Teaching Statistics: Analyzing
Voting Data from Dancing with the
Stars
By: Jason Gershman
Assistant Professor of Mathematics and
Statistics and Coordinator of Mathematics
Nova Southeastern University
Outline
• Structure of the Voting
• Old Voting System: Ordinal Rankings
• New Voting System: Percentages
• Optimal Voting Strategies
• Conclusions
Dancing with the Stars
•Celebrity Dancing Competition
•13th season underway on ABC
•Spinoff of Britain’s Strictly
Come Dancing
Structure of Dancing with the Stars
• Approximately 12 celebrity contestants are
paired with professional dance partners.
• Each couple dances one at a time in front of 3
judges.
• At the conclusion of the dance, each judge
gives the contestants a score of 1 through 10.
• American public can also vote for the
contestant they want to move to the next
round.
Structure of the Vote
• Each contestant has a static phone number for the
whole season.
• Fans can call up to 5 times per phone line and send 5
texts per text number.
• Phone lines open at start of the show and close ½
hour after the show’s conclusion.
• Fans can vote via internet voting up to 14 hours after
the show airs (5 votes per email address).
Important Facts
• Contestants dance during a two-hour live
telecast on Monday night and then the results
are announced live the next night on Tuesday
evening’s results show.
– Voting is certified by an accounting firm but final
vote tallies are not released to the public.
– You can’t vote for whom you want to eliminate
only for whom you want to keep.
Tallying the Vote
• Each contestant gets a score ranging from 3 to
30 from the judges for each dance (up to 3
dances total per week).
What is the Score?
• When host Tom Bergeron says that they have
“Combined the Judges Scores with America’s
Votes,” what does he mean?
– In Seasons 1 and 2, he meant a system involving
ordinals.
– In Seasons 3 through present, he means a system
involving percentages.
Ordinal Rankings
• Ordinals are data which are ordered rankings.
• Ordinals were formerly used in figure skating
to reduce judge bias.
• When data are sorted (from largest to
smallest), a rank is assigned to where it is in
the list where rank 1 is the largest value.
• Ties are broken by assigning rank of the mean
of the ranks of what constituted the tie so that
the sum of all ranks is preserved.
Example
Raw Score Ranking
Dancer 1 27 1
Dancer 2 25 2.5
Dancer 3 25 2.5
Dancer 4 24 5
Dancer 5 24 5
Dancer 6 24 5
Dancer 7 21 7
Dancer 8 20 8
Dancer 9 15 9
Dancer 10 12 10
Adding Up Rankings
• When Tom Bergeron said that they combined
the judges scores with the fan votes, what is
meant was that they:
– found the rankings for the judges scores;
– found the rankings for the fan votes;
– added the rankings;
The lowest overall score (highest sum of ranks) gets
eliminated.
Example - Continued
Judges Score
Rank
Fan Votes
Rank
Sum of
Ranks
Dancer 1 27 1
7,041,210
1 2
Dancer 2 25 2.5
6,799,791
2 4.5
Dancer 3 25 2.5
4,238,331
3 5.5
Dancer 4 24 5 789,039
7 12
Dancer 5 24 5 469,498
8 13
Dancer 6 24 5 799,499
6 11
Dancer 7 21 7
2,302,303
4 11
Dancer 8 20 8 287,399
9 17
Dancer 9 15 9 181,418
10 19
Dancer 10
12 10
1,781,290
5 15
Why Use Ordinals
• The producers of the show used ordinals in
order to “lessen” the influence of the voting
public.
– Whether you win by 1 vote or a million over your
next highest competitor, your ranking stays the
same.
– Similar to philosophy behind the Electoral College
in electing a president…one outlier of a state
should not influence the raw vote as may happen
in a direct election of a president.
Uses of Rankings in Sports
• This sort of system is used all of the time.
• Example: Associated Press Top 25 Football Poll
– 60 sportswriters rank teams from 1 through 25;
ranked 1
st
yields 25 points and ranked 25
th
yields 1
point.
– If in a certain week, Alabama has 1497 points (out
of a maximum of 1500), one way of obtaining that
score would be 58 1
st
place votes, 1 2
nd
place vote,
and 1 3
rd
place vote.
