📚 GCSE Eduqas Politics: Quick Revision Formulae and Theorems | GCSE Eduqas 政治：公式定理速查手册
This revision handbook collects the essential formulae, quantitative theorems and numerical tools required for GCSE Eduqas Politics. Understanding these concepts allows you to analyse electoral outcomes, party systems and parliamentary arithmetic with precision. All relationships are expressed in clear language and universally accepted notation.
这本速查手册汇总了 GCSE Eduqas 政治课程必备的公式、定量定理与计算工具。理解这些概念能帮助你精确分析选举结果、政党体制和议会算术。所有关系式皆以清晰语言和通用符号表达。
1. Duverger's Law | 杜瓦杰法则
Duverger's Law states that single-member district plurality (SMDP) electoral systems tend to produce a stable two-party system. The mechanical effect concentrates seats in the hands of two large formations, while the psychological effect discourages voters from supporting smaller parties, whom they see as unlikely to win.
杜瓦杰法则指出，单一选区相对多数决制度倾向于催生稳定的两党制。机械效应将席位集中在两大政党手中，而心理效应则会降低选民支持小党的意愿，因为小党看似获胜无望。
Empirically, the law is observed in the United Kingdom's House of Commons, where First-Past-the-Post translates a near two-party competition into single-party majority governments most of the time.
从经验上看，这一法则体现在英国下议院：简单多数制在大多数情况下将接近两党对决的选票格局转化为单一政党多数政府。
2. The Hare Quota | 黑尔限额
The Hare quota is the simplest electoral quota used to determine how many votes a candidate or list needs to win one seat under the Single Transferable Vote (STV) or largest remainder methods. It is defined as total valid votes divided by the total number of seats to be filled.
黑尔限额是最简单的选举配额，用于在可转移单票制或最大余数法中确定候选人或政党名单赢得一个席位所需的最低票数。其定义为总有效票数除以待填补的总席位数。
Hare quota = Total valid votes ÷ Total seats
黑尔限额 = 总有效票数 ÷ 总席位数
If a constituency elects 4 representatives and 200 000 valid votes are cast, the Hare quota is 50 000 votes. This high threshold makes it harder for very small independent groupings to gain representation compared with the Droop quota.
如果一个选区选举4名代表，投出200 000张有效票，黑尔限额就是50 000票。这一较高的门槛使得极小的独立团体比使用特罗普限额时更难获得代表权。
3. The Droop Quota | 特罗普限额
The Droop quota, widely used in STV elections, is designed to prevent surplus votes from being wasted. It is calculated by dividing the total valid votes by one more than the number of seats and then adding one, with the result rounded down to an integer.
特罗普限额广泛用于可转移单票制选举，旨在避免剩余选票浪费。计算方法是：总有效票数除以（席位数+1），再加1，最后向下取整。
Droop quota = ⌊ (Total valid votes ÷ (Seats + 1)) + 1 ⌋
特罗普限额 = ⌊ (总有效票数 ÷ (席位数+1)) + 1 ⌋
use it requires fewer votes to secure election than the Hare quota, the Droop quota slightly favours broader representation.
同样在4席选区、200 000票的情况下，特罗普限额为 ⌊(200 000 ÷ 5) + 1⌋ = 40 001 票。由于其当选所需票数少于黑尔限额，特罗普限额略有利于更广泛的代表性。
4. d'Hondt Method Seat Allocation | 德洪特席位分配公式
The d'Hondt method is a highest averages formula used in party-list proportional representation. For each party, a series of quotients is generated by dividing its total vote by 1, 2, 3, and so on up to the number of seats to be allocated. Seats are awarded one by one to the highest quotient in each round.
德洪特法是一种用于政党名单比例代表制的最高平均数公式。计算时，每个政党的总得票依次除以1、2、3……直至待分配席位数，每轮将1个席位分配给商数最高的政党。
| Party | Votes | ÷1 | ÷2 | ÷3 | ÷4 |
|---|---|---|---|---|---|
| A | 100 000 | 100 000* | 50 000* | 33 333* | 25 000* |
| B | 80 000 | 80 000* | 40 000* | 26 667* | 20 000 |
| C | 30 000 | 30 000 | 15 000 | 10 000 | 7 500 |
In the table, asterisks mark the seven highest quotients when 7 seats are allocated; party A receives 4 seats, party B receives 3 seats, and party C receives none. The d'Hondt method tends to benefit larger parties.
