The authors have declared that no competing interests exist.
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.
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, https://www.resultadossep.eleccionesgenerales2021.pe/SEP2021/).
However, on June 8th the right-wing conservative candidate Keiko Fujimori denounced there had been irregularities in favor of Pedro Castillo
For this reason, we performed a study based on statistical techniques according to the Benford’s Law
The forensic analysis was performed with the same computational methodology employed both in the Covid-19 registered cases study
The p-value(χ2), p-value(Man) and p-value(FW), refer collectively to as p-values, were calculated in the following way
Where i goes from 1 to 9 (excluding zero). With these probabilities, the Pearson value (χ
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 0.05
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 relation
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 relation
However, it is recommended to check Freedman’s original paper
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 fraud
Finally, we want to validate this methodology according to the total number of voters inscribed in twenty five Peru states according the ONPE data,
In
We show the p-values for the total of all registered votes in the electoral roll of Peru in the
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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 |
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p-Valor(χ2) | 0,19 | 0,97 | 0,53 |
p-Valor(Man) | 0,21 | 0,44 | 0,22 |
L |
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.