjqz
 7 etaoinsrhldcumfpgwybvkxjqz
-7 etaoinsrhldcumfpgwybvkxjqz
-6 etaoinsrhldcumfpgwybvkxjqz
-5 etaoinsrhldcumfpgwybvkxjqz
-4 etaoinsrhldcumfpgwybvkxjqz
-3 etaoinsrhldcumfpgwybvkxjqz
-2 etaoinsrhldcumfpgwybvkxjqz
-1 etaoinsrhldcumfpgwybvkxjqz

Two-Letter Sequence (Bigram) Counts

Now we turn to sequences of letters: consecutive letters anywhere within a word. In the list below are the 50 most frequent two-letter sequences (which are called "bigrams"):
BI  COUNT   PERCENT bar graph
TH  100.3 B (3.56%)  TH
HE   86.7 B (3.07%)  HE
IN   68.6 B (2.43%)  IN
ER   57.8 B (2.05%)  ER
AN   56.0 B (1.99%)  AN
RE   52.3 B (1.85%)  RE
ON   49.6 B (1.76%)  ON
AT   41.9 B (1.49%)  AT
EN   41.0 B (1.45%)  EN
ND   38.1 B (1.35%)  ND
TI   37.9 B (1.34%)  TI
ES   37.8 B (1.34%)  ES
OR   36.0 B (1.28%)  OR
TE   34.0 B (1.20%)  TE
OF   33.1 B (1.17%)  OF
ED   32.9 B (1.17%)  ED
IS   31.8 B (1.13%)  IS
IT   31.7 B (1.12%)  IT
AL   30.7 B (1.09%)  AL
AR   30.3 B (1.07%)  AR
ST   29.7 B (1.05%)  ST
TO   29.4 B (1.04%)  TO
NT   29.4 B (1.04%)  NT
NG   26.9 B (0.95%)  NG
SE   26.3 B (0.93%)  SE
HA   26.1 B (0.93%)  HA
AS   24.6 B (0.87%)  AS
OU   24.5 B (0.87%)  OU
IO   23.5 B (0.83%)  IO
LE   23.4 B (0.83%)  LE
VE   23.3 B (0.83%)  VE
CO   22.4 B (0.79%)  CO
ME   22.4 B (0.79%)  ME
DE   21.6 B (0.76%)  DE
HI   21.5 B (0.76%)  HI
RI   20.5 B (0.73%)  RI
RO   20.5 B (0.73%)  RO
IC   19.7 B (0.70%)  IC
NE   19.5 B (0.69%)  NE
EA   19.4 B (0.69%)  EA
RA   19.3 B (0.69%)  RA
CE   18.4 B (0.65%)  CE
LI   17.6 B (0.62%)  LI
CH   16.9 B (0.60%)  CH
LL   16.3 B (0.58%)  LL
BE   16.2 B (0.58%)  BE
MA   15.9 B (0.57%)  MA
SI   15.5 B (0.55%)  SI
OM   15.4 B (0.55%)  OM
UR   15.3 B (0.54%)  UR

Below is a table of all 26 × 26 = 676 bigrams; in each cell the orange bar is proportional to the frequency, and if you hover you can see the exact counts and percentage. There are only seven bigrams that do not occur among the 2.8 trillion mentions: JQ, QG, QK, QY, QZ, WQ, and WZ. If you look closely you see they are shown as deleted.


N-Letter Sequences (N-grams)

What are the most common n-letter sequences (called "n-grams") for various values of n? You can see the 50 most common for each value of n from 1 to 9 in the table below. The counts and percentages are not shown, but don't worry -- you'll get lots of counts in the next section.