上表中星号标注了分配7个席位时最大的7个商数；A党获4席，B党获3席，C党无席。德洪特法往往有利于大党。
5. First-Past-the-Post Plurality Rule | 简单多数制规则
Under First-Past-the-Post (FPTP), the candidate with the largest number of votes in each single-member constituency is elected. There is no need to secure an absolute majority; a simple plurality suffices. Mathematically, the winning condition per seat is simply Votes(Candidate) > Votes(Any other candidate).
在领先者当选制度下，每个单议席选区中得票最多的候选人即告胜出，无需获得绝对多数，相对多数即可。从数学上看，每个席位的获胜条件为：候选人得票数 > 所有其他候选人的得票数。
This formula frequently produces 'wasted votes' and can allow a candidate to win with well under 50 % of the vote where the opposition is fragmented. National seat shares are thus not proportional to national vote shares.
这一规则常常产生“废票”，并可能在反对派分散时让得票率远低于50%的候选人当选。因此全国席位占比与全国得票占比不成比例。
6. Electoral Threshold | 选举门槛定理
Many proportional representation systems impose a legal threshold, a minimum share of the vote a party must obtain nationally or regionally before it is allowed to take part in the seat allocation. A typical statutory threshold is 5 % of valid votes.
许多比例代表制设定了法定门槛，即政党在全国或地区范围内必须达到的最低得票率，方可参与席位分配。典型的法定门槛为有效票数的5%。
Eligibility condition: Party vote share ≥ Threshold (e.g. 5 %)
资格条件：政党得票率 ≥ 门槛（例如5%）
In the UK, no statutory threshold applies to general elections, but the effective threshold imposed by FPTP constituencies is very high, often well above 35 %. This explains why parties like the Liberal Democrats are historically underrepresented in the Commons relative to their vote share.
英国大选没有法定门槛，但简单多数制在选区层面产生的实际门槛极高，通常远高于35%。这解释了为何自由民主党等党派在议会中的席位占比长期低于其得票占比。
7. Swing Calculation | 摇摆幅度计算
Swing measures the net movement of voters between the two main parties from one election to the next. The Butler swing is the standard formula used in British psephology: take the change in the vote share of Party A and the change in the vote share of Party B, then halve the difference if the parties are moving in opposite directions, or more generally calculate the average change.
摇摆幅度衡量两次选举之间选民在两个主要政党间的净流动。巴特勒摇摆是英国选举学使用的标准公式：取甲党得票率变化和乙党得票率变化，若两党反向变动则取差值的一半，更一般的做法是计算平均变化量。
Swing (A to B) = (Change in B's vote share − Change in A's vote share) ÷ 2
A→B 摇摆幅度 = (B党得票率变化 − A党得票率变化) ÷ 2
If Labour's vote share rises by 3 percentage points and the Conservatives' om Conservative to Labour is ( (−2) − (+3) ) ÷ 2 = −2.5, but by convention we report a 2.5 % swing to Labour. Uniform swing models then translate this into predicted seat changes.
若工党得票率上升3个百分点，保守党下降2个百分点，保守党至工党的摇摆为 ( (−2) − (+3) ) ÷ 2 = −2.5，但惯例上我们报告2.5%的摇摆倾向工党。均匀摇摆模型随后将此转化为席位变化预测。
8. Effective Number of Parties (ENP) Index | 有效政党数目指数
The Laakso-Taagepera index measures how many effective parties compete in an election, weighted by their electoral strength. It is calculated as the reciprocal of the sum of the squared vote shares of all parties.
拉克索-塔格佩拉指数衡量一次选举中有效竞争政党的数量，并按选举实力加权。计算方法是所有政党得票率平方和的倒数。
ENP = 1 ÷ Σ(pᵢ²) where pᵢ is the vote share of party i
有效政党数 ENP = 1 ÷ Σ(pᵢ²) ，其中 pᵢ 为政党i的得票率
For a pure two-party system with each party winning 50 % ted party system, such as the Netherlands, regularly records ENP values above 5. The index helps quantify the fragmentation or concentration of party systems over time.
