📚 IGCSE CCEA Politics: Formula and Theorem Quick Reference Handbook | IGCSE CCEA 政治：公式定理速查手册
This revision handbook collects the key formulas, indexes and ‘laws’ used in CCEA IGCSE/GCSE Politics. They help you quantify elections, compare voting systems and analyse party power with clear numerical tools.
本复习手册汇集 CCEA IGCSE/GCSE 政治中常用的关键公式、指数与定律。它们帮助你用清晰的数字工具量化选举、比较投票制度并分析政党权力。
1. Electoral Quotas: Hare and Droop | 选举配额：黑尔配额与德鲁普配额
A quota is the minimum number of votes a candidate needs to win a seat under proportional representation, especially in the Single Transferable Vote system. Two quotas are widely used in CCEA Politics.
配额是指在比例代表制中，尤其是在单一可转移投票制下，候选人赢得一个席位所需的最低票数。CCEA 政治中广泛使用两种配额。
Hare quota Q = V ÷ S
Droop quota Q = ⌊ V ÷ (S + 1) ⌋ + 1
Here V is the number of valid votes cast and S is the number of seats to be filled. The Hare quota is simpler, while the Droop quota is lower and prevents too many candidates reaching the quota in early counts.
其中 V 是有效票数，S 是待填补席位数。黑尔配额较简单，而德鲁普配额较低，可防止过多候选人在早期计票中达到配额。
If 1,000 valid votes are cast for 5 seats, the Hare quota is 1,000 ÷ 5 = 200 votes. The Droop quota is the whole-number part of 1,000 ÷ 6 plus 1, giving 166 + 1 = 167 votes.
若有 1,000 张有效票，选举 5 个席位，黑尔配额为 1,000 ÷ 5 = 200 票。德鲁普配额为 1,000 ÷ 6 的整数部分加 1，即 166 + 1 = 167 票。
2. The d’Hondt Divisor Method | 东特除数法
The d’Hondt method is a highest-average formula used to allocate seats to party lists in proportional representation. After a party wins each seat, its vote total is divided by a larger divisor, and the next seat goes to the highest quotient.
东特法是一种最高平均数公式，用于在比例代表制中向政党名单分配席位。一个政党每赢得一个席位，其总票数就除以更大的除数，下一个席位分配给商数最高的政党。
Quotient = V ÷ (s + 1)
V is the party’s total votes and s is the number of seats it has already won. Divisor values are therefore 1, 2, 3, 4 and so on. This method slightly favours larger parties.
V 是政党的总票数，s 是该党已赢得的席位数。因此除数为 1、2、3、4 等。该方法略微有利于大党。
| Divisor | Party A: 1,000 | Party B: 800 | Party C: 300 |
|---|---|---|---|
| 1 | 1,000 – seat 1 | 800 – seat 2 | 300 |
| 2 | 500 – seat 3 | 400 – seat 4 | 150 |
| 3 | 333 | 267 | 100 |
In this 4-seat example, Party A wins 2 seats, Party B wins 2 seats, and Party C wins none. The Northern Ireland Assembly uses the d’Hondt formula to allocate ministerial positions after elections.
在这个 4 席例子中，A 党获得 2 席，B 党获得 2 席，C 党没有席位。北爱尔兰议会选举后使用东特公式分配部长职位。
3. The Sainte-Laguë Divisor Method | 圣拉格除数法
The Sainte-Laguë method is another highest-average formula, but its divisors increase more quickly: 1, 3, 5, 7 and so on. This makes it more proportional for smaller parties than d’Hondt.
圣拉格法是另一种最高平均数公式，但其除数增长更快：1、3、5、7 等。这使得它比东特法对较小政党更公平。
Quotient = V ÷ (2s + 1)
Here s is the number of seats already won by the party. Because the divisors jump from 1 to 3, a small party’s first quotient stays competitive for longer, reducing the large-party bonus.
这里的 s 是该党已赢得的席位数。由于除数从 1 跳到 3，小党的第一个商数能在较长时间内保持竞争力，从而减少大党红利。
- d’Hondt divisors: 1, 2, 3, 4…
- 圣拉格除数：1、3、5、7…
- Sainte-Laguë divisors: 1, 3, 5, 7…
- 东特法更偏向大党；圣拉格法更利于多党比例性。
4. Gallagher Index of Disproportionality | 加拉格尔比例偏差指数
The Gallagher index measures how far an election result deviates from perfect proportionality. It compares each party’s vote share with its seat share.
加拉格尔指数衡量选举结果与完美比例代表制的偏离程度。它将每个政党的得票率与席位率进行比较。
LSq = √( ½ × Σ (vᵢ – sᵢ)² )
In this formula vᵢ is party i’s percentage of votes and sᵢ is party i’s percentage of seats. A score of 0 means perfect proportionality; higher scores mean a more distorted result.
在此公式中，vᵢ 是政党 i 的得票百分比，sᵢ 是政党 i 的席位百分比。得分为 0 表示完全比例；得分越高表示结果越失真。
For example, if a party wins 40% of votes but 55% of seats, the difference is 15 percentage points. The Gallagher index squares that difference, so large gaps are penalised more heavily than small ones.
例如，若一个政党赢得 40% 的选票却获得 55% 的席位，差值为 15 个百分点。加拉格尔指数将该差值平方，因此大差距比小差距受罚更重。
5. Effective Number of Parties | 有效政党数
The effective number of parties is a measure of how many parties are relevant in a party system, weighted by their size. It is usually calculated from vote shares or seat shares.