1 2grams  3grams  4-grams   5-grams    6-grams     7-grams      8-grams       9-grams   
e     th     the     tion     ation     ations     present     differen     different
t     he     and     atio     tions     ration     ational     national     governmen
a     in     ing     that     which     tional     through     consider     overnment
o     er     ion     ther     ction     nation     between     position     formation
i     an     tio     with     other     ection     ication     ifferent     character
n     re     ent     ment     their     cation     differe     governme     velopment
s     on     ati     ions     there     lation     ifferen     vernment     developme
r     at     for     this     ition     though     general     overnmen     evelopmen
h     en     her     here     ement     presen     because     interest     condition
l     nd     ter     from     inter     tation     develop     importan     important
d     ti     hat     ould     ional     should     america     ormation     articular
c     es     tha     ting     ratio     resent     however     formatio     particula
u     or     ere     hich     would     genera     eration     relation     represent
m     te     ate     whic     tiona     dition     nationa     question     individua
f     of     his     ctio     these     ationa     conside     american     ndividual
p     ed     con     ence     state     produc     onsider     characte     relations
g     is     res     have     natio     throug     ference     haracter     political
w     it     ver     othe     thing     hrough     positio     articula     informati
y     al     all     ight     under     etween     osition     possible     nformatio
b     ar     ons     sion     ssion     betwee     ization     children     universit
v     st     nce     ever     ectio     differ     fferent     elopment     following
k     to     men     ical     catio     icatio     without     velopmen     experienc
x     nt     ith     they     latio     people     ernment     developm     stitution
j     ng     ted     inte     about     iffere     vernmen     evelopme     xperience
q     se     ers     ough     count     fferen     overnme     conditio     education
z     ha     pro     ance     ments     struct     governm     ondition     roduction
      as     thi     were     rough     action     ulation     mportant     niversity
      ou     wit     tive     ative     person     another     rticular     therefore
      io     are     over     prese     eneral     importa     particul     nstitutio
      le     ess     ding     feren     system     interes     epresent     ification
      ve     not     pres     hough     relati     nterest     represen     establish
      co     ive     nter     ution     ctions     elation     increase     understan
      me     was     comp     roduc     ecause     rmation     individu     nderstand
      de     ect     able     resen     becaus     mportan     ndividua     difficult
      hi     rea     heir     thoug     before     product     dividual     structure
      ri     com     thei     press     ession     formati     elations     knowledge
      ro     eve     ally     first     develo     communi     nformati     struction
      ic     per     ated     after     evelop     lations     politica     something
      ne     int     ring     cause     uction     ormatio     olitical     necessary
      ea     est     ture     where     change     certain     universi     hemselves
      ra     sta     cont     tatio     follow     increas     function     themselve
      ce     cti     ents     could     positi     relatio     informat     plication
      li     ica     cons     efore     govern     special     niversit     anization
      ch     ist     rati     contr     sition     process     iversity     according
      ll     ear     thin     hould     merica     against     lication     differenc
      be     ain     part     shoul     direct     problem     experien     operation
      ma     one     form     tical     bility     nstitut     structur     ifference
      si     our     ning     gener     effect     politic     determin     rganizati
      om     iti     ecti     esent     americ     ination     ollowing     organizat
      ur     rat     some     great     public     univers     followin     ganizatio

N-gram Counts by Word Length and Position within Word

Finally we are ready to break out the results by n-gram length, by position within word (as we did for letter counts), and also by word length. You will be able to get counts for, say, the number of times the bigram "he" appears in positions 2 through 3 of 4-letter words, for example. This is the kind of tables provided by Mayzner, but with 37 million times more data (and with a few more columns). The tables are large, so we present them in separate files; for each n-gram length from n=1 to n=9, we offer a Google Fusion Table file; you can browse the table online, or download it (with the "File > Download" menu item). We also offer all the files rolled up into a .zip file, or in a fusion table folder:

NTypesMentionsFusion TableFile Size
1263,563,505,777,820ngrams120 KB
26692,819,662,855,499ngrams2280 KB
38,6532,098,121,156,991ngrams32 MB
442,1711,507,873,312,542ngrams46 MB
593,7131,070,193,846,800ngrams510 MB
6114,565742,502,715,592ngrams610 MB
7104,610494,400,907,903ngrams78 MB
882,347308,690,305,624ngrams85 MB
959,030182,032,364,549ngrams93 MB
Fusion Table Folder
11 MB

N-gram column notation

Each column is given a name of the form "wordlength / start : end". For example, "4/2:3" means that the column counts the number of ngrams that occur in 4 letter words (such as "then"), and only in position 2 through 3 (such as the "he" in "then"). We aggregate counts with a notation involving a "*": the notation "*/2:3" refers to the second through third position within words of any length; "4/*" refers to any start positions in words of length 4; and "*/*" means any start position within words of any length. Finally, we also aggregate counts for positions near the ends of words: the notation "*/-3:-2" means the third-to-last through second-to-last position in words of any length (for example, this would be the bigram "he" for the words "hen", "then", "lexicographer", and "greatgrandfather").

Closing Thoughts

Technology has certainly changed. Here's where you would typically see a comparison saying that if you punched the 743 billion words one to a card and stacked them up, then assuming 100 cards per inch, the stack would be 100,000 miles high; nearly halfway to the moon. But that's silly, because the stack would topple over long before then. If I had 743 billion cards, what I would do is stack them up in a big building, like, say, the
Vehicle Assembly Building (VAB) at Kennedy Space Center, which has a capacity of 3.6 million cubic meters. The cards work out to only 2.9 million cubic meters; easy peasy; room to spare. And an IBM model 84 card sorter could blast through these at a rate of 2000 cards per minute, which means it would only take 700 years per pass (but you'd need multiple passes to get the whole job done).

Aren't you glad I'm providing these tables online, rather than on cards? If you use these tables to do some interesting analysis, leave a comment to let us know. Enjoy!

Peter Norvig