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Journal of Model Based Research

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  • Was there voter fraud in the 2021 Peru Presidential Elections?

    Raúl Isea 1   Jesús Isea 1  

    1Fundación Instituto de Estudios Avanzados, Hoyo de la Puerta, Baruta, Venezuela.


    This paper performs a forensic study of the Peru’s presidential election on June 6th, 2021 between Pedro Castillo and Keiko Fujimori, where ex-candidate Keiko Fujimori claimed there had been irregularities. We calculate three p-values that help us determine if there was fraud. The consensus of the results indicates that there was no manipulation of the results.

    Author Contributions
    Received 09 Sep 2021; Accepted 18 Oct 2021; Published 21 Oct 2021;

    Academic Editor: Åžafak Taktak, Psychiatry Department Ahi Evran University Education and Research Hospita, Turkey.

    Checked for plagiarism: Yes

    Review by: Single-blind

    Copyright ©  2021 Raúl Isea, et al.

    Creative Commons License     This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

    Competing interests

    The authors have declared that no competing interests exist.


    Raúl Isea, Jesús Isea (2021) Was there voter fraud in the 2021 Peru Presidential Elections?. Journal of Model Based Research - 1(4):1-5.

    Download as RIS, BibTeX, Text (Include abstract )

    DOI 10.14302/issn.2643-2811.jmbr-21-3954


    On April 11th, 2021 the first round presidential election was held with 24 political parties, where it was projected as potential winners the Peru Libre candidate Pedro Castillo and the Fuerza Popular candidate Keiko Fujimori with 2,724,752 and 1,930,762 votes, respectively. The second round was held on June 6th, 2021, where Pedro Castillo was chosen as winner with 8,835,579 votes against the 8,791,521 obtained by Fujimori. As a matter of fact, the National Office of Electoral Processes (known as ONPE, by Spanish initials) declared that there was 17,620,000 valid votes, 121,477 blank votes and 25.43% of abstention (details in ONPE,

    However, on June 8th the right-wing conservative candidate Keiko Fujimori denounced there had been irregularities in favor of Pedro Castillo1, and even claimed that the elections had to be held again2 Nevertheless, Keiko Fujimori recognized the victory of her opponent on July 26th , and finally the ONPE declared as winner to Pedro Castillo.

    For this reason, we performed a study based on statistical techniques according to the Benford’s Law3 to determine if there actually was any manipulation in results. This methodology has been employed in some many studies3, 4, 5, 6, 7. Moreover, this law has even been used in presidential elections as explained by Mebane8 and Deckert et al9 as explained in the next section.


    The forensic analysis was performed with the same computational methodology employed both in the Covid-19 registered cases study6, and in the recent US elections held in 20207. This methodology is based in the Benford’s Law, and we determine three critical values.

    The p-value(χ2), p-value(Man) and p-value(FW), refer collectively to as p-values, were calculated in the following way6, 7. The first value [ p-value(χ2) ] comes from the occurrence frequency of first digit of the data (left to right), which is based on the Benford’s Law of the first digit, and the probability should follow the following mathematical relation10:

    Where i goes from 1 to 9 (excluding zero). With these probabilities, the Pearson value (χ2) is calculated to detect discrepancies in data, which is given by:


    where P(k) and b(k) are the distributions obtained from the votes and the expected from Benford’s Law, respectively. Thanks to this value, it was possible to determine the p-value(χ2) which indicates us whenever data is correct, as long as it is greater than or equal to 0.056, 7.

    The next value, p-value(Man), employs the Mantissa Arc test, and to do so we must find the mass center of data according to the following mathematical relation10:


    where the xi are the votes to validate, and N is the total number of them. We then calculated the L2 term given by:

    where L2 should be almost zero, it means, while greater than zero, it is possible to suspect a manipulation of the elections.

    So the p-value(Man) equals to:

    Finally, the p-value (FW) is known as the Freedman-Watson test (FW), designed to compare discrete distributions based on the following mathematical relation11:


    However, it is recommended to check Freedman’s original paper11to know every mathematical detail of this last equation.