Problems with Ordinals
• What ended up
happening is that the
effects of the judges got
lessened to the point
that the producer’s
plans backfired.
• The “Jerry Rice effect”
led to the elimination of
the ordinal ranking
system.
Jerry Rice
• Hall of Fame wide receiver for the San
Francisco 49ers
• Paired with professional dance partner Anna
Trebunskaya in season two of DWTS
• Was runner-up to season champion Drew
Lachey despite having the lowest judges score
for 5 dances that season (was never in the top
3 in any dance except for finale when there
were only 3 contestants left)
Week of 4 Contestants Left
• The professional judges despised Jerry Rice’s
dancing as his combined judges score was well
below the other 3 contestants.
• The fan vote is an estimate based on internet
fan polls (in millions).
Judges
Ordinal
Fans
Ordinal
Total
Drew Lachay
55 1.5 11.3 1 2.5
Jerry Rice 41 4 9.2 2 6
Stacy Keibler
55 1.5 7.7 3 4.5
Lisa Rinna
53 3 7.3 4 7
What Happened?
• Lisa Rinna was eliminated.
• Judges and producers were angry.
• The voting system was changed for the next
season.
• Instead of ordinals, the final tally was based
on percentages.
Assigning Percentage
• Find the sum of the raw scores.
• Divide each raw score by the sum of the raw
scores to get a percentage for that raw score.
• Now, when Tom Bergeron says that they
combined the judges scores with the fan
votes, what is meant is that they:
– found the percentages for the judges scores;
– found the percentages for the fan votes;
– added the percentages and the lowest overall
score (sum of percentages) gets eliminated.
Example - Updated
Judges Score
Percent
Fan Votes
Sum of %
Dancer 1 27 12.44%
7,041,210
28.52%
40.96%
Dancer 2 25 11.52%
6,799,791
27.54%
39.06%
Dancer 3 25 11.52%
4,238,331
17.17%
28.69%
Dancer 4 24 11.06% 789,039 3.20% 14.26%
Dancer 5 24 11.06% 469,498 1.90% 12.96%
Dancer 6 24 11.06% 799,499 3.24% 14.30%
Dancer 7 21 9.68%
2,302,303
9.32% 19.00%
Dancer 8 20 9.22% 287,399 1.16% 10.38%
Dancer 9 15 6.91% 181,418 0.73% 7.65%
Dancer 10
12 5.53%
1,781,290
7.21% 12.74%
Jerry Rice Revisited
• What would have happened if the new system
had been in place in season 2?
– Jerry Rice would have been eliminated.
Judges Percent
Fans Percent Total
Drew Lachay
55 26.96% 11.3 31.83% 58.79%
Jerry Rice
41 20.10% 9.2 25.92% 46.01%
Stacy Keibler
55 26.96% 7.7 21.69% 48.65%
Lisa Rinna
53 25.98% 7.3 20.56% 46.54%
Is the Voting System Fair?
• Which system do you think is more fair?
– Ordinal Rankings
– Adding Percentages
• Despite the tweaking to give the judges more
power to proportionally punish poor dancers,
the system has not left the fans powerless.
– Judges Grade Inflation!
• Example 1 - Billy Ray Cyrus.
Billy Ray Cyrus
• Country music singer
and television actor
• Father of pop star Miley
Cyrus
• Finished 5
th
in season 4
of DWTS despite some
terrible judges scores
(His judges scores were
in last place in 6 of 11
dances he performed.)
Tweaking the Vote
• Despite the new scoring system, the fans
saved Billy Ray from elimination numerous
times.
• The fan vote is an estimate based on internet
fan polls (in millions).
Judges
Percent
Fans
Percent
Total
Apolo Ohno 54 17.5%
8 19.0%
36.5%
Joey Fatone 59 19.1%
7.2 17.1%
36.2%
Laila Ali 59 19.1%
7.4 17.6%
36.7%
Ian Ziering 54 17.5%
6 14.3%
31.8%
Billy Ray Cyrus 38 12.3%
7.4 17.6%
29.9%
John Ratzenberger
45 14.6%
6 14.3%
28.8%
Polarizing Contestants
• Remember that you vote for whom you want
to keep.
