This diary will likely have a very small audience. It takes off from my recent experience as a Democratic challenger in the 2014 recount of the 20th Michigan Senate district election.
The Democratic candidate, 60th District Rep. Sean McCann lost to Republican 61st District Rep. Margaret O’Brien by 60 votes on election night, reduced to 59 following the county canvass. McCann decided to ask for a recount, which in Michigan is not automatic, although the cost to the candidate requesting a recount is minimal, only $10 per precinct recounted. The recount took place Dec 8 through 10. O’Brien picked up and additional 15 net votes in the recount, and McCann 13, so the final margin was 61 votes.
There are many interesting anecdotes and experiences that are not part of this diary. This intends to lay out the mathematical aspects of a recount. The numerical conclusions reached probably do not apply beyond Michigan, but the mathematical basis will apply anywhere actual ballots are recounted.
Recount Math
An election between candidate A and candidate B is close enough that it is recounted. Let an “error” be defined as a change in the determination of a single ballot in this race, such that it makes a difference of one vote in the margin (difference in vote totals) between A and B. (We would hope that the recount is more accurate than the original tally on election day. However, for the purpose of this analysis, whether the actual error occurs on election day or in the recount is immaterial.) Let k be the number of errors that increase A’s margin, and n be the total number of errors in the recount, or some subset of the recount such as a precinct. Delta is defined as the change in the margin between A and B from the election to the recount, where a move in favor of A is arbitrarily defined as positive. Then delta = (errors favoring A) – (errors favoring B) = k - (n-k) = 2k – n.
The probability of exactly k out of n errors favoring A is given by the binomial probability function:
For k = 0, 1, 2, …, n, where:
We will assume that there is no process in the recount vs. the election favoring one candidate over the other, so that p = (1-p) = 0.5. Some have claimed this may not be true, but we will not concern ourselves here with this question. So the above simplifies to:
Then the expected value of delta is:
And the variance of delta is:
(I have verified that V(delta) = n for a number of cases, but do not have a general proof.)
Thus the standard deviation of delta is:
Now, for a (losing) candidate facing the decision whether to ask for a recount of a close election, the problem is how to estimate the probability that delta will be large enough to reverse the loss. The standard deviation of delta, which is equal to the square root of the number of errors, can be used to estimate this. However, we don’t know in advance of the recount how many errors there will be. We must estimate this by: n = (error rate) * (total votes in election), where the error rate is measured from a comparable recount in the past.
The 2014 recount of the 20th Michigan Senate district election provides an opportunity to estimate the error rate for future recounts held under conditions similar to those in this recount.
The enumerated errors that we see in the report of the recount represent a lower bound on the total number of errors present, because it is quite likely that there are cases where a positive and a negative error occur in the same precinct for the same candidate, thus canceling each other out. Since there is no label on a ballot showing which candidate it was determined for on Election Day, there is no possible way to directly detect these “hidden error pairs”. However, we can use the formula above that STD DEV(delta) = n^(1/2), and square the measured uncancelled errors (delta) in the report at the lowest level of aggregation (each candidate in each precinct). The sum of these squares across all candidates and precincts will estimate n, the total number of errors.
errors/pct |
O'Brien |
McCann |
Wenke |
Total |
(errors/pct)^2 |
Estimated total errors |
-2 |
1 |
1 |
0 |
2 |
4 |
8 |
-1 |
4 |
10 |
2 |
16 |
1 |
16 |
0 |
194 |
185 |
207 |
586 |
0 |
0 |
1 |
18 |
19 |
8 |
45 |
1 |
45 |
2 |
0 |
3 |
1 |
4 |
4 |
16 |
3 |
1 |
0 |
0 |
1 |
9 |
9 |
#votes |
error rate |
+errors |
21 |
25 |
9 |
55 |
|
70 |
77304 |
0.00091 |
-errors |
6 |
12 |
2 |
20 |
|
24 |
|
0.00031 |
net |
15 |
13 |
7 |
35 |
|
46 |
|
0.00060 |
total |
27 |
37 |
11 |
75 |
|
94 |
|
0.00122 |
Table 1. Summary of changes by precinct in the 20th Senate recount, and resulting calculated error rate.
