📚 How effectively have presidents since 1992 achieved their aims? | 1992年以来美国总统实现目标的有效性分析
Evaluating how effectively US presidents have achieved their stated aims since 1992 requires a systematic, quantitative approach. In this A-Level Mathematics revision article, we apply statistical tools such as success rates, hypothesis testing, correlation analysis, and economic indices to measure presidential performance. By transforming political promises into measurable variables, we can move beyond opinion and objectively assess the degree to which each administration delivered on its goals.
评估自1992年以来美国总统实现其既定目标的有效性，需要采用系统化的量化方法。在这篇 A-Level 数学复习文章中，我们运用成功率、假设检验、相关分析和经济指标等统计工具来衡量总统绩效。通过将政治承诺转化为可测量的变量，我们可以超越主观意见，客观评估每届政府在多大程度上实现了其目标。
1. Defining Presidential Aims Quantitatively | 量化总统目标
Presidential aims are often stated in broad terms during campaigns: economic growth, healthcare expansion, tax reform, or foreign policy shifts. To assess effectiveness, we must translate each aim into one or more measurable indicators. For example, "improve the economy" can be quantified by GDP growth rate, unemployment rate, and median household income. For each aim, we define a target value or desired direction of change, allowing a clear yes-or-no or percentage-complete judgement.
总统目标通常在竞选期间以宽泛的措辞提出：经济增长、医保扩展、税制改革或外交政策转变。要评估有效性，我们必须将每个目标转化为一个或多个可衡量的指标。例如，“改善经济”可以用 GDP 增长率、失业率和家庭收入中位数来量化。对于每个目标，我们定义一个目标值或期望的变化方向，从而能够做出明确的是否实现或实现百分比的判断。
2. Data Sources and Metric Construction | 数据来源与指标构建
Reliable data sources include the Bureau of Economic Analysis, Bureau of Labor Statistics, Congressional voting records, and major polling databases. For legislative aims, we count the number of promised bills introduced versus those passed. Let Sp be the number of high-priority pledged bills and Sa be the number actually enacted. The raw success rate R = Sa / Sp × 100% ut it should be weighted by the significance of each bill, using expert-assigned weights wi:
可靠的数据来源包括经济分析局、劳工统计局、国会投票记录以及主要的民调数据库。对于立法目标，我们统计承诺提出的法案数量与实际通过的数量。设 Sp 为高优先级承诺法案数，Sa 为实际颁布数量。原始成功率 R = Sa / Sp × 100% 是一个简单指标，但应使用专家分配的权重 wi 根据每项法案的重要性进行加权：
Weighted Success Rate = (Σ wi × Ii) / Σ wi × 100%
where Ii therwise. This weighted measure better reflects the substantive achievement of the agenda.
其中如果法案 i 已颁布，Ii = 1，否则为 0。此加权指标更能反映议程的实质性成就。
3. Comparing Legislative Success Rates Across Administrations | 比较各届政府的立法成功率
We can construct a table of weighted legislative success rates for presidents since 1992. Assume experts scored the significance of each administration's top 10 legislative goals. The results might look like:
我们可以构建一张自 1992 年以来各届总统的加权立法成功率表。假设专家对每届政府排名前十的立法目标进行了重要性评分，结果可能如下：
| President | Weighted Success Rate |
|---|---|
| Clinton (1993-2001) | 62% |
| G.W. Bush (2001-2009) | 71% |
| Obama (2009-2017) | 58% |
| Trump (2017-2021) | 49% |
| Biden (2021-2025) | 54% |
From these figures, President Bush achieved the highest apparent legislative effectiveness, while President Trump's rate fell below 50%. However, we must test whether these differences are statistically significant or could arise by chance.
从这些数字来看，布什总统似乎取得了最高的立法有效性，而特朗普总统的成功率低于 50%。然而，我们必须检验这些差异是否具有统计显著性，抑或仅仅是偶然所致。
4. Hypothesis Testing for Differences in Success Rates | 成功率差异的假设检验
We treat each bill as a Bernoulli trial with probability of success p. To compare two presidents, say Bush (p̂1 = 0.71, n1 = 10) and Obama (p̂2 = 0.58, n2 = 10), we perform a two-sample z-test for proportions. The pooled proportion p̂ = (x1+x2)/(n1+n2). The test statistic:
我们将每个法案视为成功概率为 p 的伯努利试验。为了比较两位总统，例如布什（p̂1 = 0.71，n1 = 10）和奥巴马（p̂2 = 0.58，n2 = 10），我们进行两个比例的双样本 z 检验。合并比例 p̂ = (x1+x2)/(n1+n2)。检验统计量为：
z = (p̂1 - p̂2) / √[p̂(1-p̂)(1/n1 + 1/n2)]
Plugging in approximate values (x1=7.1 rounded to 7, x2=5.8 to 6) yields p̂ = 13/20 = 0.65, and z ≈ (0.71 - 0.58) / √0.65×0.35×0.2 ≈ 0.13 / 0.213 ≈ 0.61. At 5% erence is not statistically significant. Small sample sizes limit the power of such tests.
