Comparative Election Fraud Detection

Abstract

Elections in Russia are widely believed to be fraudulent in various ways, a claim some support especially by looking at voter turnout, others by looking at vote counts' digits. We use polling station level data from the Russian Duma elections of 2003 and 2007 and presidential elections of 2004 and 2008 to examine how several methods for diagnosing election fraud complement one another. The methods include estimating the distribution of turnout, measuring the relationship between turnout and party support and testing for vote counts' second digits following the distribution implied by Benford's Law. Anomalies the methods detect are worse by the end of the period under study than at the beginning. The digit test detects anomalies beyond those suggested by a simple idea that turnout in many places was fraudulently inflated.

Full text

Comparative Election Fraud Detection
∗
Walter R. Mebane, Jr.
†
Kirill Kalinin
‡
August 7, 2009
Abstract
Elections in Russia are widely believed to be fraudulent in various ways, a claim some
support especially by looking at voter turnout, others by looking at vote counts’ digits. We
use polling station level data from the Russian Duma elections of 2003 and 2007 and
presidential elections of 2004 and 2008 to examine how several methods for diagnosing
election fraud complement one another. The methods include estimating the distribution of
turnout, measuring the relationship between turnout and party support and testing for vote
counts’ second digits following the distribution implied by Benford’s Law. Anomalies the
methods detect are worse by the end of the period under study than at the beginning. The digit
test detects anomalies beyond those suggested by a simple idea that turnout in many places
was fraudulently inflated.
∗
Prepared for presentation at the Annual Meeting of the American Political Science Association, Toronto, Canada,
Sept 3–6, 2009. A previous version wes presented at the 2009 Annual Meeting of the Midwest Political Science
Association.
†
Professor, Department of Political Science and Department of Statistics, University of Michi-
gan, Haven Hall, Ann Arbor, MI 48109-1045 (E-mail: wmebane@umich.edu).
‡
Ph.D. student, Department of Political Science, University of Michigan (E-mail:
kkalinin@umich.edu).
1

Introduction
The least one can say about national elections in Russia over the most recent election cycles is
that they have become less competitive, with fewer political parties presenting candidates for both
the Duma (federal parliament) and the presidency. The United Russia [UR] party, associated with
Vladimir Putin, has unquestionably become increasingly dominant. But many observers go
further and argue that during the recent period Russian elections have become increasingly unfree
and unfair. Myagkov and Ordeshook (2008) argue that during the past 15 years “falsifications in
the form of stuffed ballot boxes and artificially augmented election counts” have become
prevalent throughout the country (see also Myagkov, Ordeshook, and Shaikin 2008, 2009).
Since 2000 there have been increases in institutional barriers that suppress electoral
competition between parties and candidates: tougher registration requirements; the rise of a
parliamentary threshold for parties; cancellation of protest voting; and cancellation of the
minimum electoral threshold. In the electoral campaign there has been excessive positive
informational and financial support of candidates favored by the Kremlin along with negative
campaigning against alternative candidates and parties. Administrative changes decreased the
transparency of elections. Electoral commission activities became more closed off from the
public, with independent public observation being canceled and the rights of the legal observers
being frequently violated. The OSCE identified serious problems in the 2004 election (OSCE
Office for Democratic Institutions and Human Rights 2004), and by 2008 problems had become
so severe that international observer groups declined to observe the election (OSCE Office for
Democratic Institutions and Human Rights 2008).
Lubarev, Buzin, and Kynev (2007) argue that administrative changes since 2000 increased the
extent to which officials from all levels of the government participated in election administration.
Since 2003 both federal and regional administrative resources as well as the mass media have
been deployed in favor of UR, which thereby received substantial informational advantages, and
against its main competitor the KPRF (Communist party) (Buzin and Lubarev 2008). The
Kremlin now controls the appointments of regional executives (each governor can be fired for
1

