zipf's law for indian languages

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Zipf's Law for Indian LanguagesB. D. Jayaram a & M. N. Vidya aa Central Institute of Indian Languages , Karnataka, IndiaPublished online: 10 Oct 2008.

To cite this article: B. D. Jayaram & M. N. Vidya (2008) Zipf's Law for Indian Languages,Journal of Quantitative Linguistics, 15:4, 293-317, DOI: 10.1080/09296170802326640

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Zipf’s Law for Indian Languages*

B. D. Jayaram and M. N. VidyaCentral Institute of Indian Languages, Karnataka, India

ABSTRACT

The present paper attempts to study the application of Zipf’s law for Indian languages. Itexamines the rank-frequency distribution in four Indian languages representing twoIndo-Aryan and two Dravidian languages. The sample texts were drawn from fivedifferent genres viz., aesthetics, commerce, natural physical and professional sciences,official and media languages, and social sciences. The rank-frequency distributions wereanalysed for fitting the distribution by using Altmann Fitter software where it fitted thetruncated zeta distribution defined as

Px ¼1

xaT; x ¼ 1; 2; . . . ;R

where R is the truncation parameter and T is the normalizing constant. The analysisshows that rank-frequency distribution follows Zipf’s law.

INTRODUCTION

In the domain of quantitative linguistics, word counting is perhaps themost popular discipline. This is supported by the fact that words aresimply ‘‘given’’ in the text: everybody can count them but especiallycomputers. Only genuine linguists consider the fact that a word is a verycomplex entity whose definition is not unique. Nevertheless, counting‘‘words’’ and preparing frequency dictionaries belong to the daily jobs of

*Address correspondence to: B. D. Jayaram, Research Officer, Central Institute of IndianLanguages, Manasagangotri, Hunsar Road, Mysore 570006, Karnataka, India.E-mail: jayaram/[email protected]

Journal of Quantitative Linguistics2008, Volume 15, Number 4, pp. 293–317DOI: 10.1080/09296170802326640

0929-6174/08/15040293 � 2008 Taylor & Francis

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computer linguists performing them without even any preliminaryhypothesis. However, the counting of words always has a practicalpurpose: it is useful for language teaching and grammatical studies. Thetheoretical use of word frequencies presupposes at least one hypothesiswhose validity should be corroborated by the word count. The founderof this kind of thinking was Zipf (1935, 1949) who set up a number ofhypotheses concerning word frequency. Today his heritage has twodirections: frequentistic and synergetic linguistics. However, theranking of words made popular by him has been initiated by Estoup(1916). Today it is an extensive discipline, perhaps the most pursued onein quantitative linguistics (cf. Kohler, 1995; Baayen, 2001; Glottometrics3–5, 2002). Zipf’s original idea of using the harmonic series and thepower series for capturing the relationship of rank and frequency hasbeen:

1. Replaced by the zeta distribution which is simply normalized andtruncated if necessary.

2. Replaced by a number of different models all of which represent amodification or a generalization of his original idea.

There is such an abundance of models that one automatically suspectsthat the researchers did not work with homogeneous (but mixed) data, orthat they did not care for a fruitful definition of the word, or that theysimply tried to find the best fitting distribution not caring for thetheoretical background.Word counting in Indian languages has a short and rather poor

tradition. There are some older publications on Bengali (Dabbs, 1966),Gujarati (Pandit, 1965), Hindi (Ghatage, 1964) and Urdu (Barker et al.,1969), Malayalam (Ghatage, 1994), Oriya (Kelkar, 1994) and somenewer ones, but the discipline as a whole begins to flourish due to theemployment of computers (cf. Jayaram, 2005; Jayaram & Rajyashree,2001).In the present study we want to examine the rank-frequency

distribution in four Indian languages and show:

1. That it abides by Zipf’s law; no modification is necessary.2. That the parameter a of the distribution strongly depends on the

repeat rate of the empirical distribution which can be used forestimation purposes.

294 B. D. JAYARAM & M. N. VIDYA

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DATA SOURCE

The sample texts in the present investigation are drawn from the IndianLanguage Written Corpora maintained by the Central Institute of IndianLanguages,Mysore. This corpus is of a general type covering all the genres ofthe language. The criteria considered for the categorization of the languageinto different genres were informational, administrative, instructional, andimaginative. Thus the data were collected under five main categories:

1. aesthetics,2. commerce,3. natural physical and professional sciences,4. official and media languages, and5. social sciences.

These were further subdivided into 76 text categories. The period of thecorpus was restricted to one decade, i.e. texts published between 1981 and1990 were included and it represents the contemporary Indian language(Jayaram & Rajyashree, 2001).

METHOD AND DATA FOR THE PRESENT INVESTIGATION

From the corpus of different Indian languages available at the CentralInstitute of Indian Languages, four languages were selected to representIndo-Aryan and Dravidian families of languages. The languages are Hindiand Marathi representing Indo-Aryan, and Kannada and Telugu represent-ing Dravidian. The sample text files, varying from five to 12, were selected atrandom from each genre of the four languages depending on the number ofsample texts available under each genre. The data from each text wasanalysed to obtain rank and frequency of words by developing a programwritten in Perl language. This rank-frequency data was further analysed forfitting the distribution by using Altmann-Fitter software.Adhering to the principle of economy we restricted our search to the

original Zipfian approach and fitted iteratively the right-truncated zetadistribution defined as

Px ¼1

xaT; x ¼ 1; 2; . . . ;R ð1Þ

ZIPF’S LAW FOR INDIAN LANGUAGES 295

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where R is the truncation parameter at the right-hand side of thedistribution and T is the normalizing constant

T ¼XR

j¼1j�a

which can be expressed by means of Riemann’s zeta function or byLerch’s function.