有效政党数衡量一个政党制度中有多少个具有实际影响力的政党，并按政党规模加权。它通常根据得票率或席位率计算。
N = 1 ÷ Σ pᵢ²
Here pᵢ is each party’s share as a decimal, for example 0.50 for 50%. Squaring small parties reduces their influence, so the formula gives more weight to larger parties.
这里的 pᵢ 是每个政党的份额，以小数表示，例如 50% 记作 0.50。将小党份额平方会降低其影响，因此该公式赋予大党更大权重。
If two parties each have 50% of the vote, the calculation is N = 1 ÷ (0.25 + 0.25) = 2.00. If one party has 50%, a second has 30% and a third has 20%, then N = 1 ÷ (0.25 + 0.09 + 0.04) = 1 ÷ 0.38 = 2.63.
若两个政党各得 50% 选票，则 N = 1 ÷ (0.25 + 0.25) = 2.00。若一个政党得 50%、第二个得 30%、第三个得 20%，则 N = 1 ÷ (0.25 + 0.09 + 0.04) = 1 ÷ 0.38 = 2.63。
6. Duverger’s Law | 迪韦尔热定律
Duverger’s Law states that single-member district plurality systems, such as First Past the Post, tend to produce a two-party system. In contrast, proportional representation tends to produce multi-party systems.
迪韦尔热定律指出，单一选区相对多数制，如简单多数制，往往会产生两党制。相反，比例代表制往往会产生多党制。
There are two main explanations. The mechanical effect means small parties win few seats because they cannot finish first in individual districts. The psychological effect means voters abandon small parties to avoid wasting their votes.
主要有两种解释。机械效应指小党因无法在单个选区中排名第一而赢得很少席位。心理效应指选民为避免浪费选票而放弃小党。
This is a tendency rather than an absolute rule. Some FPTP systems, such as Canada or the UK, still have significant third parties at regional level.
这是一种趋势而非绝对法则。有些简单多数制国家，如加拿大或英国，在地区层面仍有重要的第三党。
7. The Cube Law of Elections | 选举立方定律
The cube law is an empirical rule that links vote shares to seat shares in two-party, single-member district systems. It shows how the winning party’s seat majority is exaggerated.
立方定律是一条经验法则，描述两党制单一选区制度中得票率与席位率的关系。它表明获胜政党的席位多数会被放大。
S₁ ÷ S₂ = (V₁ ÷ V₂)³
Here S₁ and S₂ are the seat totals of the two major parties, while V₁ and V₂ are their vote totals. The exponent 3 means a modest vote lead can become a much larger seat lead.
其中 S₁ 和 S₂ 是两个主要政党的席位总数，V₁ 和 V₂ 是它们的得票总数。指数 3 意味着较小的得票优势可能变成大得多的席位优势。
If Party A wins 55% of two-party votes and Party B wins 45%, the vote ratio is 55 ÷ 45 = 1.22. Cubing this gives about 1.83, so the seat ratio is close to 65:35 rather than 55:45.
若 A 党赢得两党选票的 55%，B 党赢得 45%，得票比为 55 ÷ 45 = 1.22。将其立方后约为 1.83，因此席位比接近 65:35，而不是 55:45。
8. Michels’ Iron Law of Oligarchy | 米歇尔斯寡头铁律
Michels’ Iron Law of Oligarchy claims that all large organisations, including democratic political parties, eventually develop a ruling elite. Power concentrates in the hands of a small leadership group.
米歇尔斯寡头铁律认为，所有大型组织，包括民主政党，最终都会形成统治精英。权力集中在少数领导集团手中。
Leaders acquire specialist knowledge, control internal communication and build professional staff. Ordinary members often lack time or information, so they become passive followers.
领导者掌握专业知识，控制内部沟通，并建立职业团队。普通成员往往缺乏时间或信息，因此成为被动追随者。
This theorem is useful for evaluating party democracy. It suggests that internal party elections may not be enough to prevent oligarchy unless there are strong accountability mechanisms.
该定理对评估政党民主很有用。它表明，除非存在强有力的问责机制，否则党内选举可能不足以防止寡头化。
9. Median Voter Theorem and Left-Right Scale | 中位选民定理与左右谱系
The median voter theorem says that in a two-party election where voters are arranged on a single left-right dimension, both parties have an incentive to move towards the median voter’s position.
中位选民定理指出，在选民按单一左右维度排列的两党选举中，两党都有动力向中位选民的立场靠拢。
The median voter is the voter in the middle when all voters are ordered from left to right. Half of voters are to the left of this position and half are to the right.
中位选民是将所有选民从左到右排列后位于中间的选民。一半选民在该位置左侧，一半在右侧。
Ideological distance = |x – y|
On a scale from -10 (far left) to +10 (far right), the formula above gives the distance between two party or voter positions. Smaller distances mean closer policy competition.
在从 -10（极左）到 +10（极右）的尺度上，上述公式给出两个政党或选民立场之间的距离。距离越小，表示政策竞争越接近。
10. Turnout and Majority Thresholds | 投票率与多数门槛
Voter turnout is the percentage of registered voters who cast a ballot in an election. It is a basic indicator of political participation and legitimacy.
投票率是在一次选举中投了票的登记选民所占的百分比。它是政治参与和合法性的基本指标。
Turnout T = (Votes cast ÷ Registered electorate) × 100%
If 60,000 people vote from a registered electorate of 100,000, turnout is 60%. Political apathy is often measured as 100% minus turnout, giving 40% non-voting in this case.
若登记选民为 100,000 人，其中 60,000 人投票，则投票率为 60%。政治冷漠常以 100% 减去投票率来衡量，此时为 40% 未投票。
Required votes = ⌈R × N⌉Published by TutorHao | IGCSE 政治 Revision Series | aleveler.com
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