“loss of trust”), which has made them responsible for delivering “recommended” electoral figures
to the Kremlin. After gubernatorial elections were canceled in December 2004, by the spring of
2007, 70 of 85 governors had announced they were participating in the party of power (Gel’man
2007). Thus, it appears that the entire regional state apparatus is now at the service of the party of
power, making it one large electoral “political machine.” This operation is characterized by
control over the mass media, administrative pressure on both opposition and voters, and possibly
falsifications as well. In fact, state officials have excessive control over all levels of electoral
commissions, including precincts (UIKs), territorial (TIKs) and regional commissions.
Allegations point to a wide variety of methods used to distort reported votes (Kalinin 2008).
Many of these methods relate to voter turnout and so as markers for fraud may appear ambiguous
to the extent they resemble efforts to boost genuine electoral support. In the 2004 presidential
election, due to the effectiveness of administrative resources and the popularity of Putin, the
outcome of presidential elections was essentially predetermined. As a result, neither of the
opposition parties was promoting its leader as their candidate (Buzin and Lubarev 2008, 26).
Nonetheless a wide variety of methods was used to increase turnout, including forced voting by
absentee certificates. According to Buzin and Lubarev, in 214 territories the 2004 election was
the first election where the turnout rate and the share of votes for the winner simultaneously
exceeded 90 percent (Buzin and Lubarev 2008, 26). This phenomenon occurred frequently in
republics as well as in Rostovskaya, Tumenskaya, Chukotskii and Yamalo-Nenetskii AO.
Voters in Russia do not personally register to vote, but all eligible voters are assigned to
specific UIKs depending on where they live. There is a permanent gap between the number of
real voters and the “listed voters”—the average number of unaccounted voters in Russia is 2–5
million people—and on election days there are always extensive corrections of the voter lists
(Arbatskaya 2004). The large-scale character of these corrections depends on the specific ways
the voter lists are formed, methods that differ in different territories. Arbatskaya argues that this
correction of voter lists phenomenon opens the door for administrative tyranny, violating the
democratic rights of citizens Arbatskaya (2004, 224–226).
2

The federal elections of 2007 and 2008 took place under different conditions. The president’s
popularity remained high, and the “party of power” controlled not only the Duma but also many
federal legislatures, encompassing the majority of regional heads, mayors of big cities, and other
representatives of political, administrative and economic elites. Putin had promised not to change
the Constitution to allow himself to stand for reelection, so he was preparing for his successor.
Between 2003 and 2007, the Duma election was changed from a mixed system to a system based
entirely on proportional representation. The UR party list was headed by Putin, and it also
included a majority of the governors. In the absence of any viable political competitors and
Putin’s unique position in UR’s list (he was the only one in its federal list), the 2007 federal
elections were labeled as referendum for the all-national Leader. The lack of competition and the
absence of the “against all” (Protiv vseh) option on the ballot—this protest voting option was
prohibited after the 2004 election—produced a danger of low turnout. According to Buzin and
Lubarev, the main task of federal authorities in 2007 and 2008 was twofold: to provide the victory
of Kremlin candidates, and to provide high turnout (Buzin and Lubarev 2008, 184, 257-258).
Therefore Buzin and Lubarev claim that vote falsifications were not solely about shifting votes
from one candidate to another, but rather about simultaneously increasing the number of votes
and the number of voters. These goals can be implemented by “stuffing” the ballot boxes (vbros)
or “adding figures to protocols” (pripiska) (Buzin and Lubarev 2008, 184).
Like Myagkov et al. (2009), Lubarev et al. (2007) and Buzin and Lubarev (2008) argue that
direct falsifications played a much larger role in the federal elections of 2007 and 2008 than they
had in the federal elections and regional elections of the 1990s and early 2000s. Buzin and
Lubarev (2008) present electoral data, observer reports and multiple stories from observers and
ordinary voters that illustrate the growth of crude falsifications and their widespread character, a
pattern they refer to as “mass administrational electoral technology.” Buzin and Lubarev (2008)
conclude that compared to all other elections, the elections of 2007 and 2008 showed that direct
falsifications started to affect the results of elections, by affecting the distribution of votes
between the candidates.
3

