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Borda_distance_correlation.mw
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Borda Distance Correlation is a measure of [[candidate correlation]] proposed by Ken Kuhlman and Dan Bishop.
== Definitions ==
The '''Borda distance''' between two candidates on one ballot is the absolute value of the difference of their [[Borda count|Borda]] scores on that ballot.
The '''total Borda distance''' between two candidates is the sum of their Borda distances on each ballot.
The '''average Borda distance''' is the total Borda distance divided by the number of ballots.
The '''most-correlated pair''' of candidates is the one with the lowest average Borda distance.
== Example ==
{{Tenn_voting_example}}
On the Memphis>Nashville>Chattanooga>Knoxville ballots, the Borda scores are:
* Memphis: 3
* Nashville: 2
* Chattanooga: 1
* Knoxville: 0
Thus, the Borda distances on these ballots are:
* between Memphis and Nashville: abs(3-2) = 1
* between Memphis and Chattanooga: abs(3-1) = 2
* between Memphis and Knoxville: abs(3-0) = 3
* between Nashville and Chattanooga: abs(2-1) = 1
* between Nashville and Knoxville: abs(2-0) = 2
* between Chattanooga and Knoxville: abs(1-0) = 1
Similarly, on the Nashville>Chattanooga>Knoxville>Memphis ballots, the Borda distances are:
* between Nashville and Chattanooga: 1
* between Nashville and Knoxville: 2
* between Nashville and Memphis: 3
* between Chattanooga and Knoxville: 1
* between Chattanooga and Memphis: 2
* between Knoxville and Memphis: 1
On the Chattanooga>Knoxville>Nashville>Memphis ballots:
* between Chattanooga and Knoxville: 1
* between Chattanooga and Nashville: 2
* between Chattanooga and Memphis: 3
* between Knoxville and Nashville: 1
* between Knoxville and Memphis: 2
* between Nashville and Memphis: 1
And on the Knoxville>Chattanooga>Nashville>Memphis ballots:
* between Knoxville and Chattanooga: 1
* between Knoxville and Nashville: 2
* between Knoxville and Memphis: 3
* between Chattanooga and Nashville: 1
* between Chattanooga and Memphis: 2
* between Nashville and Memphis: 1
In summary,
<table border align="center">
<tr align="center">
<th>candidate pair</th>
<th>42%</th>
<th>26%</th>
<th>15%</th>
<th>17%</th>
<th>avg.</th>
</tr>
<tr align="center">
<th align="left">Memphis & Nashville</th>
<td>1</td>
<td>3</td>
<td>1</td>
<td>1</td>
<td>1.52</td>
</tr>
<tr align="center">
<th align="left">Memphis & Chattanooga</th>
<td>2</td>
<td>2</td>
<td>3</td>
<td>2</td>
<td>2.15</td>
</tr>
<tr align="center">
<th align="left">Memphis & Knoxville</th>
<td>3</td>
<td>1</td>
<td>2</td>
<td>3</td>
<td>2.33</td>
</tr>
<tr align="center">
<th align="left">Nashville & Chattanooga</th>
<td>1</td>
<td>1</td>
<td>2</td>
<td>1</td>
<td>1.15</td>
</tr>
<tr align="center">
<th align="left">Nashville & Knoxville</th>
<td>2</td>
<td>2</td>
<td>1</td>
<td>2</td>
<td>1.85</td>
</tr>
<tr align="center">
<th align="left">Chattanooga & Knoxville</th>
<td>1</td>
<td>1</td>
<td>1</td>
<td>1</td>
<td>1.00</td>
</tr>
</table>
The [[clone]] pair of Chattanooga and Knoxville has the lowest average Borda distance (1.00), so is the most-correlated pair. Memphis and Knoxville have the highest average Borda distance (2.33) and so are the least-correlated pair.