RESULTS AND DISCUSSIONS

The analysis of data was carried out using Altmann-Fitter software.Altmann-Fitter is an interactive program for fitting theoretical univariatediscrete probability functions to empirical frequency distributions usingmethods of iterative optimization based on the simplex algorithm byNelder-Mead. Fitting starts with the common point estimates and isoptimized successively. Goodness of fit is determined by the chi-squaretest and the associated probability. An example of fitting is shown in thetable in the Appendix.The empirical repeat rate has been computed for each text according to

RR ¼ 1

N 2

XR

x¼1f 2x ð2Þ

where N is the number of all words in the text (¼ text length, number oftokens), R is the maximal rank, i.e. the number of types, identical withthe truncation parameter of the Zipf distribution, and fx is the absolutefrequency at rank x. This index lies in the interval h1/R, 1i but in long-taildata it is usually so small that ‘‘optically’’ it is not very expressive.Besides, it is not directly comparable. Therefore one usually takes the‘‘relative repeat rate’’ according to McIntosh (1967), defined as

RRrel ¼1�

ffiffiffiffiffiffiffiRRp

1� 1=ffiffiffiffiRp ð3Þ

The values of RRrel can be found in the last column of all tables.The results are presented in the following 20 tables analysing the texts

for each language and genre separately (four languages and five genres).


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The tables present the values of parameters a and R (representing thetruncation parameter, i.e. the number of types), the corresponding degrees offreedom, w2 values with probability, the sample text size (N), repeat rate andrelative repeat rate. It can be seen from the result that all texts from eachlanguage and each genre follow the right-truncated zeta distribution exceptfor one text in the category of Telugu–Commerce. The degrees of freedomstrongly differ from the number of types (R) because many classes have beenpooled in order to attain the necessary expected frequency.In order to show the output of the exact analysis the data from one text

has been analyzed in detail by giving the rank of each word and itsfrequency and also the expected frequency obtained by using the right-truncated zeta distribution. The result of this analysis is shown in thegraph, which indicates that rank-frequency data follows Zipf’s first law.Detailed analysis of one sample text of aesthetics from Hindi is shown in

the Appendix and graphically presented in Figure A1 (bilogarithmic scale).

THE BEHAVIOUR OF PARAMETER a

Having obtained very positive fittings of the right-truncated Zipf (¼ zeta)distribution to 195 data sets from four languages, we see that theparameter a variates in the interval h0.46, 0.88i. We know that it dependson the form of the empirical distribution but its estimation is not simplebecause of the normalizing constant which is itself a function of a. Inaddition, it must be estimated in each case separately if one does not usea software package. On the other hand, each parameter of the

Table 1. Fitting of right-truncated zeta distribution of Hindi language: Aesthetics.

Text no. a R df w2 P(w2) N RR RRrel

Aes 03 0.7890 865 782 112.24 1.0000 2757 0.0072 0.9471Aes 04 0.8119 893 712 84.320 1.0000 2362 0.0101 0.9304Aes 11 0.8088 638 569 94.360 1.0000 2055 0.0099 0.9376Aes 12 0.8269 550 535 76.890 1.0000 2139 0.0116 0.9321Aes 15 0.7589 653 580 78.190 1.0000 1849 0.0074 0.9510Aes 16 0.7679 1136 904 108.87 1.0000 2737 0.0074 0.9417Aes 26 0.8095 710 605 60.020 1.0000 2110 0.0101 0.9344Aes 27 0.7911 1045 820 137.86 1.0000 2566 0.0082 0.9382Aes 28 0.8617 845 758 114.01 1.0000 3224 0.0118 0.9230Aes 31 0.8387 779 642 100.27 1.0000 2343 0.0115 0.9261


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Table 2. Fitting of right-truncated zeta distribution of Hindi language: Commerce.


Comm. 160 0.7632 741 616 34.50 1.0000 1894 0.0088 0.9405Comm. 164 0.7633 699 581 32.93 1.0000 1778 0.0093 0.9393Comm. 173 0.7989 611 608 113.87 1.0000 2311 0.0113 0.9313Comm. 295 0.7921 678 578 47.53 1.0000 1932 0.0105 0.9334Comm. 296 0.8077 721 694 94.07 1.0000 2656 0.0091 0.9395Comm. 297 0.7975 765 665 58.58 1.0000 2306 0.0100 0.9339Comm. 298 0.7694 813 668 56.27 1.0000 2069 0.0090 0.9380Comm. 299 0.7871 844 711 74.18 1.0000 2347 0.0088 0.9387Comm. 300 0.8384 520 517 167.78 1.0000 2317 0.0122 0.9304Comm. 301 0.8560 546 543 84.34 1.0000 2434 0.0143 0.9199

Table 3. Fitting of right-truncated zeta distribution of Hindi language: Natural physicaland professional sciences.


Npps 013 0.8510 714 627 76.80 1.0000 2510 0.0122 0.9242Npps 014 0.7817 795 702 200.57 1.0000 2374 0.0065 0.9534Npps 021 0.7780 976 813 81.29 1.0000 2619 0.0073 0.9451Npps 022 0.7805 773 736 115.06 1.0000 2612 0.0074 0.9483Npps 023 0.8322 823 690 116.36 1.0000 2532 0.0100 0.9325Npps 210 0.8064 759 653 78.03 1.0000 2293 0.0095 0.9366Npps 211 0.8045 1072 900 142.80 1.0000 3131 0.0071 0.9443Npps 212 0.7752 907 745 70.19 1.0000 2347 0.0077 0.9439Npps 213 0.7901 943 799 87.22 1.0000 2684 0.0078 0.9424Npps 215 0.8015 748 699 106.77 1.0000 2588 0.0085 0.9424

Table 4. Fitting of right-truncated zeta distribution of Hindi language: Official and medialanguages.


Oml 149 0.7177 712 566 56.70 1.0000 1515 0.0070 0.9520Oml 150 0.7311 560 460 31.44 1.0000 1294 0.0087 0.9466Oml 151 0.7536 732 593 54.02 1.0000 1744 0.0081 0.9451Oml 152 0.7803 850 704 81.62 1.0000 2255 0.0084 0.9408Oml 154 0.8166 970 842 132.09 1.0000 3100 0.0080 0.9410Oml 155 0.7985 725 638 92.83 1.0000 2235 0.0085 0.9429Oml 156 0.7992 750 665 99.46 1.0000 2353 0.0083 0.9433Oml 157 0.7803 908 763 67.13 1.0000 2482 0.0079 0.9427Oml 158 0.7769 727 606 55.34 1.0000 1922 0.0087 0.9417


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Table 5. Fitting of right-truncated zeta distribution of Hindi language: Social science.


