📚 Modelling Changes under Labour, 1997–2010 | 1997–2010 工党执政期间变化的数学建模
Between 1997 and 2010, Labour governments led by Tony Blair and Gordon Brown introduced major changes in public spending, welfare and macroeconomic policy. This article applies A-level Mathematics techniques to quantify and interpret those changes.
1997 年至 2010 年间，由托尼·布莱尔和戈登·布朗领导的工党政府在公共支出、福利和宏观经济政策方面推行了重大变革。本文运用 A-level 数学技巧对这些变化进行量化和解读。
1. Why Use Mathematics to Study Political Change? | 为什么用数学研究政治变化
History tells us what changed; mathematics tells us by how much, how fast and whether the trend was statistically meaningful.
历史告诉我们发生了什么变化，数学则告诉我们变化了多少、变化有多快以及趋势在统计上是否显著。
Common tools include percentage change, compound growth rates, index numbers, time series and regression lines.
常用工具包括百分比变化、复合增长率、指数、时间序列和回归直线。
2. Key Variables and Data Sources | 关键变量与数据来源
We focus on GDP, unemployment, education spending, NHS funding and income inequality over the period 1997–2010.
我们重点关注 1997–2010 年间的国内生产总值、失业率、教育支出、NHS 资金和收入不平等。
Reliable sources include the Office for National Statistics and the World Bank, although figures may be revised over time.
可靠来源包括英国国家统计局和世界银行，但数据可能随时间修正。
| Variable / 变量 | 1997 value / 1997 年值 | 2010 value / 2010 年值 |
|---|---|---|
| UK nominal GDP / 英国名义 GDP | £0.87 trillion / 0.87 万亿英镑 | £1.57 trillion / 1.57 万亿英镑 |
| Unemployment rate / 失业率 | 7.2% | 7.8% |
| Education spending / 教育支出 (share of GDP / 占 GDP 比重) | 4.9% | 6.0% |
| NHS spending / NHS 支出 | £37 billion / 370 亿英镑 | £121 billion / 1210 亿英镑 |
The table uses approximate ONS-based figures for illustration, so examination answers should use the data provided in the question.
上表使用基于英国国家统计局的近似数据进行说明，因此考试答案应使用题目中给出的数据。
3. Percentage Change | 百分比变化
Percentage change = (V₁ − V₀) ÷ V₀ × 100%
If NHS spending rose from £37 billion to £121 billion, the percentage change is (121 − 37) ÷ 37 × 100% ≈ 227%.
如果 NHS 支出从 370 亿英镑增加到 1210 亿英镑，百分比变化为 (121 − 37) ÷ 37 × 100% ≈ 227%。
This shows a large nominal increase, but it does not separate inflation or population growth.
这表明名义上大幅增长，但并没有剔除通货膨胀或人口增长的影响。
Percentage change is the most direct way to compare two values of the same variable over time.
百分比变化是比较同一变量在两个时间点数值的最直接方法。
4. Compound Annual Growth Rate (CAGR) | 复合年增长率
CAGR = (V_end ÷ V_start)^(1/n) − 1
For GDP growing from £0.87 trillion in 1997 to £1.57 trillion in 2010, n = 13 years.
对于 GDP 从 1997 年的 0.87 万亿英镑增长到 2010 年的 1.57 万亿英镑，n = 13 年。
CAGR = (1.57 ÷ 0.87)^(1/13) − 1 ≈ 0.0460, so about 4.6% per year.
CAGR = (1.57 ÷ 0.87)^(1/13) − 1 ≈ 0.0460，即每年约 4.6%。
CAGR smooths out year-by-year volatility and gives a single annual growth rate for the whole period.
CAGR 平滑了逐年波动，给出整个时期的单一年度增长率。
5. Index Numbers with a Base Year | 以基年构建指数
Index value = (Value in year t ÷ Value in base year) × 100
Setting 1997 as the base year, the 2010 NHS spending index is (121 ÷ 37) × 100 ≈ 327, meaning spending was 327% of the 1997 level.
以 1997 年为基年，2010 年 NHS 支出指数为 (121 ÷ 37) × 100 ≈ 327，即支出是 1997 年水平的 327%。
Index numbers make it easier to compare variables measured in different units on the same scale.
指数使不同单位的变量可以在同一尺度上进行比较。
6. Real Versus Nominal Values | 实际值与名义值
Real value = (Nominal value ÷ Price index) × 100
Nominal spending includes inflation; real spending removes the effect of price rises so that volume changes can be compared.
名义支出包含通货膨胀；实际支出剔除物价上涨的影响，以便比较数量的变化。
If NHS nominal spending rose by 227% but the GDP deflator rose by 48%, real growth would be approximately (3.27 ÷ 1.48) − 1 ≈ 1.21, or 121%.
如果 NHS 名义支出增长 227%，而 GDP 平减指数上升 48%，实际增长约为 (3.27 ÷ 1.48) − 1 ≈ 1.21，即 121%。
7. Unemployment Rate as a Proportion | 失业率的比例计算
Unemployment rate = (Unemployed ÷ Labour force) × 100%
A rise from 7.2% to 7.8% is an increase of 0.6 percentage points, not 0.6 percent.
从 7.2% 上升到 7.8% 是上升了 0.6 个百分点，而不是 0.6%。
The percentage increase in the unemployment rate is (7.8 − 7.2) ÷ 7.2 × 100% ≈ 8.33%.
失业率的百分比增幅为 (7.8 − 7.2) ÷ 7.2 × 100% ≈ 8.33%。
Confusing percentage points with percentage increase is a common A-level error.
混淆百分点和百分比增幅是 A-level 考试中的常见错误。
8. Time Series and Moving Averages | 时间序列与移动平均
3-point moving average at t = (x_{t−1} + x_t + x_{t+1}) ÷ 3
Moving averages smooth short-term fluctuations and reveal underlying trends in annual spending or unemployment figures.
移动平均可以平滑短期波动，揭示年度支出或失业数据的基本趋势。
For three consecutive unemployment rates 7.2%, 7.5% and 7.8%, the moving average is (7.2 + 7.5 + 7.8) ÷ 3 ≈ 7.5%.
对于连续三年的失业率 7.2%、7.5% 和 7.8%，移动平均为 (7.2 + 7.5 + 7.8) ÷
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