Myagkov et al. (2009), Lubarev et al. (2007) and Buzin and Lubarev (2008) all emphasize
what they claim is fraudulent voter turnout. Buzin and Lubarev (2008) state that along with
stuffed ballot boxes, the easiest and the most popular technique is to change figures in UIKs’
protocols by UIKs or even more often by TIKs (territories). Buzin and Lubarev are astonished by
how widespread direct falsifications are and about the “courage” of falsificators, who appear
confident that they are supported by administration and courts (Buzin and Lubarev 2008, 177). As
Buzin and Lubarev point out “the insolence with which the protocols are changed in TIKs,
knowing that the copy is already given out to observers, is explained by their confidence in
impunity, being assured that falsificators and law machinery, including courts, are acting
together” (Buzin and Lubarev 2008, 177). They argue that the federal elections of 2007 and 2008
showed widespread discrepancies between data derived from UIKs and official data produced by
TIKs and the Gas VIBORI system (the internet-accessible election reporting system).
In this paper we use UIK-level data from the 2003 and 2007 Duma elections and the 2004 and
2008 presidential elections to show that it is useful to augment analysis of Russian elections that
focuses on voter turnout statistics with information about the distribution of the second significant
digits in UIK-level vote counts.
Tests of vote counts based on the so-called second-digit Benford’s Law (2BL) distribution
have figured prominently in work on election forensics (Mebane 2006a,b, 2007b,a, 2008b). The
analysis in Mebane (2007a) ultimately focuses on the conditional means of the second digits in
collections of vote counts, measuring how these means differ from the means expected according
to the 2BL distribution. The conditioning factors in that analysis, which examined data from the
2006 election in Mexico, were the partisan affiliations of mayors in Mexican municipalities.
Mebane (2008a) and Kalinin (2008) combined an examination of UIK vote counts second-digit
conditional means with outlier detection methods (Mebane and Sekhon 2004) to try to diagnose
which of several hypothesized methods for fraud may have affected the votes reported for Russian
presidential candidates in 2004 and 2008.
4

Nonparametric Regression 2BL Test
The 2BL test used in this paper involves comparing the arithmetic mean of the vote counts’
second digits to the mean value expected if the digits are 2BL-distributed. This test adapts an idea
used in Grendar, Judge, and Schechter (2007)’s analysis that focuses on the first significant digit
and is intended to identify what they describe as generalized Benford distributions. Grendar et al.
suggest that data that do not conform to Benford’s law may have first digits that match a member
of a specified class of exponential families. Mebane (2006b) argues that vote counts in general do
not have digits that match Benford’s law at all. In particular, the distribution of the first digits of
vote counts is undetermined. Mebane (2006b) demonstrates a pair of naturalistic models that
produce simulated vote counts with second digits but not first digits that are distributed roughly as
specified by Benford’s law. Nonetheless we can use the mean of the second digits to test how
closely the digits match the 2BL distribution. According to Benford’s law, the expected relative
frequency q
j
with which the second significant digit is j is (rounded)
(q
0
, . . . , q
9
) = (.120, .114, .109, .104, .100, .097, .093, .090, .088, .085). Given 2BL-distributed
counts, the value expected for the second-digit mean
¯
j is approximately
¯
j
B
=
P
9
j=0
jq
j
= 4.18 7 .
Mebane (2006b) and Mebane (2007a) suggest that vote counts whose second digits follow the
2BL distribution are unproblematic, while departures from the 2BL distribution indicate that some
kind of manipulation has occurred. Whether the manipulation the second-digit test may detect
constitutes any kind of fraud is something that needs to be established by additional evidence.
The test is, first, whether
¯
j differs from
¯
j
B
and, second, whether it differs in a way that
depends on observed conditioning factors. The conditioning factor in the current analysis is
reported voter turnout, measured as the proportion of registered voters who voted at each UIK.
1
For vote counts y
i
observed for UIKs indexed by i, we nonparametrically regress the second
digits on the turnout proportion x
i
. To estimate the nonparametric regressions we use the package
sm (Bowman and Azzalini 1997) for the statistical programming environment R (R Development
1
Specifically the value we use to measure turnout is the sum of the number of ballots given out to voters before
election day, the number given to voters in polling places and the number given to voters outside of polling places
divided by the number of registered voters.
5