SS 005 0.8736 678 592 116.02 1.0000 2497 0.0134 0.9197SS 006 0.8110 719 645 121.15 1.0000 2368 0.0088 0.9413SS 007 0.8436 909 752 63.83 1.0000 2816 0.0117 0.9226SS 008 0.7755 586 558 92.48 1.0000 1929 0.0084 0.9474SS 009 0.8009 697 667 79.16 1.0000 2492 0.0093 0.9392SS 010 0.8079 820 650 84.77 1.0000 2116 0.0104 0.9305SS 190 0.7793 775 637 91.46 1.0000 2014 0.0081 0.9441SS 191 0.8048 671 567 68.92 1.0000 1934 0.0100 0.9362SS 192 0.8113 581 535 57.14 1.0000 1978 0.0107 0.9352SS 193 0.7847 581 578 121.64 1.0000 2332 0.0087 0.9461

Table 6. Fitting of right-truncated zeta distribution of Kannada language: Aesthetics.


Aes 07 0.5900 2724 2081 176.1259 1.0000 4434 0.0016 0.9790Aes 08 0.5640 2781 2136 159.2632 1.0000 4393 0.0013 0.9825Aes 09 0.5888 2222 1734 124.2403 1.0000 3733 0.0016 0.9806Aes 10 0.6187 2613 2094 135.8057 1.0000 4828 0.0017 0.9780Aes 24 0.5589 2896 2239 172.7607 1.0000 4588 0.0011 0.9852Aes 25 0.5910 2635 2088 175.8098 1.0000 4559 0.0013 0.9838Aes 26 0.5806 2072 1702 92.2136 1.0000 3716 0.0016 0.9820Aes 44 0.5687 1281 977 84.3413 1.0000 2000 0.0026 0.9765Aes 45 0.5323 2747 2163 128.2723 1.0000 4304 0.0010 0.9874Aes 46 0.5457 3077 2359 186.6336 1.0000 4723 0.0010 0.9860Aes 47 0.5350 2726 2078 174.5346 1.0000 4084 0.0013 0.9834Aes 48 0.4626 1623 1237 81.9487 1.0000 2219 0.0013 0.9887

Table 7. Fitting of right-truncated zeta distribution of Kannada language: Commerce.


Comm 01 0.7008 1664 1364 109.0306 1.0000 3713 0.0042 0.9590Comm 02 0.7121 1738 1532 160.2482 1.0000 4508 0.0032 0.9669Comm 10 0.7114 1782 1543 135.0751 1.0000 4483 0.0035 0.9635Comm 19 0.6519 833 714 48.6209 1.0000 1787 0.0041 0.9699Comm 20 0.7218 1755 1526 133.0404 1.0000 4556 0.0038 0.9610Comm 21 0.6359 790 628 41.6713 1.0000 1455 0.0047 0.9655Comm 22 0.7360 1764 1500 150.4831 1.0000 4554 0.0042 0.9582Comm 23 0.7285 1738 1532 166.4100 1.0000 4685 0.0036 0.9635Comm 30 0.6892 2005 1667 113.9220 1.0000 4499 0.0032 0.9646Comm 31 0.7039 1920 1644 190.3819 1.0000 4672 0.0028 0.9694


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Table 8. Fitting of right-truncated zeta distribution of Kannada language: Naturalphysical and professional sciences.



Table 9. Fitting of right-truncated zeta distribution of Kannada language: Official andmedia language.


Oml 075 0.7696 1635 1377 88.2640 1.0000 4485 0.0059 0.9464Oml 080 0.6370 2579 2043 215.8699 1.0000 4829 0.0017 0.9784Oml 081 0.6071 1830 1452 104.8726 1.0000 3247 0.0020 0.9778Oml 082 0.6004 873 670 63.0916 1.0000 1435 0.0044 0.9665Oml 186 0.6401 1806 1431 119.5314 1.0000 3382 0.0023 0.9747

Table 10. Fitting of right-truncated zeta distribution of Kannada language: Social science.




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Table 11. Fitting of right-truncated zeta distribution of Marathi language: Aesthetics.



Table 12. Fitting of right-truncated zeta distribution of Marathi language: Commerce.


Comm 027 0.7256 1400 1316 203.0357 1.0000 4128 0.0036 0.9656Comm 028 0.6923 2386 1938 226.8475 1.0000 5191 0.0023 0.9724Comm 029 0.6843 1412 1245 111.7723 1.0000 3424 0.0032 0.9692Comm 038 0.6924 1607 1439 177.2989 1.0000 4078 0.0027 0.9725Comm 046 0.6995 1458 1404 218.6351 1.0000 4186 0.0029 0.9718Comm 052 0.6844 1628 1343 94.3626 1.0000 3549 0.0032 0.9678Comm 149 0.5931 1547 1308 76.4122 1.0000 2946 0.0020 0.9805Comm 150 0.6390 1523 1346 105.6338 1.0000 3372 0.0023 0.9768Comm 151 0.7343 1702 1537 144.9855 1.0000 4843 0.0041 0.9596Comm 154 0.6480 1719 1445 103.5503 1.0000 3601 0.0023 0.9753

Table 13. Fitting of right-truncated zeta distribution of Marathi language: Naturalphysical and professional sciences.




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Table 14. Fitting of right-truncated zeta distribution of Marathi language: Official andmedia languages.


Oml 288 0.6604 2079 1649 133.6021 1.0000 4060 0.0025 0.9718Oml 289 0.6746 2312 1874 141.6499 1.0000 4831 0.0024 0.9716Oml 290 0.5983 2319 1829 152.3543 1.0000 4025 0.0015 0.9817Oml 291 0.6694 1957 1569 114.9293 1.0000 3954 0.0027 0.9700Oml 292 0.6820 2197 1805 153.5596 1.0000 4765 0.0025 0.9711Oml 293 0.5610 2006 1601 97.1879 1.0000 3337 0.0014 0.9843Oml 294 0.6578 1931 1550 105.8837 1.0000 3825 0.0026 0.9715Oml 295 0.6691 2322 1904 129.0378 1.0000 4895 0.0022 0.9730Oml 296 0.6674 1970 1550 130.9973 1.0000 3836 0.0027 0.9695Oml 297 0.6747 2278 1813 150.8252 1.0000 4605 0.0025 0.9707

Table 15. Fitting of right-truncated zeta distribution of Marathi language: Social science.



Table 16. Fitting of right-truncated zeta distribution of Telugu language: Aesthetics.




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Table 17. Fitting of right-truncated zeta distribution of Telugu language: Commerce.


