In proportional representation systems there are five key mathematical formulae through which votes are translated into seats. In list PR systems, see List PR, there are 'highest average' methods (D'Hondt and Sainte-Lagu ) and 'largest remainder' methods (Hare and Droop). The single transferable vote form of PR, see Single Transferable Vote, almost always uses the Droop quota.
Examples of each method are given below:
| Table 1 Seat Allocation by Two Highest-Average Formulas in a Six-Member District with Four Parties |
| Party |
Votes
(v) |
D'Hondt Allocation
|
Total
Seats |
| A |
42 000 |
42 000 (1) |
21 000 (3) |
14 000 (6) |
3 |
| B |
31 000 |
31 000 (2) |
15 500 (5) |
10 333 |
2 |
| C |
15 000 |
15 000 (4) |
7 500 |
|
1 |
| D |
12 000 |
12 000 |
|
|
0 |
| |
|
|
|
|
|
| TOTAL |
100 000 |
|
|
|
|
|
| Party | Votes
(v) | Modified Sainte-Lague Allocation
|
Total
Seats |
| A |
42 000 |
30 000 (1) |
14 000 (3) |
8 400 |
2 |
| B |
31 000 |
22 143 (2) |
10 333 (5) |
6 200 |
2 |
| C |
15 000 |
10 714(4) |
5 000 |
|
1 |
| D |
12 000 |
8 571 (6) |
|
|
1 |
| |
|
|
|
|
|
| TOTAL |
100 000 |
|
|
|
|
| Table 2 Seat Allocation by Two Largest-Remainder Formulas in a Six-Member District with Four Parties
|
| Party |
Votes |
Hare Quotas (a) |
Full Quota Seats |
Remaining Seats |
Total Seats |
| A |
42 000 |
2.52 |
2 |
0 |
2 |
| B |
31 000 |
1.86 |
1 |
1 |
2 |
| C |
15 000 |
0.90 |
0 |
1 |
1 |
| D |
12 000 |
0.72 |
0 |
1 |
1 |
| |
|
|
|
|
|
| TOTAL |
100 000 |
6.00 |
3 |
3 |
6 |
|
| Party |
Votes |
Droop Quotas (b) |
Full Quota Seats |
Remaining Seats |
Total Seats |
| A |
42 000 |
2.94 |
2 |
1 |
3 |
| B |
31 000 |
12.17 |
2 |
0 |
2 |
| C |
15 000 |
1.05 |
1 |
0 |
1 |
| D |
12 000 |
0.84 |
0 |
0 |
0 |
| |
|
|
|
|
|
| TOTAL |
100 000 |
7.00 |
5 |
1 |
6 |
| Table 3 Seat Allocation by Single Transferable Vote in a Three-Member District with Five Candidates |
| Droop Quota = [ 100 / ( 3 + 1 ) ] + 1 = 26 |
| Preferences |
| 23 ballots P, Q, T |
| 23 ballots P, R, S
|
| 16 ballots Q, R
|
| 5 ballots R, S
|
| 20 ballots S, T
|
| 8 ballots T, Q, R
|
| 5 ballots T
|
|
| Candidate |
First Count |
Second Count |
Third Count |
| P |
46 |
- 20 = 26 |
26 |
| Q |
16 |
+ 10 = 26 |
26 |
| R |
5 |
+ 10 = 15 |
+ 8 = 23 |
| S |
20 |
20 |
20 |
| T |
13 |
13 |
- 13 = 0 |
| Nontransferable |
- |
0 |
+ 5 = 5 |
|
Candidates elected: P, Q, and R
|