Core Team 2005).
Turnout, Votes and Manipulations in Russia 2003–2008
Start by considering some of the facts about the distribution of turnout in recent Russian elections
that support suspicions that the elections were, increasingly, affected by fraud. Figures 1 and 2
display kernel density estimates
2
for UIK-level turnout in the Duma elections of 2003 and 2007
and the presidential elections of 2004 and 2008.
3
Following Myagkov et al. (2009), we consider
separately data from republics and data from other regions (“oblasts”). The figures mirror results
presented by Buzin and Lubarev (2008, Appendix, Illustration 38), which they attribute to S. A.
Shpilkin.
*** Figures 1 and 2 about here ***
The progression of figures shows worse distributions in 2007 and 2008 than in the earlier two
years. The distributions are also worse in the presidential election years than in the Duma election
years. The top row of Figure 1 shows the distribution for 2003. For both oblasts and republics
there is a spike of UIKs with turnout at or very near 100 percent. A higher proportion of UIKs in
the republics than in the oblasts have this feature. But in oblasts most of the UIKs have turnout
following a relatively smooth unimodal distribution, and in republics many of the UIKs have
turnout following such a distribution. In 2004 (the second row of 1), the proportion of UIKs with
turnout near 100 percent increases noticeably in oblasts and very substantially in republics. In
oblasts the distribution also exhibits spikes at locations corresponding to the excess of turnout
values at values of 70%, 80% and 90% noticed by Shpilkin and Shulgin (Buzin and Lubarev
2008, 201). The distributions for 2007 (top row of Figure 2) shows spikes of UIKs at or near 100
percent turnout similar to those observed in 2004. In the distribution for oblasts, spikes are
apparent at round number percentages of turnout above 60%. The distributions for 2008 (bottom
row of Figure 2) have proportions of UIKs with turnout at or near 100 percent comparable to
2
These densities are computed using R’s density() function.
3
All vote and turnout data were downloaded from the website of the Central Election Commission of the Russian
Federation, http://www.vybory.izbirkom.ru/region/izbirkom.
6

2004. The distribution for oblasts shows very pronounced spikes at round number percentages of
turnout, and in the distribution for republics a spike is evident near 75% turnout.
Buzin and Lubarev (2008, 201) argue that the only acceptable explanation for the spiked
distributions is a wide-spread adjustment of turnout to specific “rounded” figures. Inspecting the
last digits of the original UIK-level turnout counts adds to the impression that many of them are
faked. If the turnout counts reflected the natural complex of processes that cause people to vote or
not to vote, we would expect the counts’ last digits to be uniformly distributed (i.e., each digit
zero through nine would occur equally often) (Beber and Scacco 2008). Table 1 shows that the
distribution of the last digits in the actual turnout counts from 2003–2008 is very often far from
uniform. The table shows for each digit the signed square root of the discrepancy between the
observed frequency of the digit and the frequency of 0.1 expected if the distribution is uniform.
4
A value of 2.0 or greater in magnitude represents a significant discrepancy. The table shows that
there are always too many zeros, with one exception too few nines, and usually too many fives.
Year 2003 for UIKs in oblasts is the only situation where neither the number of fives nor the
number of nines is significantly discrepant from the expected uniform distribution, and that subset
of UIKs is the only one for which the overall Pearson chi-square statistic is not statistically
significant at the .05 test level. As measured by the overall chi-squared statistics, the extent of the
discrepancy from the uniform expectation increases monotonically as one moves from 2003 to
2008. Turnout fakery seems to be much worse at the end of the time period than at the beginning.
*** Table 1 about here ***
Myagkov et al. (2009) emphasize the way turnout is associated with votes for the party of
power at the rayon level, and Buzin and Lubarev (2008, 204) discuss similar kinds of relationships
using UIK-level data. Both discussions make the point that where turnout is very high, support
for UR tends also to be very high, and support for other parties—notably the KPRF—tends to be
relatively low. Figures 3–6 illustrate these relationships for these two parties. These figures show
a solid line representing the nonparametric regression of the vote proportions on the turnout
4
If p
j
is the observed frequency of digit j and N is the number of UIKs, then the signed square root statistic is
sign(p
j
− 0.1)N[(p
j
− 0.1)
2
/0.1]
1/2
.
7