Comm 038 0.6113 2462 2054 186.9583 1.0000 4779 0.0014 0.9825Comm 039 0.6592 2164 1872 230.3849 1.0000 4869 0.0019 0.9776Comm 040 0.6278 2984 2484 200.1874 1.0000 5969 0.0015 0.9787Comm 100 0.6918 2039 1774 211.7489 1.0000 4964 0.0024 0.9725Comm 101 – – – – – 6316 – –Comm 114 0.6014 2618 2130 160.3616 1.0000 4806 0.0014 0.9823Comm 115 0.6271 2863 2291 182.7469 1.0000 5363 0.0016 0.9782Comm 156 0.7597 2011 1775 248.1304 1.0000 5900 0.0036 0.9611Comm 425 0.8238 1029 1026 369.9993 1.0000 4427 0.0068 0.9468Comm 426 0.7122 665 606 58.7884 1.0000 1765 0.0061 0.9594

Table 18. Fitting of right-truncated zeta distribution of Telugu language: Naturalphysical and professional sciences.



Table 19. Fitting of right-truncated zeta distribution of Telugu language: Official andmedia languages.


Oml 76 0.5991 1733 1405 88.9548 1.0000 3136 0.0021 0.9775Oml 77 0.5471 1225 981 56.2165 1.0000 1999 0.0020 0.9829Oml 78 0.5024 654 518 37.0058 1.0000 984 0.0031 0.9823Oml 79 0.5920 1603 1310 80.6293 1.0000 2899 0.0021 0.9786Oml 80 0.6261 1393 1179 104.4937 1.0000 2817 0.0025 0.9757Oml 81 0.6327 1460 1208 69.8876 1.0000 2889 0.0027 0.9733Oml 82 0.5932 1628 1322 116.8656 1.0000 2921 0.0018 0.9823Oml 83 0.6095 1438 1236 90.2426 1.0000 2886 0.0021 0.9797Oml 84 0.6006 1607 1352 82.8308 1.0000 3081 0.0021 0.9790Oml 85 0.6082 1508 1252 79.8901 1.0000 2870 0.0024 0.9758


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distribution must be expressible by means of some functions, e.g.moments. But in long-tail data like word frequencies, moments are notvery useful as characteristics. Hence we try to find a simple function ofthe empirical data which could be associated with the parameter a. Tothis end we consider the relative repeat rate as given in equation (2) andempirically in the last columns of all tables.If one inserts the points in a co-ordinate system, one sees at once that a

is evidently linked with RRrel. The supposed dependence has the form ofa concave monotone decreasing curve. In order to obtain this curve weuse the Wimmer-Altmann approach of modelling and consider therelative rate of change of the dependent variable y (here a) proportionalto the change of the independent variable x (here RRrel) which has aminimum (m) and a maximum (M) and set

y0

y¼ b

x�m� a

M� xð4Þ

which results in

y ¼ CðM� xÞaðx�mÞb ð5Þ

Setting the integration constant C¼ 1 in order to save a parameter wecan get different curves according to our aim. If we insert the empiricalminimum and maximum of RRrel we obtain

y ¼ ð0:993� xÞ0:1478ðx� 0:915Þ�0:0532

yielding a determination coefficient D¼ 0.885.

Table 20. Fitting of right-truncated zeta distribution of Telugu language: Social science.




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If we consider the minimum as zero, we obtain

y ¼ ð0:993� xÞ0:1258x�2:2825; yielding D ¼ 0:89

but if we let m be simply a constant, we obtain

y ¼ ð0:999� xÞ0:2611ðxþ 1:4638Þ0:5837 ð6Þ

yielding D¼ 0.91. The values of a computed according to this lastformula are presented in the third column of Table 21.

Table 21. Relative repeat rate RRrel and parameter a in 195 texts of four Indianlanguages.

RRrel a acomp

0.9197 0.8736 0.85880.9199 0.8560 0.85820.9226 0.8436 0.85110.9230 0.8617 0.85000.9242 0.8510 0.84670.9261 0.8387 0.84150.9304 0.8119 0.82910.9304 0.8384 0.82910.9305 0.8079 0.82890.9313 0.7989 0.82650.9321 0.8269 0.82410.9325 0.8322 0.82290.9334 0.7921 0.82020.9339 0.7975 0.81860.9344 0.8095 0.81710.9352 0.8113 0.81460.9362 0.8048 0.81150.9366 0.8064 0.81020.9376 0.8088 0.80700.9380 0.7694 0.80570.9382 0.7911 0.80500.9387 0.7871 0.80340.9392 0.8009 0.80180.9393 0.7633 0.80140.9395 0.8077 0.80080.9405 0.7632 0.79750.9408 0.7803 0.79640.9410 0.8166 0.7958

RRrel a acomp

0.9413 0.8110 0.79480.9417 0.7679 0.79340.9417 0.7769 0.79340.9424 0.7901 0.79100.9424 0.8015 0.79100.9427 0.7803 0.79000.9429 0.7985 0.78930.9433 0.7992 0.78790.9439 0.7752 0.78570.9441 0.7793 0.78510.9443 0.8045 0.78440.9451 0.7780 0.78150.9451 0.7536 0.78150.9461 0.7847 0.77790.9464 0.7696 0.77680.9466 0.7311 0.77610.9468 0.8238 0.77530.9471 0.7890 0.77420.9474 0.7755 0.77310.9483 0.7805 0.76980.9510 0.7589 0.75940.9518 0.7733 0.75620.9520 0.7177 0.75540.9534 0.7817 0.74980.9535 0.7367 0.74930.9537 0.7773 0.74850.9571 0.7668 0.73410.9582 0.7360 0.7292

(continued )


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RRrel a acomp

0.9590 0.7008 0.72560.9594 0.7122 0.72380.9596 0.7343 0.72290.9599 0.7298 0.72150.9610 0.7218 0.71630.9611 0.7597 0.71590.9614 0.7096 0.71440.9635 0.7114 0.70420.9635 0.7285 0.70420.9646 0.6892 0.69860.9648 0.7034 0.69760.9649 0.6860 0.69710.9651 0.7096 0.69610.9655 0.6359 0.69400.9656 0.7145 0.69350.9656 0.7256 0.69350.9664 0.6989 0.68920.9665 0.6004 0.68870.9669 0.7121 0.68650.9672 0.6850 0.68490.9678 0.6844 0.68160.9680 0.6807 0.68050.9687 0.6395 0.67660.9692 0.6843 0.67380.9694 0.7039 0.67260.9695 0.6674 0.67200.9695 0.6882 0.67200.9699 0.6519 0.66970.9700 0.6694 0.66910.9702 0.6574 0.66800.9704 0.7008 0.66680.9707 0.6747 0.66500.9709 0.6557 0.66380.9709 0.6804 0.66380.9709 0.6935 0.66380.9711 0.6820 0.66260.9711 0.5776 0.66260.9712 0.6720 0.66200.9714 0.6723 0.66080.9715 0.6578 0.66020.9716 0.6746 0.65960.9718 0.6995 0.65870.9718 0.6604 0.6587

