Table 1.
Percentage of votes favoring Chávez and voting turnout.
Figure 1.
Second-Digit Benford's law and proportions on electoral units.
Top panels: Electoral units with 10 or more votes for Chávez. Bottom panels: Electoral units with 100 or more votes for Chávez. Left panels: Presidential elections and referenda previous to 2004. Right panels: Elections and referenda between 2004 and 2012. The proportions of the 2005 Parliamentary Elections are partially out of the y-axis range.
Table 2.
Benford's test statistics based on electoral units with 10 or more votes for Chávez.
Table 3.
Benford's test statistics based on electoral units with 100 or more votes for Chávez.
Figure 2.
Second-Digit Benford's law and proportions on polling centers.
Left panel: Presidential elections and referenda previous to 2004. Right panel: Elections and referenda between 2004 and 2012.
Table 4.
Benford's test statistics based on polling centers.
Figure 3.
Election fingerprints: 3D-histograms of the number of electoral units for a given voter turnout (x-axis) and the percentage of votes for Chávez (y-axis).
Left panel: 1998 Presidential elections. Right panel: 2004 Recall Referendum. Color represents the number of electoral units with corresponding (x,y)-coordinates.
Figure 4.
Gaussian quadratic classifier: The black line represents the decision boundary.
Each electoral unit represented by a blue circle has been classified as an observation of the Gaussian fit model based on 1998 data. Otherwise, it is represented by one red x. In both elections, the units are clustered around their respective averages of turnout and votes for Chávez. By excluding some units with turnout between 60% and 80%, and low support for Chávez (less than 20%), the scatterplots appear to be normally distributed.
Figure 5.
A high percentage of electoral units from the 2006 and 2012 elections and the 2004 and 2009 referenda cannot be classified as observations of the Gaussian fit model based on 1998 data.
The scatterplots have many units with high turnout and high support for Chávez.
Figure 6.
The scatterplot of the 2007 referendum and the 2010 parliamentary elections has a shape similar to the 2004 case.
However, the set of electoral units close to the top right corner is less dense (2010) or negligible (2007). Additionally, their [%] Mod.98 values are considerably high, as well as the percentage of electoral units classified into the 2004 model. These elections seem to fit a true mixture model.
Figure 7.
The scatter plots of the 1999 referenda and the 2005 elections are mainly characterized by the low voter turnout and high votes for Chávez (Figure 6).
This shape is a consequence of the low opposition turnout in 1999 and its almost total absence in 2005.
Table 5.
Percentage of electoral units classified as observations from the election fingerprint model of 1998.
Figure 8.
Cumulative number of votes favoring Chávez as a function of turnout.
The shape of every referendum/election is a sigmoid that reaches a plateau at the maximal vote count for Chávez. The curves of 2004, 2006, and 2012 increase close to complete turnout.
Figure 9.
The distributions of Z-scores of different elections collapse on a t-student with 3 degrees of freedom.
Only the 2000 elections show slightly heavier tails.
Figure 10.
The standardized differences of the 1999 referenda and the 1998, 2000 and 2005 elections (wide black lines) are well embedded within/in the 99% normal confidence interval at large sample sizes.
Standardized differences of fair elections computed from a hierarchical bootstrap model (thin blue lines) also verify the expected behavior under H1.
Figure 11.
The standardized differences based on official results from 2004 onwards reach values higher than any simulation.
They are well above the 99% normal confidence interval. These elections provide strong evidence against H1. The more irregular distributions of votes occurred on electoral units where the vote counting was significantly favorable to Chávez.
Figure 12.
The correlation between irregular variations in centers and bias in the vote counting can be visualized by computing the proportion of votes favoring Chávez as a function of the inter-annual variation.
The curves are centered, by subtracting the overall percentage of votes obtained by Chávez en each case.
Figure 13.
Percentage of votes favoring Chávez as a function of the inter-annual variation in the number of voters registered at polling centers.
The official results of these elections are reached at extremely large values of the inter-annual variation. At small values, up to 69% of the total valid votes, the results are tight for the 2004 Recall Referendum. At moderate levels, up to the 79% of the total valid votes, the results are adverse for Chávez in the 2012 Presidential Elections.