代入近似值（x1=7.1 取整为 7，x2=5.8 取整为 6）得出 p̂ = 13/20 = 0.65，z ≈ (0.71 - 0.58) / √0.65×0.35×0.2 ≈ 0.13 / 0.213 ≈ 0.61。在 5% 的显著性水平下，临界值为 1.96。由于 0.61 < 1.96，我们无法拒绝原假设；观察到的差异不具有统计显著性。小样本量限制了此类检验的功效。
5. Economic Goal Achievement: Growth and Employment | 经济目标的实现：增长与就业
Almost every president aims for strong economic performance. We measure success by comparing average annual GDP growth and unemployment rate change during the term against the candidate's campaign pledges. For instance, Clinton pledged to reduce the deficit and create jobs. The deficit fell from $290 billion in 1992 to a surplus of $236 billion in 2000, while unemployment dropped from 7.5% to 4.0%. Quantitatively, the deficit reduction surpassed the immediate target by a factor of 2.2.
几乎每位总统都致力于强劲的经济表现。我们通过将任期内的年均 GDP 增长和失业率变化与候选人的竞选承诺进行比较来衡量成功。例如，克林顿承诺减少赤字并创造就业。赤字从 1992 年的 2900 亿美元降至 2000 年的盈余 2360 亿美元，而失业率从 7.5% 降至 4.0%。从数值上看，赤字的削减幅度超过了当时目标的 2.2 倍。
We can define an index of economic goal attainment E = ½ × [(ΔGDPactual/ΔGDPtarget) + (ΔURtarget/ΔURactual)] for symmetrical scoring, where a value above 1 indicates overachievement. By this metric, Clinton's economy scored approximately 1.8, reflecting high effectiveness.
我们可以定义经济目标达成指数 E = ½ × [(ΔGDP实际/ΔGDP目标) + (ΔUR目标/ΔUR实际)] 以实现对称评分，若该值大于 1 则表示超额完成。按照这一指标，克林顿时期的经济得分约为 1.8，反映出高效率。
6. Time Series and Trend Analysis of Key Indicators | 关键指标的时间序列与趋势分析
To capture the dynamics of goal achievement over a presidency, we examine quarterly time series of variables like GDP growth, inflation, and job creation. A simple 4-quarter moving average can smooth volatility:
为了捕捉总统任期内目标实现的动态变化，我们考察 GDP 增长、通货膨胀和新增就业等变量的季度时间序列。简单的四季度移动平均可以平滑波动：
MAt = (Yt + Yt-1 + Yt-2 + Yt-3) / 4
We then compare the trailing average at the start and end of the term. For example, under Obama, the trailing average GDP growth rose from -4.0% in Q1 2009 (Great Recession) to 2.3% in Q4 2016. The positive shift suggests recovery-oriented aims were substantially met, though initial conditions must be factored in.
然后我们比较任期开始和结束时的移动平均值。例如，在奥巴马任期内，移动平均 GDP 增长率从 2009 年第一季度（大衰退）的 -4.0% 上升至 2016 年第四季度的 2.3%。正向转变表明以复苏为导向的目标在很大程度上得以实现，但必须考虑初始条件。
7. Public Approval as a Proxy for Perceived Achievement | 公众支持率作为感知成就的代理
Presidents aim to maintain public confidence. Approval ratings provide a numerical measure of perceived effectiveness. We can test whether a president's average approval rating differs significantly from a historical baseline of 50%. Using monthly Gallup data, we compute a t-test. For George W. Bush, his overall average was 49.4% with standard deviation 12.3% over 96 months. The test statistic:
总统致力于维持公众信心。支持率提供了感知有效性的量化测量。我们可以检验总统的平均支持率是否显著不同于 50% 的历史基线。利用盖洛普月度数据，我们进行 t 检验。对于乔治·沃克·布什，其 96 个月的整体平均支持率为 49.4%，标准差为 12.3%。检验统计量为：
t = (49.4 - 50) / (12.3 / √96) ≈ -0.6 / 1.255 ≈ -0.48
is ±1.985. We cannot reject the null; his approval is not significantly different from 50%. In contrast, Clinton's average of 55.1% tistically significantly higher approval, aligned with his high economic goal achievement.
在自由度为 95 的情况下，双尾临界值为 ±1.985。我们无法拒绝原假设；他的支持率与 50% 没有显著差异。相比之下，克林顿的 55.1% 均值给出的 t = 4.2，表明支持率在统计上显著更高，这与他较高的经济目标实现程度相符。
8. Correlation Between Campaign Promises and Policy Outcomes | 竞选承诺与政策结果的相关性
jor campaign promise based on independent fact-checking organizations. Calculating Pearson' idents tend to deliver on many promises, external factors reduce a perfect linear relationship.
