📚 Year 12 WJEC Politics: Formula and Theorem Quick Reference Handbook | Year 12 WJEC 政治：公式定理速查手册
This quick-reference handbook gathers the essential formulas, theorems, and indices embedded in the Year 12 WJEC Government and Politics course – covering electoral systems, party behaviour, and voting analysis. Mastering these rules will sharpen your essay arguments and data-response answers.
本速查手册汇集了 Year 12 WJEC 政府与政治课程中的核心公式、定理和指数——涵盖选举制度、政党行为和投票分析。掌握这些规则将提升你在论述题和数据分析题中的答题能力。
1. Duverger's Law | 迪韦尔热定律
Duverger's Law holds that single-member district plurality (SMDP) electoral systems, like the one used for UK general elections, strongly favour a two-party system. This is not a mathematical equation but a powerful empirical regularity.
迪韦尔热定律指出，单一选区相对多数制（SMDP）——如英国大选采用的制度——强烈倾向于产生两党制。这不是一个数学等式，而是一条强有力的经验规律。
The law operates through a 'mechanical effect' (only the largest party in each constituency wins the seat, under-representing smaller parties) and a 'psychological effect' (voters avoid wasting their votes on third parties, and elites coalesce into two broad electoral blocs).
该定律通过“机械效应”（每个选区只有得票最多的政党获得席位，小党代表性不足）和“心理效应”（选民不愿把票浪费在第三党身上，精英向两大选举阵营靠拢）发挥作用。
2. Median Voter Theorem | 中间选民定理
Developed by Anthony Downs, the median voter theorem predicts that in a two-party system where voters' preferences can be ordered along a single left-right dimension, both parties will converge towards the policy position of the median voter to maximise their chance of election.
由安东尼·唐斯发展的中间选民定理预测，在两党制中，若选民的偏好可沿单一的左右维度排序，两党都会向中间选民的政策立场靠拢，以最大化胜选机会。
If a party remains too far to the left or right, its rival can capture more votes by moving closer to the centre. This model helps explain why Labour and the Conservatives often occupy similar ground on key economic issues in UK elections.
如果一个政党离中心过左或过右，对手便可因更靠近中心而赢得更多选票。该模型有助于解释英国选举中工党和保守党为何在核心经济议题上经常占据相似立场。
3. Electoral Quotas: Hare and Droop | 选举配额：黑尔配额与德鲁普配额
Under the Single Transferable Vote (STV) used in Northern Ireland Assembly elections, a candidate must reach a quota to be elected. Two main formulas exist:
在北爱尔兰议会选举使用的单记可转让投票制（STV）下，候选人必须达到配额才能当选。主要有两种公式：
Hare Quota: Q = V ÷ S
Droop Quota: Q = [V ÷ (S + 1)] + 1
total number of seats to be filled. The Droop quota is the more common formula in UK STV contests because it prevents more candidates from reaching the quota than there are seats. The result is rounded down to the nearest whole number.
其中 V = 总有效票数，S = 应选席次。德鲁普配额是英国 STV 选举中更常用的公式，因为它能防止多于席位数的候选人达到配额。计算结果向下取整。
4. Divisor Methods: d'Hondt and Sainte-Laguë | 除数法：东特法与圣拉古法
These divisor formulas allocate seats in list-based proportional representation systems. Votes for each party are divided by a series of divisors; seats are awarded to the parties with the highest resulting quotients.
这些除数法公式用于名单比例代表制中的席位分配。各政党所得票数除以一系列除数；席位按最高商数分配。
d'Hondt method divisors: 1, 2, 3, 4, … This slightly favours larger parties. It is used for European Parliament elections (when the UK was a member), Welsh Senedd regional lists, and London Assembly seats.
东特法除数：1, 2, 3, 4, … 这一方法略微偏向大党。曾用于英国欧洲议会选举、威尔士议会地区名单和伦敦议会席位。
Sainte-Laguë method divisors: 1, 3, 5, 7, … This provides a more proportional outcome and gives smaller parties a better chance. It is used in some European countries and for Scottish local government by-elections.
圣拉古法除数：1, 3, 5, 7, … 这一方法产生更比例化的结果，更有利于小党。若干欧洲国家和苏格兰地方政府补选使用此方法。
5. Effective Number of Parties (Laakso-Taagepera Index) | 有效政党数（拉克斯-塔格佩拉指数）
This index measures how many 'relevant' parties exist in a party system, weighting them by their strength. It allows comparison across countries and electoral systems.
该指数衡量一个政党体系中存在多少“相关”政党，并根据实力加权，便于跨国家和选举制度比较。
Effective Number of Parties = 1 ÷ (p₁² + p₂² + p₃² + …)
s) held by party i. If one party holds all the seats, ENP = 1. If two parties each hold 50% In a multiparty system with many small parties, ENP rises above the raw count of parties.