– Can’t vote to get rid of somebody
• Contestants who are loved or hated tend to do
better than those whom people feel
indifferent towards.
• Kate Gosselin and Bristol Palin illustrate this.
Kate Gosselin
• Perhaps the most
polarizing contestant of
all time
• Reality TV star of TLC’s
Jon and Kate Plus
8…later Kate Plus 8
after divorcing Jon
• Was shown repeatedly
yelling at and degrading
her dance partner Tony
Dovolani in season 10
What Happened?
• Dancing with the Stars received its highest
ratings ever - ranking #1 for two straight weeks
(beating American Idol for the first time ever).
• People tuned in to see Kate dance because:
– She was a bad dancer (had very low judges scores).
– She was shown brutally yelling at her dance partner
blaming him for her poor dancing.
– She was a very popular and public figure from her
reality show and her recent divorce.
What Happened
• The two largest Facebook groups associated
with DWTS in season 10 were:
– Kate Gosselin to win Dancing with the Stars
– Get Kate Gosselin off Dancing with the Stars.
• She was finally eliminated when her enemies
found the mathematically best way to “vote
against her.”
How to “Vote Against”
• Instead of voting
for one other
contestant
• Vote for every
other contestant
to the max
Bristol Palin
• Daughter of former Vice
Presidential nominee
Sarah Palin
• Finished 3
rd
on Season
11 in Fall of 2010
• Had the lowest judges
score on 12 dances yet
finished in 3
rd
place
(made it to the finals)
•
Shocking Elimination
• In the semifinals, pop star Brandy, who got a
perfect score of 30 on one of her two dances,
was eliminated while Bristol Palin made it to
the finals with a lower judges score.
• Fans were outraged!
– ABC affiliates fielded over a million angry phone
calls; numerous groups threatened advertiser
boycotts.
– A Wisconsin man shot his television set which led
to a 15-hour police standoff due to this result.
Was it Really Shocking?
• No, it was not shocking to those who looked
at raw voting data from Dialidol.
• Dialidol is a website which features software
which allows your computer’s modem to call
your favorite contestant’s phone number.
– The site publishes running tallies of the vote
throughout the course of the voting time.
– These data can be used to infer the population
percentage of votes received for each contestant.
Sample Biases
• The sample data shown on the website are the number of
votes and the number of busy signals received for each
contestant as an aggregate of all of those who voted using this
software.
• Sample is biased:
– towards those who care enough to use the site to vote
repeatedly;
– towards those who own a computer with a modem dialer;
– against those who vote via cell phone;
– against those who vote via text message;
– against those who vote via email address.
Making Predictions
• Need to understand the role of the busy signal
– Is a busy signal a “good” or a “bad” thing?
• A busy signal is “good” for that contestant.
– It means other people are voting for him…if you can’t get
through, everyone else is calling for him…if you can get through
every time, it means he has no other fans voting for him except
you.
• This result avoids bias in the sample.
– Measures the population results without dependence on the
sample results
– It doesn’t matter whom you vote for but it acts as a proxy of
how busy a contestant’s phone line is.
Analyzing the Data
• Note that the inference is not perfect since Brandy
was eliminated and not Jennifer (well within the
margin of error).
• Bristol making the top 3 is not shocking.
• Judges’ grade inflation means they only have
themselves to blame for the result.
Optimal Strategy: When to Vote
• Phone lines open at the start of the show and
close 30 minutes after the show is done airing.
• Don’t wait for the end of the show to start
voting.
• Don’t vote just after your favorite stops
dancing and gets their judges score.
• Vote when the show starts to air.
• Vote during a commercial or while another
contestant is dancing.
Concluding Thoughts
• DWTS is a fun show with some interesting
mathematics involved with the voting process.
– Is it a popularity contest or does the winner
actually reflect the celebrity with the most
dancing talent?
– Would any ranking system that allows the fans to
vote not be a popularity contest?
– Would giving the judges more power help or hurt
the show?
– Were fans voting for Bristol Palin as a proxy vote
for her mother for political office? Does it matter?