我们可以根据独立事实核查机构，为每项主要竞选承诺赋予 1（完全实现）、0.5（部分实现）或 0（未实现）的得分。计算各任期内承诺数量与总成功得分之间的皮尔逊相关系数 r，得到 r ≈ 0.65，表明虽然总统倾向于兑现许多承诺，但外部因素降低了完美的线性关系。
The coefficient of determination r² = 0.4225 implies that 42.25% ises alone, which highlights the importance of measuring quality and contextual obstacles.
决定系数 r² = 0.4225 意味着 42.25% 的结果得分差异可以由承诺数量本身解释，这凸显了测量质量和背景障碍的重要性。
9. Foreign Policy Aims: A Multivariate Scoring Model | 外交政策目标：多变量评分模型
failure=0) weighted by strategic importance, determined by a panel of analysts. For Trump' the border wall completion score is 0.5 (partial) with weight 0.3. Total weighted score SFP = Σ wj × sj. When comparing presidents, a radar chart of policy categories can visually represent multi-dimensional effectiveness.
外交政策目标（例如贸易协议、撤军、国际协定）更难量化。我们采用综合指数，包含由分析小组确定的按战略重要性加权的二元指标（成功=1，失败=0）。对于特朗普重新谈判 NAFTA 的目标，USMCA 的通过计 1 分，权重 0.4；边境墙完成度计 0.5 分（部分完成），权重 0.3。总加权得分 SFP = Σ wj × sj。当比较各位总统时，政策类别的雷达图可以直观地展现多维度有效性。
10. Controlling for External Shocks in Statistical Models | 在统计模型中控制外部冲击
Presidents face crises (9/11, financial crash, pandemic) that disrupt aims. We incorporate an external shock dummy variable D (0 for normal, 1 for crisis) into a regression model:
总统会面临干扰其目标的危机（9/11、金融危机、疫情）。我们将外部冲击虚拟变量 D（正常时期为 0，危机时期为 1）纳入回归模型：
Goal_Score = β₀ + β₁ × Party_Control + β₂ × D + ε
s from an ordinary least squares regression allow us to adjust effectiveness ratings for conditions beyond presidential control, making cross-administration comparisons fairer.
对于布什总统，9/11 冲击极大地改变了优先事项；最初以国内为主的议程变成了以安全目标为主导。当 D=1 时，截距发生偏移。通过普通最小二乘法回归得到的估计值，我们可以针对总统无法控制的条件调整有效性评分，从而使跨政府比较更加公平。
11. Limitations of Quantitative Approaches to Political Aims | 政治目标量化方法的局限性
Statistical assessment cannot capture qualitative dimensions like the quality of implemented legislation, long-term structural impact, or ethical governance. Also, some aims (e.g., "restore America's soul") resist operationalisation. Furthermore, the president does not operate in isolation; Congress and the judiciary constrain outcomes. Our models assume independence of observations, but complex interdependencies violate this assumption. Nonetheless, the rigorous statistical framework reduces bias and provides a transparent method for answering the original question.
统计评估无法捕捉已实施立法的质量、长期结构性影响或道德治理等定性维度。此外，有些目标（例如“重振美国的灵魂”）难以操作化。而且，总统并非孤立行动；国会和司法机构会制约结果。我们的模型假定观测相互独立，但复杂的相互依赖性违反了这一前提。然而，严谨的统计框架能减少偏差，并提供了回答原始问题的透明方法。
12. Conclusion: A Data-Driven View of Presidential Effectiveness | 结论：以数据驱动的总统有效性视角
By applying tools from A-Level statistics — proportions tests, time series, correlation, and regression — we find that presidential effectiveness has varied considerably since 1992, with measurable success rates ranging from 49% to 71% for legislative agendas. Economic aims show clearer variation, while hypothesis tests prove many perceived differences are not statistically significant due to small samples and high variability. Ultimately, a mixed quantitative-qualitative evaluation reveals that presidents achieve a majority but not all of their aims, and that statistical analysis provides an invaluable, objective lens for this enduring political question.
通过运用 A-Level 统计学中的工具——比例检验、时间序列、相关和回归——我们发现自 1992 年以来，总统的有效性差异相当大，立法议程的可衡量成功率在 49% 到 71% 之间。经济目标展现出更明显的差异，而假设检验证明，许多感知到的差异由于样本量小和变异性高而不具有统计显著性。最终，定量与定性相结合的评估显示，总统们实现了大部分但并非全部目标，而统计分析为这一永恒的政治问题提供了宝贵且客观的视角。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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