RRrel a acomp

0.9720 0.6821 0.65710.9724 0.6923 0.65470.9725 0.6924 0.65400.9725 0.6918 0.65400.9728 0.6373 0.65210.9728 0.6821 0.65210.9730 0.6691 0.65090.9732 0.5791 0.64960.9733 0.6327 0.64860.9734 0.6603 0.64830.9735 0.6563 0.64770.9738 0.6718 0.64570.9739 0.6632 0.64510.9747 0.6401 0.63980.9750 0.6339 0.63780.9752 0.6476 0.63640.9752 0.5771 0.63640.9752 0.7009 0.63640.9753 0.6480 0.63570.9755 0.6913 0.63440.9756 0.6529 0.63370.9757 0.6261 0.63300.9758 0.6082 0.63230.9761 0.6441 0.63020.9762 0.6424 0.62950.9764 0.6565 0.62810.9765 0.5687 0.62740.9767 0.6698 0.62600.9768 0.6390 0.62520.9769 0.6558 0.62450.9770 0.6104 0.62380.9771 0.6636 0.62310.9773 0.6538 0.62160.9775 0.5984 0.62010.9775 0.6121 0.62010.9775 0.5991 0.62010.9776 0.6592 0.61940.9777 0.6339 0.61870.9777 0.6448 0.61870.9778 0.6071 0.61790.9780 0.6187 0.61640.9780 0.6521 0.61640.9780 0.6478 0.6164

(continued )

Table 21. (Continued ).


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The graphical representation can be found in Figure 1.We suppose that if a rank-frequency distribution of words follows the

Zipf (zeta) distribution, then its parameter a can be estimated for anylanguage from formula (6) by simply substituting the value of the relativerepeat rate of the sample for x. This is a different approach to theestimation but it shows at the same time that the characteristics of arank-frequency distribution must be interrelated.

RRrel a acomp

0.9782 0.6271 0.61490.9782 0.6362 0.61490.9784 0.6370 0.61340.9784 0.5650 0.61340.9786 0.5920 0.61190.9787 0.6278 0.61110.9790 0.5900 0.60880.9790 0.6392 0.60880.9790 0.6006 0.60880.9792 0.5863 0.60720.9795 0.6221 0.60490.9795 0.6246 0.60490.9797 0.6095 0.60330.9797 0.6076 0.60330.9798 0.5836 0.60250.9805 0.5931 0.59680.9806 0.5888 0.59600.9808 0.6070 0.59430.9808 0.5979 0.59430.9808 0.6277 0.59430.9809 0.6037 0.59350.9816 0.6108 0.58750.9817 0.5983 0.58660.9820 0.5806 0.58400.9821 0.6351 0.58310.9822 0.4824 0.58230.9823 0.6014 0.58140.9823 0.5024 0.58140.9823 0.5932 0.58140.9825 0.5640 0.57960.9825 0.6113 0.57960.9829 0.5471 0.5759

RRrel a acomp

0.9831 0.5516 0.57410.9834 0.5350 0.57130.9834 0.5374 0.57130.9836 0.6069 0.56940.9837 0.4940 0.56850.9838 0.5910 0.56750.9843 0.5610 0.56270.9843 0.5855 0.56270.9844 0.5691 0.56170.9846 0.5482 0.55970.9852 0.5589 0.55360.9852 0.4809 0.55360.9860 0.5457 0.54520.9862 0.5292 0.54300.9862 0.4925 0.54300.9874 0.5323 0.52940.9874 0.4825 0.52940.9877 0.4716 0.52590.9880 0.4953 0.52220.9887 0.4626 0.51350.9929 0.4634 0.4485

Table 21. (Continued ).


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CONCLUSION

The Indian languages, namely Hindi, Marathi, Kannada, andTelugu, follow Zipf’s law with only one exception which must beexamined separately. The parameter a varies across languages andacross categories. It determines the course of the distribution andmust be related to various other characteristics of the data. In thiscase we have chosen the relative repeat rate. The above results arebased on several sample texts drawn from different genres in eachlanguage.The result describes quantitatively the regularities one can find with

regard to the relation of the rank and frequency of the words indifferent Indian languages. The detailed study of parameter a and itsassociation with the relative repeat rate on many more Indianlanguages may yield new insights in the mechanism of word frequencydistributions.

Fig.1. Dependence of the parameter a on the relative repeat rate in four Indian languages.


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REFERENCES

Baayen, R. H. (2001). Word Frequency Studies. Dordrecht: Kluwer.Barker, M. A., Hamdani, J. H., & Dihlavi, K. M. S. (1969). An Urdu Newspaper Word

Count. Montreal: Institute of Islamic Studies, McGill University.Dabbs, J. A. (1966). Word Frequencies in Newspaper Bengali. Texas: A & M

University.Estoup, J. B. (1916). Gammes stenographiques. Methode et exercises pour l’acquisition de

la vitesse. Paris: Institut stenographique.Ghatage, A. M. (Ed.) (1964). Phonemic and Morphemic Frequencies in Hindi. Poona:

Deccan College Postgraduate and Research Institute.Ghatage, A. M. (1994). Phonemic and Morphemic Frequency Count in Malayalam.

Mysore: Ciil.Glottometrics 3–5. (2002). To Honor G. K. Zipf. Ludenscheid: RAM.Jayaram, B. D. (2005). Corpora in Indian languages. In G. Altmann, V. Levickij & V.

Perebyinis (Eds), Problems of Quantitative Linguistics (pp. 323–329). Chernivcy:Ruta.

Jayaram, B. D., & Rajyashree, K. S. (2001). Indian language corpora development. Paperpresented in Language Engineering in South Asian Languages, NCST, Mumbai,India, 23–25 April 2001.