proportions bounded by dashed lines indicating 95% confidence bounds. A dotted line shows the
unconditional mean vote proportion. Figure 3 shows the results for UR in republics, with one plot
for the UIKs in each year. Clearly mean support for UR is much greater in UIKs where turnout is
very high. The increase in mean UR vote share from its approximate floor (for turnout roughly 50
percent) to its peak at turnout equalling 100 percent is greater in 2003 than in 2007 but also
greater in 2008 than in 2004.
5
Figure 4 shows the results for UR in oblasts. In the Duma election
years, then mean support for UR no longer peaks at turnout equal to 100 percent but instead
reaches a maximum for turnout at around 90 percent. In the presidential election years, mean
support for UR does have a maximum at the highest level of turnout. The gain in mean support
from floor to maximum is now greater in 2007 than in 2003 and in 2008 than in 2004.
*** Figures 3 and 4 about here ***
Figures 5 and 6, which show the same kinds of scatterplots and nonparametric regression
lines, in contrast show mean support for the KPRF decreasing once turnout increases beyond a
certain level. The relationships for UIKs in republics, in Figure 5, show mean KPRF support
declining throughout the distribution of turnout in 2003, but 2004, 2007 and 2008 a decline in
mean support sets in only for turnout greater than about 60 percent. The decline from ceiling to
minimum is greater in 2003 than in 2007 but greater in 2008 than in 2004. The relationships for
UIKs in oblasts, in Figure 6, show mean KPRF support declining only turnout greater than a
certain level in all four years. In 2003 and 2004 the decline begins once turnout reaches about 80
percent, but in 2007 and 2008 the decline starts when turnout reaches about 50 percent. The
ceiling to minimum declines in mean KPRF support are also larger in 2007 than in 2003 and in
2008 than in 2004.
*** Figures 5 and 6 about here ***
Reported turnout certainly looks suspicious when its distribution and the distribution of
turnout counts’ last digits are viewed on their own, and turnout is clearly related to the mean
support for UR and the KPRF. Plots computed for other parties resemble the ones shown here for
5
For 2003 we use the proportional represntation votes, to match the electoral system in place in 2007.
8

the KPRF. Such a pattern of UR tending to gain support in places where the KPRF and other
parties are tending to lose support strongly suggests that vote switching is possibly occurring.
The 2BL test may provide further evidence on this point. Simulations reported in Mebane
(2006b) and Mebane (2008a) suggest that variations from the 2BL distribution can occur both
when vote counts are artificially increased and when they are artificially reduced. As Mebane
(2008a) observes, it is unclear whether an artificial increase in vote counts will mean that the
mean second digit,
¯
j, also increases, or whether an artificial decrease implies that
¯
j decreases. But
we might expect that if substantial vote switching is occurring, we should see significant
departures from the 2BL expected mean ,
¯
j
B
, for both the receiver party and the donor party in
places where the vote switching is happening. In the current case, we might expect nonparametric
regression lines to show that the expected second digit differs from the 2Bl expected value for
both UR and the KPRF for the same values of turnout, if vote switching is taking place.
Figure 7 shows the first of a series of graphs intended to allow such parallel assessments. Each
plot in the figure shows a solid line representing the nonparametric regression of the second digits
on the turnout proportions, with a pair of dashed lines indicating the boundaries of a 95%
confidence interval. A horizontal dotted line locates
¯
j
B
= 4.187. A rug plot along the bottom of
each plot locates the observed values of turnout. The first question is whether there is any range
of turnout values for which the nonparametric regression curve’s confidence interval does not
contain
¯
j
B
. If so, we will then ask whether the same region of discrepancy is found for both UR
and KPRF.
*** Figure 7 about here ***
In all four years in republics, shown in Figure 7, the nonparametric regression curves for the
UR vote counts’ second digits have roughly the same shape, but across the years the regions
where
¯
j significantly differs from
¯
j
B
varies somewhat. In 2003,
¯
j differs significantly from
¯
j
B
only for turnout in the interval roughly (0.45–0.6).
6
In this region,
¯
j >
¯
j
B
. In 2007,
¯
j differs from
¯
j
B
for roughly the same values of turnout in the same way, but
¯
j also is significantly less than
¯
j
B
6
Magnification is probably needed to see the differences discussed in this section.
9