其中 pᵢ 是政党 i 所占席位（或选票）的比例。若一个政党拥有全部席位，ENP = 1。若两党各占 50%，ENP = 2。在拥有诸多小党的多党制中，ENP 会大于政党实际数量。
6. Butler Swing | 巴特勒摇摆
Butler swing measures the net shift of voters between the two main parties (usually Conservative and Labour) across an election. It is a standard tool for analysing UK election results and predicting seat changes.
巴特勒摇摆衡量选票在两大政党（通常是保守党和工党）之间的净转移，是分析英国选举结果和预测席位变化的标准工具。
Swing = (ΔA + ΔB) ÷ 2
Where ΔA is the percentage-point increase in Party A's vote share and ΔB is the percentage-point decrease in Party B's share (or vice versa). For example, if Labour' s by 3 points and the Conservative share falls by 3 points, the swing from Con to Lab is 3%. The formula gives (3 + 3)/2 = 3%.
其中 ΔA 是 A 党得票率的百分比上升幅度，ΔB 是 B 党得票率的下降幅度（反之亦然）。例如，工党得票率上升 3 个百分点，保守党下降 3 个百分点，从保守党到工党的摇摆为 3%。公式计算为 (3+3)/2 = 3%。
7. Pedersen Index of Electoral Volatility | 佩德森选举波动性指数
The Pedersen index quantifies net change in a party system from one election to the next. It captures the total proportion of voters who switched parties or caused seat shares to change.
佩德森指数量化政党体系从一次选举到下一次的净变化，捕捉了转换政党或导致席位份额变化的总选民比例。
V = (1/2) Σ |Δsᵢ|
Here, Δsᵢ is the change in the vote or seat share of party i between two elections. The absolute values are summed and halved. An index of 0 means no change; a high index indicates massive voter realignment.
此处 Δsᵢ 是政党 i 在两次选举之间选票或席位份额的变化。将绝对值加总后除以2。指数为 0 表示无变化；指数很高则意味着大规模选民重组。
8. Banzhaf Power Index | 班扎夫权力指数
The Banzhaf index measures an actor's power in a weighted voting system – for instance, a committee or international organisation – by counting how often that actor is a 'critical' member whose vote can swing a winning coalition to a losing one.
班扎夫指数通过计算一个行为者在加权投票中成为“关键”成员的频率——即其投票能让一项获胜联盟转为失败——来衡量该行为者的权力，常用于委员会或国际组织。
= (Number of coalitions where i is critical) ÷ (Total critical defections of all members)
Although this index does not feature directly in the Year 12 specification, it underpins analysis of voting power in bodies like the EU Council and helps illustrate why simple seat counts can be deceptive.
虽然该指数不直接出现在 Year 12 课纲中，但它支撑着对欧盟理事会等机构表决权的分析，有助于说明为何单纯计算席位数量可能具有误导性。
9. Voter Turnout Calculation | 投票率计算
Turnout is a simple yet vital statistic for understanding democratic participation. It appears in Unit 2 when evaluating participation and representation.
投票率是理解民主参与的一项简单而关键的统计数据，在第二单元评价参与和代表时经常出现。
Turnout (%) = (Total votes cast ÷ Size of registered electorate) × 100
Always check whether 'valid votes' or 'total ballot papers' is being used; spoilt papers are usually excluded from turnout figures in UK general elections but may be included in some referendums.
务必检查是使用“有效票数”还是“总投票数”；英国大选的投票率通常排除废票，但某些公投可能将其计入。
10. The Cube Law | 立方定律
Historically, the Cube Law described the relationship between votes and seats in two-party SMDP systems, especially in the UK. It states that the ratio of seats won by two parties approximates the cube of their vote ratio.
从历史上看，立方定律描述了在两党 SMDP 制度中选票与席位的关系，尤其在英国。它指出两党赢得的席位之比约等于其选票之比的立方。
Seatsₐ ÷ Seatsᵦ ≈ (Votesₐ ÷ Votesᵦ)³
For example, if Party A wins 52% of votes and Party B 48% ratio is 52/48 ≈ 1.083. Its cube is about 1.27, predicting a seat ratio of roughly 1.27:1 – a winner' but the concept remains a useful illustration of the disproportionality embedded in FPTP.
例如，若 A 党赢得 52% 的选票、B 党赢得 48%，选票比为 52/48 ≈ 1.083。其立方约为 1.27，预测席位比约为 1.27:1——大党由此获得“胜者红利”。由于多党竞争和地区差异，该定律在近年英国选举中有所弱化，但它仍是说明简单多数制不均衡性的有效工具。
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