    So, there would be no suspicion of manipulation of the election, if any of the p-values is greater than or equal to 0.05. Nevertheless, if all three values are less than 0.05, it is a sign of inconsistency or fraud6, 7.

    Finally, we want to validate this methodology according to the total number of voters inscribed in twenty five Peru states according the ONPE data, i.e. in the states of Amazonas, Ancash, Apurimac, Arequipa, Ayacucho, Cajamarca, Callao, Cusco, Huancavelica, Huanuco, Ica, Junin, La Libertad, Lambayeque, Lima, Loreto, Madre de Dios, Moquegua, Pasco, Piura, Puno, San Martin, Tacna, Tumbes, and Ucayal. Finally we analyzed the second round election results grouped by political party.


    In Table 1 we present all the data collected for the study according to the ONPE results. In this table we can observe that Pedro Castillo loss in six of twenty five states (ie., Callao, La Libertad, Lambayeque, Lima, Piura, and Tumbes). In the Loreto state, for instance, Castillo won only for 5,350 votes (which represent about 1.46% of the electoral roll), and in the Ica state while the votes difference was a little higher (5626 votes ahead for Castillo), this difference is only 1.09% of the electoral register in this last state. In contrast, there were some states where the difference was way higher, such as Puno: where Castillo won over with more than 580 thousand votes (a difference of almost 75% of the electoral roll).The most favorable state for Fujimori was Lima, where she outnumbered Castillo by almost two million votes, which only represent 27,67% of difference in between them.

    We show the p-values for the total of all registered votes in the electoral roll of Peru in the Table 2, and also the analysis of the votes obtained by Pedro Castillo and Keiko Fujimori, respectively. As we can see in this table, there’s no evidence of data manipulation as the all surpass the threshold value of 0.05. Furthermore, the L2value must be close to zero and there are also similarities between all of them (see Table 2)

    Table 1. Peru’s states where is indicated the number of electors for each state, and the votes obtained for Pedro Castillo and Keiko Fujimori.
    State Total voters Votes for Pedro Castillo Votes for Keiko Fujimori
    Amazonas 161,890 98,716 52,913
    Ancash 597,055 314,394 233,325
    Apurimac 209,560 160,943 36,737
    Arequipa 900,759 549,681 299,759
    Ayacucho 287,140 223,383 49,130
    Cajamarca 690,285 456,128 190,041
    Callao 642,766 195,098 403,813
    Cusco 718,117 561,406 116,299
    Huancavelica 174,567 139,498 26,243
    Huanuco 367,857 229,059 114,648
    Ica 515,652 231,546 225,920
    Junin 693,301 377,083 271,117
    La Libertad 1,022,886 376,424 570,558
    Lambayeque 711,954 274,662 387,053
    Lima 6,418,172 2,127,809 3,903,451
    Loreto 366,268 176,864 171,514
    Madre de Dios 76,770 50,244 20,533
    Moquegua 114,448 78,009 28,926
    Pasco 130,700 80,358 42,140
    Piura 996,743 363,786 560,618
    Puno 733,093 624,592 76,280
    San Martin 430,319 222,029 177,108
    Tacna 219,577 150,672 57,187
    Tumbes 131,348 41,464 80,064
    Ucayali 258,435 115,356 126,116

    Table 2. The p-values for the total of all registered votes in the electoral roll of Peru, the votes received by Pedro Castillo and Keiko Fujimori, according to the data indicated in Table 1.
      Total votes of the electoral register Votes obtained for Pedro Castillo Votes obtained for Keiko Fujimori
    p-Valor(χ2) 0,19 0,97 0,53
    p-Valor(Man) 0,21 0,44 0,22
    L2 0,06 0,03 0,06
    p-Valor(FW) 0,19 0,88 0,44


    This paper determined a forensic analysis of the Peru’s presidential elections on June 6, 2021. We determined three p-values that can help us determine if there have been manipulations of the results. The results indicated that there is no fraud. Moreover, we show how valid is the methodology when we analyzed the electoral register in twenty five states, and therefore, we concluded that there was no fraud in Peru election.


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    2.AFP. (2021) Presidential hopeful Fujimori repeats Peru vote fraud claims.France 24.
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