for turnout in the interval roughly (0.7–0.9). In 2004,
¯
j differs from
¯
j
B
for turnout in the interval
roughly (0.6–0.8), and in that interval
¯
j <
¯
j
B
. For 2008,
¯
j <
¯
j
B
significantly for turnout in
roughly (0.65–0.9), and
¯
j >
¯
j
B
significantly for turnout greater than roughly 0.975. The graphs
for UIKs in oblasts, shown in Figure 8, show roughly the same pattern as in the republics for 2003
and 2007. For 2004, the oblasts graph shows
¯
j >
¯
j
B
significantly for turnout in the interval
roughly (0.3–0.5) and
¯
j <
¯
j
B
significantly for turnout in roughly (0.55–0.8). The graph for 2008
is similar. None of the graphs for oblasts shows a significant discrepancy between
¯
j and
¯
j
B
at the
very highest levels of turnout.
*** Figure 8 about here ***
Comparing these figures to those for the second digits of the KPRF vote counts, we see in the
plot for republics (Figure 9) very different patterns. In 2003,
¯
j <
¯
j
B
significantly for turnout in
two intervals, roughly (0.3–0.55) and greater than 0.75. The results for 2004 are approximately
similar. In 2007 the lower interval shrinks to roughly (0.55–0.6) and the upper interval is also
slightly smaller (greater than about 0.85). In 2008,
¯
j >
¯
j
B
significantly for turnout in roughly
(0.6–0.7), while
¯
j <
¯
j
B
significantly for turnout greater than about 0.9.
*** Figure 9 about here ***
The intervals of turnout for which
¯
j 6=
¯
j
B
significantly for UR overlap with the intervals for
which
¯
j 6=
¯
j
B
significantly for the KPRF in all four years. For 2003, 2004 and 2007, the overlaps
occur for turnout values in the vicinity of 0.5, and
¯
j >
¯
j
B
for UR but
¯
j <
¯
j
B
for the KPRF. For
2008 overlaps occur for most of the turnout values greater than about 0.65, and once again
¯
j −
¯
j
B
has opposite signs for UR and for the KPRF. Such patterns strongly suggest vote switching.
Notably, in every year except 2008, the pattern that suggests vote switching occurs for moderate
levels of turnout and not at the highest levels. The pattern of
¯
j for the KPRF in republics in the
earlier years clearly suggest something irregular was happening in the highest turnout UIKs.
Perhaps, as in some of the simulations reported by Mebane (2006b) and Mebane (2008a), vote
switching was also occurring in those years but it did not rise to levels sufficient to trigger a 2BL
signal in the receiver party’s vote counts.
10

The results for
¯
j for the KPRF for UIKs in oblasts, shown in Figure 10, are similar to those for
republics. With minor differences the comparison between those conditional means and the
conditional means for UR suggest the same kind of conclusion.
*** Figure 10 about here ***
Conclusion
Anomalies the methods detect are worse by the end of the period under study than at the
beginning. The second-digit test detects anomalies beyond those suggested by a simple idea that
turnout in many places was fraudulently inflated.
11

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tial Election, 14 March 2004: OSCE/ODIHR Election Observation Mission Report.” Warsaw,
June 2, 2004.
OSCE Office for Democratic Institutions and Human Rights. 2008. “OSCE/ODIHR Regrets that
Restrictions Force Cancellation of Election Observation Mission to Russian Federation.” Press
release, Warsaw, February 7, 2008.
R Development Core Team. 2005. R: A Language and Environment for Statistical Computing. R
Foundation for Statistical Computing. Vienna, Austria. ISBN 3-900051-07-0.
URL http://www.R-project.org
14