Kelkar, A. R. (1994). Phonemic and Morphemic Frequency Count in Oriya. Mysore:Ciil.

Kohler, R. (1995). Bibliography of Quantitative Linguistics. Amsterdam: Benjamins.McIntosh, R. P. (1967). An index of diversity and the relation to certain concepts of

diversity. Ecology, 48, 392–404.Pandit, P. W. (1965). Phonemic and Morphemic Frequencies of the Gujarati Language.

Poona: Deccan College Postgraduate and Research Institute.Zipf, G. K. (1935). The Psycho-biology of Language. Boston: Houghton Mifflin.Zipf, G. K. (1949). Human Behavior and the Principle of the Least Effort. An Introduction

to Human Ecology. New York: Hafner.


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APPENDIX: AN EXAMPLE OF FITTING

X(i) f(i) Nf(i)

1 100 177.1312 75 102.5093 74 74.44194 67 59.32445 59 49.74696 58 43.08127 53 38.14728 45 34.33239 34 31.285410 32 28.789611 29 26.703912 27 24.932013 27 23.406014 26 22.076615 25 20.906916 25 19.868917 23 18.940818 21 18.105519 19 17.349320 18 16.661221 18 16.031922 17 15.454123 17 14.921524 16 14.428725 16 13.971326 16 13.545627 16 13.148228 14 12.776229 14 12.427330 14 12.099331 13 11.790332 13 11.498633 13 11.222734 12 10.961535 12 10.713636 11 10.478137 11 10.254038 11 10.040539 11 9.836840 11 9.642241 10 9.456142 10 9.2780

X(i) f(i) Nf(i)

43 10 9.107444 10 8.943645 9 8.786546 9 8.635447 9 8.490148 9 8.350249 9 8.215550 9 8.085551 9 7.960252 8 7.839153 8 7.722254 8 7.609155 8 7.499856 8 7.393957 8 7.291358 8 7.192059 8 7.095660 8 7.002161 7 6.911462 7 6.823363 7 6.737764 7 6.654565 7 6.573666 7 6.494867 7 6.418268 7 6.343669 7 6.271070 7 6.200271 7 6.131272 7 6.063973 7 5.998274 7 5.934275 7 5.871776 7 5.810677 6 5.751078 6 5.692779 6 5.635880 6 5.580181 6 5.525782 6 5.472583 6 5.420484 6 5.3694

X(i) f(i) Nf(i)

85 6 5.319586 6 5.270687 6 5.222888 6 5.175989 6 5.129990 6 5.084991 5 5.040892 5 4.997593 5 4.955094 5 4.913495 5 4.872596 5 4.832497 5 4.793198 5 4.754599 5 4.7165100 5 4.6793101 5 4.6427102 5 4.6067103 5 4.5714104 5 4.5367105 5 4.5026106 5 4.4690107 5 4.4360108 5 4.4036109 5 4.3717110 5 4.3403111 5 4.3094112 5 4.2790113 5 4.2491114 5 4.2197115 5 4.1907116 5 4.1621117 5 4.1340118 5 4.1064119 5 4.0791120 5 4.0523121 4 4.0258122 4 3.9998123 4 3.9741124 4 3.9488125 4 3.9238126 4 3.8992

(continued )


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X(i) f(i) Nf(i)

127 4 3.8750128 4 3.8511129 4 3.8275130 4 3.8043131 4 3.7813132 4 3.7587133 4 3.7364134 4 3.7144135 4 3.6926136 4 3.6712137 4 3.6500138 4 3.6292139 4 3.6085140 4 3.5882141 4 3.5681142 4 3.5482143 4 3.5287144 4 3.5093145 4 3.4902146 4 3.4713147 4 3.4527148 4 3.4343149 4 3.4161150 4 3.3981151 4 3.3803152 4 3.3627153 4 3.3454154 4 3.3282155 4 3.3113156 4 3.2945157 4 3.2779158 4 3.2616159 3 3.2454160 3 3.2294161 3 3.2135162 3 3.1979163 3 3.1824164 3 3.1670165 3 3.1519166 3 3.1369167 3 3.1221168 3 3.1074169 3 3.0929170 3 3.0785

X(i) f(i) Nf(i)

171 3 3.0643172 3 3.0502173 3 3.0363174 3 3.0225175 3 3.0089176 3 2.9954177 3 2.9820178 3 2.9688179 3 2.9557180 3 2.9428181 3 2.9299182 3 2.9172183 3 2.9046184 3 2.8922185 3 2.8798186 3 2.8676187 3 2.8555188 3 2.8435189 3 2.8316190 3 2.8198191 3 2.8082192 3 2.7966193 3 2.7852194 3 2.7739195 3 2.7626196 3 2.7515197 3 2.7405198 3 2.7296199 3 2.7187200 3 2.7080201 3 2.6974202 3 2.6868203 3 2.6764204 3 2.6660205 3 2.6557206 3 2.6456207 3 2.6355208 3 2.6255209 3 2.6156210 3 2.6057211 3 2.596212 3 2.5863213 3 2.5767214 3 2.5672

X(i) f(i) Nf(i)

215 3 2.5578216 3 2.5484217 3 2.5392218 3 2.5300219 3 2.5209220 3 2.5118221 3 2.5028222 3 2.4939223 3 2.4851224 3 2.4764225 3 2.4677226 3 2.4590227 2 2.4505228 2 2.4420229 2 2.4336230 2 2.4252231 2 2.4169232 2 2.4087233 2 2.4006234 2 2.3925235 2 2.3844236 2 2.3765237 2 2.3685238 2 2.3607239 2 2.3529240 2 2.3451241 2 2.3375242 2 2.3298243 2 2.3223244 2 2.3148245 2 2.3073246 2 2.2999247 2 2.2925248 2 2.2852249 2 2.2780250 2 2.2708251 2 2.2637252 2 2.2566253 2 2.2495254 2 2.2425255 2 2.2356256 2 2.2287257 2 2.2219258 2 2.2151

(continued )

Appendix (Continued ).


