D'Hondt method calculation: a step-by-step example

The D'Hondt formula

The method is defined by one formula, applied seat by seat: quotient = V ÷ (s + 1), where V is a party's votes and s the seats it has already won. The next seat goes to the party with the largest quotient, its s rises by one and its quotient shrinks. Repeating this until all seats are filled gives exactly the same result as the table method below, which is easier in a spreadsheet: write every party's votes divided by 1, 2, 3 … and take the largest quotients overall.

D'Hondt belongs to the highest averages methods. In US apportionment history the same rule is known as Jefferson's method.

Step 1: build the quotient table

Put the parties in columns, their votes in row 2 and the divisors 1, 2, 3 … in column A. One formula fills the whole table:

=B$2/$A3

The dollar signs do the work: B$2 keeps the votes row fixed while you copy down, $A3 keeps the divisor column fixed while you copy across. You need at least as many divisors as the largest party could win. Six are enough here, because Party A gets 4.

Step 2: find the cut-off and count the seats

The 10th-largest quotient is the smallest one that still wins a seat:

=LARGE(B3:F8,B9) returns 8,600 (Party D's votes ÷ 1).

Each party's seats are its quotients at or above the cut-off: =COUNTIF(B3:B8,">="&$B$10), copied across. Party A (41,000 votes) gets 4 seats, B (29,500) 3, C (17,800) 2, D (8,600) 1 and E (3,100) none. Always add up the seats as a check: they must equal 10.

Worked example: 10 seats for five parties by D'Hondt

The votes, the divisors and the number of seats are yellow. Rows 3 to 8 hold the quotient table, row 10 the cut-off and row 11 the seats.

Worked example: 10 seats for five parties by D'Hondt

	A	B	C	D	E	F
1	Divisor	Party A	Party B	Party C	Party D	Party E
2	Votes	41,000	29,500	17,800	8,600	3,100
3	1	41,000	29,500	17,800	8,600	3,100
4	2	20,500	14,750	8,900	4,300	1,550
5	3	13,667	9,833	5,933	2,867	1,033
6	4	10,250	7,375	4,450	2,150	775
7	5	8,200	5,900	3,560	1,720	620
8	6	6,833	4,917	2,967	1,433	517
9	Seats to fill	10
10	Cut-off (10th largest)	8,600
11	Seats	4	3	2	1	0
12	Total seats	10
13	Vote share	41.0%	29.5%	17.8%	8.6%	3.1%
14	Seat share	40.0%	30.0%	20.0%	10.0%	0.0%
15	Next quotient (seat 11)	8,200

Fictional parties and votes. Educational only: the guide explains how the counting method works and does not recommend any electoral system.

Reading the result: who D'Hondt favors

Compare vote shares with seat shares. Party A has 41.0% of the votes and 40% of the seats, B 29.5% and 30%, C 17.8% and 20%, D 8.6% and 10%. Party E's 3.1% is not enough for a single seat. Here the seats track the votes closely, but across many elections D'Hondt tends to favor larger parties slightly, because a small party must reach the cut-off with its full vote total while a large party clears it with a fraction of its votes.

The next quotient, 8,200 (Party A ÷ 5), shows who would win an 11th seat. Party D's quotient beat Party A's fifth quotient by only 400.

Ties, thresholds and Sainte-Laguë

Three details change real results. Ties at the cut-off make COUNTIF hand out one seat too many, so the seat total check matters: the law decides ties, for example by drawing lots or by total votes. Thresholds, such as a minimum share of the national vote, remove parties before the table is built. And Sainte-Laguë uses the divisors 1, 3, 5, 7 …, which treats smaller parties more evenly. In this example it gives the same 4 / 3 / 2 / 1 / 0 split, but in closer contests the last seat often moves to a smaller party.

For ranked ballots instead of party lists, see the Borda count in Excel.

Where the D'Hondt method is used

D'Hondt, named after the Belgian lawyer Victor D'Hondt, is widely used for party-list proportional representation, for example in parliamentary elections in Spain, Portugal, Belgium and the Netherlands and for European Parliament elections in many member states. Each country adds its own rules on districts, thresholds and ties, so use the official electoral law for any real count. This guide only explains how the arithmetic works and does not recommend any electoral system.

Common mistakes

• Too few divisors. If the largest party could win more seats than there are rows, its seats are undercounted. Add divisors until its last quotient is below the cut-off.
• Wrong dollar signs. =B$2/$A3 must lock the votes row and the divisor column, otherwise the copied table divides the wrong numbers.
• Ignoring ties at the cut-off. Two equal quotients at the cut-off give one seat too many. Check that the seats add up to the total.
• Building the table before applying the threshold. Parties below a legal threshold are removed first.
• Confusing D'Hondt with largest remainders. Hamilton's largest remainder method uses a quota and remainders, not a quotient table, and can give a different result.
• Rounding the quotients. Compare the exact quotients. Rounding can create false ties or change the order.