Table 1: Distribution of last Digits for UIK Vote Totals in Russian Elections 2003–2008
Year
2003 2004 2007 2008
Digit Repub. Oblast Repub. Oblast Repub. Oblast Repub. Oblast
0 6.2 3.0 9.9 4.7 10.5 7.7 15.4 10.5
1 −2.0 0.1 −1.8 1.3 −0.8 0.1 −1.4 −0.7
2 −0.6 −0.4 1.1 0.5 −1.3 1.7 −2.1 −0.2
3 −1.2 −1.0 −1.8 −0.7 −0.6 −1.4 −1.9 −2.1
4 0.7 −0.3 −3.3 −0.8 −3.4 −1.0 −3.7 −1.3
5 3.1 0.9 2.1 3.3 2.1 −0.2 2.7 0.8
6 −1.8 −0.9 −0.1 0.0 −2.8 −1.8 −1.3 −1.1
7 −0.2 0.9 −2.0 −2.2 −1.1 −0.9 −3.1 0.0
8 −0.7 −1.5 −0.7 −2.1 −0.3 −0.8 −1.4 −1.0
9 −3.5 −0.8 −3.4 −4.0 −2.3 −3.5 −3.2 −4.9
χ
2
L
69.9 15.4 137.2 60.9 144.2 82.0 292.2 143.0
n 17, 008 77, 305 17, 600 77, 824 17, 875 77, 92 8 17, 865 78, 383
Notes: Entries for each digit show signed square roots of chi-squared statistics implied by the null
hypothesis that the total number of votes cast at each UIK (polling station) have uniformly
distributed last digits. The χ
2
L
statistics show the overall Pearson chi-squared statistic (9 degrees
of freedom). n shows the number of UIKs.

0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Distribution of Turnout across UIKs, 2003 Oblasts
N = 77757 Bandwidth = 0.0146
Density
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Distribution of Turnout across UIKs, 2003 Republics
N = 17347 Bandwidth = 0.02346
Density
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Distribution of Turnout across UIKs, 2004 Oblasts
N = 77826 Bandwidth = 0.01452
Density
0.2 0.4 0.6 0.8 1.0
0 1 2 3 4 5 6
Distribution of Turnout across UIKs, 2004 Republics
N = 17600 Bandwidth = 0.01942
Density
Figure 1: UIK turnout, 2003 and 2004

0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.5 1.0 1.5 2.0 2.5
Distribution of Turnout across UIKs, 2007 Oblasts
N = 77930 Bandwidth = 0.01422
Density
0.0 0.2 0.4 0.6 0.8 1.0
0 1 2 3 4 5 6
Distribution of Turnout across UIKs, 2007 Republics
N = 17875 Bandwidth = 0.02052
Density
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.5 1.0 1.5 2.0
Distribution of Turnout across UIKs, 2008 Oblasts
N = 78384 Bandwidth = 0.01447
Density
0.2 0.4 0.6 0.8 1.0
0 1 2 3 4 5 6
Distribution of Turnout across UIKs, 2008 Republics
N = 17865 Bandwidth = 0.01786
Density
Figure 2: UIK turnout, 2007 and 2008

0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8
turnout proportion
Edinaya Rossiya vote proportion
0.2 0.4 0.6 0.8 1.0
0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
turnout proportion
Putin vote proportion
0.0 0.2 0.4 0.6 0.8 1.0
0.65 0.70 0.75 0.80 0.85 0.90 0.95
turnout proportion
EDINAYA ROSSIYA vote proportion
0.2 0.4 0.6 0.8 1.0
0.6 0.7 0.8 0.9
turnout proportion
Medvedev vote proportion
Figure 3: UIK United Russia vote proportion by turnout, 2003, 2004, 2007 and 2008, republics

0.0 0.2 0.4 0.6 0.8 1.0
0.30 0.35 0.40
turnout proportion
Edinaya Rossiya vote proportion
0.0 0.2 0.4 0.6 0.8 1.0
0.60 0.65 0.70 0.75
turnout proportion
Putin vote proportion
0.0 0.2 0.4 0.6 0.8 1.0
0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
turnout proportion
EDINAYA ROSSIYA vote proportion
0.0 0.2 0.4 0.6 0.8 1.0
0.55 0.60 0.65 0.70 0.75
turnout proportion
Medvedev vote proportion
Figure 4: UIK United Russia vote proportion by turnout, 2003, 2004, 2007 and 2008, oblasts