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X(i) f(i) Nf(i)

259 2 2.2083260 2 2.2016261 2 2.1950262 2 2.1883263 2 2.1818264 2 2.1752265 2 2.1688266 2 2.1623267 2 2.1559268 2 2.1496269 2 2.1433270 2 2.1370271 2 2.1308272 2 2.1246273 2 2.1185274 2 2.1124275 2 2.1063276 2 2.1003277 2 2.0943278 2 2.0883279 2 2.0824280 2 2.0766281 2 2.0707282 2 2.0649283 2 2.0592284 2 2.0534285 2 2.0478286 2 2.0421287 2 2.0365288 2 2.0309289 2 2.0254290 2 2.0199291 2 2.0144292 2 2.0089293 2 2.0035294 2 1.9981295 2 1.9928296 2 1.9875297 2 1.9822298 2 1.9769299 2 1.9717300 2 1.9665301 2 1.9614

X(i) f(i) Nf(i)

302 2 1.9563303 2 1.9512304 2 1.9461305 2 1.9411306 2 1.9360307 2 1.9311308 2 1.9261309 2 1.9212310 2 1.9163311 2 1.9114312 2 1.9066313 2 1.9018314 2 1.8970315 2 1.8923316 2 1.8875317 2 1.8828318 2 1.8782319 2 1.8735320 2 1.8689321 2 1.8643322 2 1.8597323 2 1.8552324 2 1.8507325 2 1.8462326 2 1.8417327 2 1.8373328 2 1.8328329 2 1.8284330 2 1.8241331 2 1.8197332 2 1.8154333 2 1.8111334 2 1.8068335 2 1.8026336 2 1.7983337 2 1.7941338 2 1.7899339 2 1.7857340 2 1.7816341 2 1.7775342 2 1.7734343 2 1.7693344 2 1.7652

X(i) f(i) Nf(i)

345 2 1.7612346 2 1.7572347 2 1.7532348 2 1.7492349 2 1.7453350 2 1.7413351 2 1.7374352 2 1.7335353 2 1.7296354 2 1.7258355 1 1.7219356 1 1.7181357 1 1.7143358 1 1.7105359 1 1.7068360 1 1.7030361 1 1.6993362 1 1.6956363 1 1.6919364 1 1.6883365 1 1.6846366 1 1.6810367 1 1.6774368 1 1.6738369 1 1.6702370 1 1.6666371 1 1.6631372 1 1.6595373 1 1.6560374 1 1.6525375 1 1.6491376 1 1.6456377 1 1.6421378 1 1.6387379 1 1.6353380 1 1.6319381 1 1.6285382 1 1.6252383 1 1.6218384 1 1.6185385 1 1.6152386 1 1.6119387 1 1.6086

(continued )



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X(i) f(i) Nf(i)

388 1 1.6053389 1 1.6020390 1 1.5988391 1 1.5956392 1 1.5924393 1 1.5892394 1 1.5860395 1 1.5828396 1 1.5797397 1 1.5765398 1 1.5734399 1 1.5703400 1 1.5672401 1 1.5641402 1 1.5610403 1 1.5580404 1 1.5549405 1 1.5519406 1 1.5489407 1 1.5459408 1 1.5429409 1 1.5399410 1 1.5369411 1 1.5340412 1 1.5310413 1 1.5281414 1 1.5252415 1 1.5223416 1 1.5194417 1 1.5165418 1 1.5137419 1 1.5108420 1 1.5080421 1 1.5052422 1 1.5023423 1 1.4995424 1 1.4968425 1 1.4940426 1 1.4912427 1 1.4885428 1 1.4857429 1 1.4830430 1 1.4803

X(i) f(i) Nf(i)

431 1 1.4775432 1 1.4748433 1 1.4722434 1 1.4695435 1 1.4668436 1 1.4642437 1 1.4615438 1 1.4589439 1 1.4563440 1 1.4536441 1 1.4510442 1 1.4484443 1 1.4459444 1 1.4433445 1 1.4407446 1 1.4382447 1 1.4356448 1 1.4331449 1 1.4306450 1 1.4281451 1 1.4256452 1 1.4231453 1 1.4206454 1 1.4182455 1 1.4157456 1 1.4132457 1 1.4108458 1 1.4084459 1 1.4060460 1 1.4035461 1 1.4011462 1 1.3987463 1 1.3964464 1 1.3940465 1 1.3916466 1 1.3893467 1 1.3869468 1 1.3846469 1 1.3822470 1 1.3799471 1 1.3776472 1 1.3753473 1 1.3730

X(i) f(i) Nf(i)

474 1 1.3707475 1 1.3684476 1 1.3662477 1 1.3639478 1 1.3617479 1 1.3594480 1 1.3572481 1 1.3550482 1 1.3527483 1 1.3505484 1 1.3483485 1 1.3461486 1 1.3439487 1 1.3418488 1 1.3396489 1 1.3374490 1 1.3353491 1 1.3331492 1 1.3310493 1 1.3289494 1 1.3267495 1 1.3246496 1 1.3225497 1 1.3204498 1 1.3183499 1 1.3162500 1 1.3142501 1 1.3121502 1 1.3100503 1 1.3080504 1 1.3059505 1 1.3039506 1 1.3019507 1 1.2998508 1 1.2978509 1 1.2958510 1 1.2938511 1 1.2918512 1 1.2898513 1 1.2878514 1 1.2858515 1 1.2839516 1 1.2819

(continued )



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X(i) f(i) Nf(i)

517 1 1.2800518 1 1.2780519 1 1.2761520 1 1.2741521 1 1.2722522 1 1.2703523 1 1.2683524 1 1.2664525 1 1.2645526 1 1.2626527 1 1.2607528 1 1.2589529 1 1.2570530 1 1.2551531 1 1.2532532 1 1.2514533 1 1.2495534 1 1.2477535 1 1.2458536 1 1.2440537 1 1.2422538 1 1.2404539 1 1.2385540 1 1.2367541 1 1.2349542 1 1.2331543 1 1.2313544 1 1.2296545 1 1.2278546 1 1.2260547 1 1.2242548 1 1.2225549 1 1.2207550 1 1.219551 1 1.2172552 1 1.2155553 1 1.2137554 1 1.2120555 1 1.2103556 1 1.2086557 1 1.2069558 1 1.2051559 1 1.2034

X(i) f(i) Nf(i)