对一个双方各得50%选票的纯粹两党制，ENP = 1 ÷ (0.5² + 0.5²) = 2.0。而零散的政党体制，如荷兰，ENP值通常超过5。该指数有助于量化政党体制随时间推移的碎片化或集中化程度。
9. The Median Voter Theorem | 中间选民定理
The Median Voter Theorem holds that in a two-party competition over a single policy dimension, both parties will converge on the ideological position of the median voter to maximise their electoral support. The median voter's ideal point splits the electorate exactly in half.
中间选民定理指出，在单一政策维度下的两党竞争中，两党都会向中间选民的意识形态位置靠拢，以最大化其选举支持。中间选民的理想点恰好将全体选民一分为二。
If the electorate's preferences are distributed along a left–right axis and voters are fully informed, any party that deviates too far from the median voter risks losing the centre. This theorem provides a powerful explanatory tool for why major parties often adopt similar centrist platforms in a FPTP system.
如果选民的偏好沿左右轴分布且拥有充分信息，任何偏离中间选民太远的政党都可能失去中间地带。该定理很好地解释了在简单多数制下，大党为何经常采取相似的中间路线政纲。
10. Confidence Motion Threshold | 信任动议通过门槛
In the House of Commons, a government survives a confidence vote only if it secures the support of a simple majority of all MPs. With 650 seats, a government must win at least 326 votes, assuming full attendance and no abstentions that change the effective majority.
在下议院，政府要赢得信任投票，必须获得全体议员简单多数的支持。在650个席位下，政府须赢得至少326票，前提是全勤且没有改变有效多数的弃权。
Confidence threshold = ⌊ Total seats ÷ 2 ⌋ + 1
信任门槛 = ⌊ 总席位数 ÷ 2 ⌋ + 1
The Fixed-term Parliaments Act 2011 (since repealed) required a specific super-majority for early elections, but confidence motions remained governed by the simple majority rule. This arithmetic highlights the importance of even a small numerical advantage in parliamentary politics.
2011年《固定任期议会法》（现已废除）曾要求提前大选须达到特定超多数，但信任动议始终遵循简单多数规则。这一算术凸显了议会政治中哪怕是微弱的人数优势也至关重要。
11. Vote of No Confidence Calculation | 不信任投票计数方法
A vote of no confidence is governed by the same simple majority arithmetic as a confidence motion. The opposition must organise at least 326 votes to topple a government with a working majority of zero. The calculation becomes more complex when independent MPs, the Speaker and tellers are considered.
不信任投票遵循与信任动议相同的简单多数算术。反对党须凑齐至少326票才能推翻一个有效多数为零的政府。当计入独立议员、议长和计票员时，计算会变得更为复杂。
If a government commands 320 reliable supporters and faces 325 opposition MPs plus 5 independents, the government will lose unless it can secure at least 6 votes from the crossbench. The effective majority is not simply 325 minus 320, because abstentions change the number of votes required to reach a majority of those voting.
若政府拥有320名可靠支持者，面对325名反对党议员及5名独立议员，政府将落败，除非能从中立席锁定至少6票。有效多数并非简单的325减320，因为弃权会改变实际投票时达到多数所需的票数。
12. Voter Turnout Formula | 投票率公式
Turnout is the most basic indicator of political participation. It is computed as the total number of valid votes cast divided by the electorate, expressed as a percentage. Spoilt ballots are usually not counted in the numerator when calculating usable turnout figures, though they are sometimes reported separately.
投票率是衡量政治参与的最基础指标。计算方式为实际总有效投票人数除以登记选民总数，以百分比表示。废票通常不计入分子以得出有效投票率，但有时会单独列报。
Turnout (%) = (Total votes cast ÷ Registered electorate) × 100
投票率（%）=（总投票数 ÷ 登记选民数）× 100
Turnout varies significantly by election type: UK general elections in recent cycles have seen turnouts around 65 to 69 %, whereas local elections and police and crime commissioner contests often fall below 30 %. Low turnout can affect the democratic mandate of elected representatives.
不同类型的选举投票率差异显著：英国近几轮大选投票率约在65%至69%，而地方选举及警察与犯罪事务专员竞选常跌破30%。低投票率可能影响民选代表的民主授权。
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