📚 Election Results and Their Impact on Parties and Government: A Statistical Perspective | 选举结果及其对政党与政府的影响：统计视角
In Edexcel A-Level Mathematics, statistical methods are not just abstract tools; they can be applied directly to real political data such as election results. This article explores how percentages, sampling, confidence intervals, hypothesis tests, correlation and probability models help us interpret election outcomes and their consequences for parties and government.
在 Edexcel A-Level 数学中，统计方法不仅是抽象工具，还可以直接应用于选举结果等真实政治数据。本文探讨百分比、抽样、置信区间、假设检验、相关性和概率模型如何帮助我们解读选举结果及其对政党和政府的影响。
1. From Votes to Seats: The Mathematical Core | 从选票到席位：数学核心
An election result can be summarised by two key quantities: vote share and seat share. Vote share is the percentage of valid votes won by a party, while seat share is the percentage of parliamentary seats it obtains. In the UK general election, the first-past-the-post system often produces a gap between these two shares.
选举结果可以用两个关键量来概括：得票率和席位率。得票率是一个政党赢得的有效选票百分比，而席位率是它获得的议会席位百分比。在英国大选中，简单多数制往往使这两个比例之间产生差距。
Vote share = (Votes for party ÷ Total valid votes) × 100%
Seat share = (Seats won ÷ Total seats) × 100%
For example, a party with 12,000 votes in a constituency where 40,000 valid votes are cast has a vote share of 30%. If that party wins 45 of 150 seats nationally, its seat share is 30% as well, but this equality is not guaranteed under all electoral systems.
例如，某党在一个选区获得 12,000 张选票，该选区的有效选票总数为 40,000 张，则其得票率为 30%。如果该党在全国 150 个席位中赢得 45 席，其席位率也是 30%，但这种相等并非在所有选举制度下都能得到保证。
2. Percentages, Swing and Change | 百分比、摇摆与变化
Swing is a standard measure of change in voter support between two elections. In a two-party contest, swing is often calculated as the average of one party's gain and the other party's loss in percentage points.
摇摆是衡量两次选举之间选民支持变化的常用指标。在两党竞争中，摇摆通常计算为一个政党得票率上升幅度与另一个政党下降幅度的平均值。
Swing = (Party A gain in percentage points + Party B loss in percentage points) ÷ 2
If a party's vote share rises from 34% to 39%, the change is +5 percentage points. A swing of 2.5 points may change many marginal seats, which is why parties focus resources on constituencies with small majorities.
如果某党得票率从 34% 上升到 39%，变化为 +5 个百分点。2.5 个百分点的摇摆可能会改变许多边缘席位，这就是政党将资源集中在多数优势较小的选区的原因。
It is important not to confuse percentage points with percentage change. A rise from 20% to 25% is an increase of 5 percentage points, but a percentage increase of 25%.
重要的是不要将百分点与百分比变化混淆。从 20% 上升到 25% 是增加了 5 个百分点，但百分比增幅为 25%。
3. Opinion Polls and Sampling Methods | 民意调查与抽样方法
Opinion polls rely on sampling. In Edexcel statistics, you learn about random, stratified, quota and cluster sampling. A poll that does not use random sampling may suffer from selection bias, so its results may not generalise to the whole electorate.
民意调查依赖抽样。在 Edexcel 统计中，你会学习随机抽样、分层抽样、配额抽样和整群抽样。未使用随机抽样的民调可能存在选择偏差，因此其结果可能无法推广到全体选民。
- Simple random sampling: every voter has an equal chance of being selected. 简单随机抽样：每个选民都有相同被选中的机会。
- Stratified sampling: the sample is divided into groups such as age, region and past vote, and a random sample is taken from each group. 分层抽样：将样本按年龄、地区和过往投票等分组，然后从每组中随机抽样。
- Quota sampling: interviewers select a fixed number of respondents from each category, but this is non-random and may introduce bias. 配额抽样：访问员从每个类别中选取固定数量的受访者，但这不是随机的，可能会引入偏差。
Stratified sampling ensures subgroups such as age, region and past vote are represented in proportion to the population. This can reduce sampling error, but it cannot remove response bias if certain groups refuse to take part.
分层抽样确保年龄、地区和过往投票等子群体按人口比例被代表。这可以减少抽样误差，但如果某些群体拒绝参与，则无法消除回答偏差。
4. Margin of Error and Confidence Intervals | 误差幅度与置信区间
A poll of n = 1000 likely voters gives an estimated support proportion p̂. The approximate 95% confidence interval is p̂ ± 1.96 × √(p̂(1 - p̂) ÷ n). This margin of error is usually around ±3 percentage points for n = 1000.
对 n = 1000 名可能选民的民调给出估计支持率 p̂。近似 95% 置信区间为 p̂ ± 1.96 × √(p̂(1 - p̂) ÷ n)。当 n = 1000 时，误差幅度通常约为 ±3 个百分点。
95% confidence interval: p̂ ± 1.96 × √( p̂(1 - p̂) ÷ n )
A 4-point lead with a 3-point margin of error may not be statistically significant, because the interval for one party overlaps the interval for the other. For instance, if Party A is at 44% and Party B at 40%, the intervals are approximately 41% to 47% and 37% to 43%, so the intervals overlap.
领先 4 个百分点且误差幅度为 3 个百分点可能并不具有统计显著性，因为一个政党的置信区间与另一个政党的置信区间重叠。例如，若政党 A 支持率为 44%，政党 B 为 40%，则区间分别约为 41% 到 47% 和 37% 到 43%，因此区间重叠。
A larger sample size reduces the margin of error because the standard error √(p̂(1 - p̂) ÷ n) becomes smaller as n increases.
较大的样本量会缩小误差幅度，因为标准误差 √(p̂(1 - p̂) ÷ n) 随着 n 增大而变小。
5. Hypothesis Testing in Election Analysis | 选举分析中的假设检验
Hypothesis testing can be used to test whether a party' re is no change, H₀: p = 0.35, and the alternative hypothesis is H₁: p > 0.35 or p ≠ 0.35.
假设检验可用于检验一个政党的支持率是否发生变化。原假设通常是没有变化，H₀: p = 0.35，备择假设为 H₁: p > 0.35 或 p ≠ 0.35。
H₀: p = p₀ H₁: p > p₀ (one-tailed) or p ≠ p₀ (two-tailed)
The test statistic can be modelled using the binomial distribution X ~ not close to 0 or 1. The normal approximation requires np > 5 and nq > 5, where q = 1 - p.
检验统计量可用二项分布 X ~ B(n, p₀) 建模，当 n 较大且 p 不接近 0 或 1 时，也可用正态分布近似。正态近似要求 np > 5 且 nq > 5，其中 q = 1 - p。
In an
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图