560 1 1.2018561 1 1.2001562 1 1.1984563 1 1.1967564 1 1.1950565 1 1.1934566 1 1.1917567 1 1.1900568 1 1.1884569 1 1.1867570 1 1.1851571 1 1.1834572 1 1.1818573 1 1.1802574 1 1.1786575 1 1.1769576 1 1.1753577 1 1.1737578 1 1.1721579 1 1.1705580 1 1.1689581 1 1.1673582 1 1.1658583 1 1.1642584 1 1.1626585 1 1.1610586 1 1.1595587 1 1.1579588 1 1.1564589 1 1.1548590 1 1.1533591 1 1.1517592 1 1.1502593 1 1.1487594 1 1.1471595 1 1.1456596 1 1.1441597 1 1.1426598 1 1.1411599 1 1.1396600 1 1.1381601 1 1.1366602 1 1.1351

X(i) f(i) Nf(i)

603 1 1.1336604 1 1.1321605 1 1.1307606 1 1.1292607 1 1.1277608 1 1.1262609 1 1.1248610 1 1.1233611 1 1.1219612 1 1.1204613 1 1.1190614 1 1.1176615 1 1.1161616 1 1.1147617 1 1.1133618 1 1.1118619 1 1.1104620 1 1.1090621 1 1.1076622 1 1.1062623 1 1.1048624 1 1.1034625 1 1.1020626 1 1.1006627 1 1.0992628 1 1.0978629 1 1.0965630 1 1.0951631 1 1.0937632 1 1.0924633 1 1.0910634 1 1.0896635 1 1.0883636 1 1.0869637 1 1.0856638 1 1.0842639 1 1.0829640 1 1.0816641 1 1.0802642 1 1.0789643 1 1.0776644 1 1.0763645 1 1.0750

(continued )



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X(i) f(i) Nf(i)

646 1 1.0736647 1 1.0723648 1 1.0710649 1 1.0697650 1 1.0684651 1 1.0671652 1 1.0658653 1 1.0645654 1 1.0633655 1 1.0620656 1 1.0607657 1 1.0594658 1 1.0582659 1 1.0569660 1 1.0556661 1 1.0544662 1 1.0531663 1 1.0519664 1 1.0506665 1 1.0494666 1 1.0481667 1 1.0469668 1 1.0456669 1 1.0444670 1 1.0432671 1 1.0419672 1 1.0407673 1 1.0395674 1 1.0383675 1 1.0371676 1 1.0359677 1 1.0347678 1 1.0335679 1 1.0323680 1 1.0311681 1 1.0299682 1 1.0287683 1 1.0275684 1 1.0263685 1 1.0251686 1 1.0239687 1 1.0228688 1 1.0216

X(i) f(i) Nf(i)

689 1 1.0204690 1 1.0192691 1 1.0181692 1 1.0169693 1 1.0158694 1 1.0146695 1 1.0135696 1 1.0123697 1 1.0112698 1 1.0100699 1 1.0089700 1 1.0077701 1 1.0066702 1 1.0055703 1 1.0043704 1 1.0032705 1 1.0021706 1 1.0010707 1 0.9999708 1 0.9987709 1 0.9976710 1 0.9965711 1 0.9954712 1 0.9943713 1 0.9932714 1 0.9921715 1 0.9910716 1 0.9899717 1 0.9888718 1 0.9877719 1 0.9867720 1 0.9856721 1 0.9845722 1 0.9834723 1 0.9824724 1 0.9813725 1 0.9802726 1 0.9792727 1 0.9781728 1 0.9770729 1 0.9760730 1 0.9749731 1 0.9739

X(i) f(i) Nf(i)

732 1 0.9728733 1 0.9718734 1 0.9707735 1 0.9697736 1 0.9686737 1 0.9676738 1 0.9666739 1 0.9655740 1 0.9645741 1 0.9635742 1 0.9625743 1 0.9614744 1 0.9604745 1 0.9594746 1 0.9584747 1 0.9574748 1 0.9564749 1 0.9553750 1 0.9543751 1 0.9533752 1 0.9523753 1 0.9513754 1 0.9503755 1 0.9494756 1 0.9484757 1 0.9474758 1 0.9464759 1 0.9454760 1 0.9444761 1 0.9434762 1 0.9425763 1 0.9415764 1 0.9405765 1 0.9395766 1 0.9386767 1 0.9376768 1 0.9366769 1 0.9357770 1 0.9347771 1 0.9338772 1 0.9328773 1 0.9319774 1 0.9309

(continued )



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X(i) f(i) Nf(i)

775 1 0.9300776 1 0.9290777 1 0.9281778 1 0.9271779 1 0.9262780 1 0.9253781 1 0.9243782 1 0.9234783 1 0.9225784 1 0.9215785 1 0.9206786 1 0.9197787 1 0.9188788 1 0.9178789 1 0.9169790 1 0.9160791 1 0.9151792 1 0.9142793 1 0.9133794 1 0.9124795 1 0.9115796 1 0.9106797 1 0.9097798 1 0.9088799 1 0.9079800 1 0.9070801 1 0.9061802 1 0.9052803 1 0.9043804 1 0.9034

X(i) f(i) Nf(i)

805 1 0.9025806 1 0.9016807 1 0.9007808 1 0.8999809 1 0.8990810 1 0.8981811 1 0.8972812 1 0.8964813 1 0.8955814 1 0.8946815 1 0.8938816 1 0.8929817 1 0.8920818 1 0.8912819 1 0.8903820 1 0.8895821 1 0.8886822 1 0.8878823 1 0.8869824 1 0.8861825 1 0.8852826 1 0.8844827 1 0.8835828 1 0.8827829 1 0.8818830 1 0.8810831 1 0.8802832 1 0.8793833 1 0.8785834 1 0.8777

X(i) f(i) Nf(i)

835 1 0.8768836 1 0.8760837 1 0.8752838 1 0.8744839 1 0.8735840 1 0.8727841 1 0.8719842 1 0.8711843 1 0.8703844 1 0.8694845 1 0.8686846 1 0.8678847 1 0.8670848 1 0.8662849 1 0.8654850 1 0.8646851 1 0.8638852 1 0.8630853 1 0.8622854 1 0.8614855 1 0.8606856 1 0.8598857 1 0.8590858 1 0.8582859 1 0.8574860 1 0.8567861 1 0.8559862 1 0.8551863 1 0.8543864 1 0.8535865 1 0.8527


Sample size N¼ 2757, a¼ 0.7891, R¼ 865, DF¼ 782, X2¼ 112.24, P*1.00.


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Fig. A1. Bilogarithmic presentation of a fitting.


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zipf's law for indian languages

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