0.0 0.2 0.4 0.6 0.8 1.0
0.10 0.15 0.20 0.25 0.30 0.35 0.40
turnout proportion
KPRF vote proportion
0.2 0.4 0.6 0.8 1.0
0.00 0.05 0.10 0.15
turnout proportion
Haritonov vote proportion
0.0 0.2 0.4 0.6 0.8 1.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12
turnout proportion
Kommunisticheskaya vote proportion
0.2 0.4 0.6 0.8 1.0
0.05 0.10 0.15 0.20 0.25 0.30
turnout proportion
Zyuganov vote proportion
Figure 5: UIK Communist vote proportion by turnout, 2003, 2004, 2007 and 2008, republics

0.0 0.2 0.4 0.6 0.8 1.0
0.02 0.04 0.06 0.08 0.10 0.12 0.14
turnout proportion
KPRF vote proportion
0.0 0.2 0.4 0.6 0.8 1.0
0.10 0.15 0.20 0.25
turnout proportion
Haritonov vote proportion
0.0 0.2 0.4 0.6 0.8 1.0
0.02 0.04 0.06 0.08 0.10 0.12 0.14
turnout proportion
Kommunisticheskaya vote proportion
0.0 0.2 0.4 0.6 0.8 1.0
0.15 0.20 0.25
turnout proportion
Zyuganov vote proportion
Figure 6: UIK Communist vote proportion by turnout, 2003, 2004, 2007 and 2008, oblasts

0.2 0.4 0.6 0.8 1.0
3.5 4.0 4.5 5.0 5.5 6.0
turnout proportion
2003 Edinaya Rossiya 2d digit
0.2 0.4 0.6 0.8 1.0
3 4 5 6 7 8 9
turnout proportion
2004 Putin 2d digit
0.2 0.4 0.6 0.8 1.0
3 4 5 6 7 8 9
turnout proportion
2007 EDINAYA ROSSIYA 2d digit
0.2 0.4 0.6 0.8 1.0
0 2 4 6 8
turnout proportion
2008 Medvedev 2d digit
Figure 7: UIK United Russia second-digit by turnout, 2003, 2004, 2007 and 2008, republics

0.2 0.4 0.6 0.8 1.0
1 2 3 4 5
turnout proportion
2003 Edinaya Rossiya 2d digit
0.2 0.4 0.6 0.8 1.0
1 2 3 4 5 6
turnout proportion
2004 Putin 2d digit
0.2 0.4 0.6 0.8 1.0
0 2 4 6 8
turnout proportion
2007 EDINAYA ROSSIYA 2d digit
0.2 0.4 0.6 0.8 1.0
1 2 3 4 5 6 7
turnout proportion
2008 Medvedev 2d digit
Figure 8: UIK United Russia second-digit by turnout, 2003, 2004, 2007 and 2008, oblasts

0.2 0.4 0.6 0.8 1.0
0 1 2 3 4
turnout proportion
2003 KPRF 2d digit
0.4 0.6 0.8 1.0
3 4 5 6 7
turnout proportion
2004 Haritonov 2d digit
0.2 0.4 0.6 0.8 1.0
0 2 4 6 8
turnout proportion
2007 Kommunisticheskaya 2d digit
0.2 0.4 0.6 0.8 1.0
0 1 2 3 4 5
turnout proportion
2008 Zyuganov 2d digit
Figure 9: UIK Communist second-digit by turnout, 2003, 2004, 2007 and 2008, republics

0.2 0.4 0.6 0.8 1.0
2 3 4 5 6
turnout proportion
2003 KPRF 2d digit
0.2 0.4 0.6 0.8 1.0
2 3 4 5 6 7
turnout proportion
2004 Haritonov 2d digit
0.2 0.4 0.6 0.8 1.0
0 1 2 3 4 5 6
turnout proportion
2007 Kommunisticheskaya 2d digit
0.2 0.4 0.6 0.8 1.0
0 1 2 3 4 5 6
turnout proportion
2008 Zyuganov 2d digit
Figure 10: UIK Communist second-digit by turnout, 2003, 2004, 2007 and 2008, oblasts