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Voting Methods & Apportionment Cheat Sheet
A printable reference covering plurality, runoff, Borda count, Condorcet, approval voting, Hamilton apportionment, and divisor methods for grades 9-12.
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Voting methods and apportionment are tools for turning individual preferences or population counts into fair group decisions. This cheat sheet helps students compare election systems, spot when methods give different winners, and understand how seats are divided among states or groups. It is useful for applied math, civics, statistics, and real-world decision making. The goal is to make each method easy to follow with clear steps and formulas. The main voting methods include plurality, instant runoff, Borda count, Condorcet comparisons, and approval voting. Apportionment methods start with a standard divisor, calculate quotas, and assign whole-number seats. Hamilton's method uses lower quotas and largest remainders, while divisor methods adjust the divisor until the total number of seats is correct. Fairness criteria help evaluate whether a method behaves reasonably in different situations.
Key Facts
- In plurality voting, the candidate with the most first-place votes wins, even if that candidate does not have a majority.
- A majority means more than half of the votes, so a candidate needs votes greater than total votes / 2.
- In Borda count with n candidates, a first-place ranking earns n - 1 points, second place earns n - 2 points, and last place earns 0 points.
- In a Condorcet comparison, compare two candidates at a time, and the candidate preferred by more voters wins that head-to-head matchup.
- In Hamilton's method, standard divisor = total population / total seats, and standard quota = group population / standard divisor.
- In Hamilton's method, each group first receives its lower quota, then leftover seats go to the groups with the largest fractional remainders.
- In Jefferson's divisor method, use modified quotas and round each quota down until the rounded total equals the number of seats.
- In Webster's divisor method, use modified quotas and round each quota to the nearest whole number until the rounded total equals the number of seats.
Vocabulary
- Preference schedule
- A table that shows how many voters ranked the candidates in each possible order.
- Plurality winner
- The candidate who receives the greatest number of first-place votes.
- Majority
- More than half of the total votes, calculated as a number greater than total votes / 2.
- Condorcet winner
- A candidate who beats every other candidate in one-on-one head-to-head comparisons.
- Standard quota
- The exact number of seats a group should receive before rounding, found by group population / standard divisor.
- Modified divisor
- An adjusted divisor used in divisor methods so rounded quotas add to the required number of seats.
Common Mistakes to Avoid
- Confusing plurality with majority is wrong because the plurality winner only needs the most first-place votes, not more than half of all votes.
- Giving the wrong Borda points is wrong because the point scale depends on the number of candidates, with n - 1 points for first place and 0 points for last place.
- Skipping head-to-head comparisons in Condorcet voting is wrong because the method requires each candidate to be compared directly against every other candidate.
- Rounding standard quotas immediately in Hamilton's method is wrong because Hamilton first uses lower quotas, then assigns remaining seats by largest remainders.
- Using the same divisor for every divisor method without checking the seat total is wrong because Jefferson, Webster, and other divisor methods may require a modified divisor.
Practice Questions
- 1 In an election with 100 voters, Candidate A gets 42 first-place votes, Candidate B gets 35, and Candidate C gets 23. Who is the plurality winner, and does that candidate have a majority?
- 2 Four candidates are ranked in a Borda count election. How many points should a candidate receive for first place, second place, third place, and fourth place?
- 3 A country has 50 seats and a total population of 2,000,000. Region X has population 360,000. Find the standard divisor and Region X's standard quota.
- 4 Explain why two fair-looking voting methods can produce different winners from the same preference schedule.