Table 1 presents a summary of results of the 2014 recount of the 20th Michigan Senate district (consisting of all of Kalamazoo County) election. The Democratic candidate, 60th District Rep. Sean McCann lost to Republican 61st District Rep. Margaret O’Brien by 60 votes on election night, reduced to 59 following the county canvass. O’Brien picked up and additional 15 net votes in the recount, and McCann 13, so the final margin was 61 votes.
Looking at the Table 1, we see that O’Brien had one precinct in which she lost two votes, four in which she lost one, 194 precincts in which her total was unchanged, 18 in which she gained one, and one precinct in which she gained three votes.
I follow the procedure outlined above to estimate that the total error rate was 0.00122. I am dividing by 77304, the number of votes actually recounted, rather than the over 80,000 votes actually cast in the election. This was the total number of votes in the recount, seven precincts having been determined to be unrecountable.
Once we know the estimated error rate, we can multiply it by numbers of votes in different size elections to find the estimated number of errors in those future recounts, taking the square root to find the standard deviation of the margin change delta, then multiplying that by the appropriate number of standard deviations using the one-tailed normal distribution to find the size of vote margin overcome at various probability levels.
Total Number of Votes in Election:
|
statewide |
US House |
MI Sen |
MI House |
Kal CC |
Prob. of Win |
3000000 |
220000 |
80000 |
30000 |
10000 |
6000 |
3000 |
1000 |
25% |
40.8 |
11.1 |
6.7 |
4.1 |
2.4 |
1.8 |
1.3 |
0.7 |
|
10% |
77.5 |
21.0 |
12.7 |
7.8 |
4.5 |
3.5 |
2.5 |
1.4 |
5% |
99.5 |
26.9 |
16.2 |
10.0 |
5.7 |
4.5 |
3.1 |
1.8 |
1% |
140.7 |
38.1 |
23.0 |
14.1 |
8.1 |
6.3 |
4.5 |
2.6 |
|
0.10% |
187.0 |
50.6 |
30.5 |
18.7 |
10.8 |
8.4 |
5.9 |
3.4 |
|
Std. Dev. |
60.5 |
16.4 |
9.9 |
6.0 |
3.5 |
2.7 |
1.9 |
1.1 |
Table 2. Probabilities of overcoming a given vote margin in elections of differing size, using error rate estimated from the 2014 recount of the 20th Michigan Senate district election.
What is striking about the numbers in Table 2 is just how small they are. For a 5% chance of overturning the result of a Michigan House election, the existing vote margin can be no larger than 10. I believe this is far smaller than most interested people would have estimated. For a Michigan Senate election, a 5% chance of overcoming a 16 vote margin – compared with the actual existing margin of 59 in the 20th recount going in.
This is not at all to say that the decision to hold the recount in the 20th was wrong, or that it was a waste of time. In the first place, we did not know that the error rate would be this low in advance. Other (albeit smaller or less comparable), recent recounts have shown error rates three to five times higher.
Secondly, our discussion so far has only concerned single-ballot errors, which while random, we can gain an understanding of their distribution. The 20th Senate recount uncovered a different kind or error, one less amenable to analysis. The county clerk had mis-recorded the total for Wenke, the Libertarian candidate, in one precinct. The paper tape had a smudge that was mis-read as 114, rather than 14. That is the sort of bulk error that is unpredictable, but it is also something that I think the County Board of Canvassers could have caught (by comparing tapes, or the electronic card). So that was an error that reduced Wenke's total by 100 votes -- a difference more than enough to reverse the election result if that had happened to be O'Brien. I think the Board of Canvassers needs to change procedures to ensure that sort of error is caught in the future, either immediately on election night, or at least during the County canvass, without needing to wait for the chance circumstance of a recount for it to come to light.
If this happens, and our future elections results are therefore more reliable, it will have justified the effort and expense of this recount.
Finally, I need to acknowledge the limitations and applicability of these results. Other recounts in other parts of Michigan may involve different election systems, election inspectors with varying levels of training, voting populations with demographic factors that lead to different voting behaviors giving rise to varying tabulator errors, etc. The ballot design may place a candidate in a recount right where the crease caused by folding AV ballots falls. There are any number of situations we can’t even foresee that could lead to an increase in error rates.
Having said all that, this was a large recount carried out under the applicable rules of the Board of Elections of a county having an urban/rural mix, average income, and demographic mix broadly similar to Michigan as a whole. The result should be helpful in providing guidance to future candidates making the recount decision.