📚 A-Level CIE Psychology: Formula & Theorem Quick Reference Handbook | A-Level CIE 心理学：公式定理速查手册
This handbook provides a concise summary of essential formulas, statistical tests, and key psychological theorems required for the Cambridge International AS & A Level Psychology (9990) syllabus. It is designed as a rapid revision tool, ensuring you can recall and apply the right quantitative method for research methods questions and theoretical explanations. Symbols are presented using standard Unicode notation to mirror examination expectations, and every concept is paired with a clear Chinese translation for bilingual learners.
本手册简明汇总了剑桥国际AS与A Level心理学（9990）教学大纲所要求的基本公式、统计检验和关键心理学定理。它旨在作为快速复习工具，确保你能在研究方法和理论解释题中准确调用适当的量化方法。所有符号均采用标准Unicode表示法，贴近考试习惯，且每个概念均配有清晰的中文翻译，方便双语学习者掌握。
1. Descriptive Statistics: Central Tendency & Dispersion | 描述统计：集中趋势与离散程度
The arithmetic mean is the most common measure of central tendency. For a set of n scores, it is computed by summing all values and dividing by the number of scores. The symbol x̄ denotes the sample mean.
算术平均数是最常用的集中趋势量数。对一组包含n个分数的数据，其计算方式为将所有数值求和后除以数据个数。符号x̄代表样本平均数。
Mean: x̄ = Σx / n
The median is the middle score when data are ordered. Its position is found with (n+1)/2. If n is even, the median is the average of the two middle values. The mode is simply the most frequently occurring score, with no formula needed.
中位数是将数据排序后居中的数值；其位置由 (n+1)/2 确定。若n为偶数，中位数则为中间两个数值的平均数。众数即为出现频次最高的数值，无需公式。
To describe spread, the range is the simplest measure: highest score minus lowest score. A more robust alternative is the interquartile range (IQR), which subtracts the first quartile (Q1) from the third quartile (Q3).
描述离散程度时，全距是最简单的量数：最大值减去最小值。更稳健的替代指标是四分位距（IQR），即第三四分位数减去第一四分位数。
Range = Xmax − Xmin IQR = Q3 − Q1
Variance and standard deviation quantify how scores deviate from the mean. For a sample, the unbiased estimator uses n−1 in the denominator.
方差与标准差量化了分数相对于平均数的偏离程度。对样本而言，无偏估计量在分母中使用 n−1。
Sample variance: s² = Σ(x − x̄)² / (n − 1)
Sample standard deviation: s = √[Σ(x − x̄)² / (n − 1)]
In examinations, you may be asked to compute these manually or interpret output. Remember that standard deviation retains the original units, making it particularly useful when comparing variability across datasets.
考试中可能会要求你手动计算这些值或解读输出结果。请记住，标准差保留了原始单位，因此在比较不同数据集的变异性时格外有用。
2. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s rho (ρ) measures the strength and direction of association between two ranked variables. It is a non‑parametric test suitable for ordinal data or interval data that violate normality. The test ranks each set of scores, finds the difference d between each pair of ranks, and applies the formula below.
斯皮尔曼等级相关系数ρ用于测量两个等级变量之间关联的强度与方向。它是一种非参数检验，适用于顺序数据或违背正态性的等距数据。该方法对两组分数分别排序，计算每对等级之差d，并代入下述公式。
ρ = 1 − (6 Σd²) / (n(n² − 1))
Here, d is the difference between a participant’s two ranks, and n is the number of pairs. The coefficient ranges from −1 (perfect negative correlation) through 0 (no correlation) to +1 (perfect positive correlation).
其中d为一位被试的两个等级之差，n为配对数量。相关系数范围从−1（完全负相关）、0（无相关）到+1（完全正相关）。
You must compare the calculated ρ to a critical value from the Spearman table at your chosen significance level (usually 0.05). If ρ is greater than or equal to the critical value, the correlation is statistically significant.
你须将计算所得的ρ值与斯皮尔曼临界值表中选定显著性水平（通常为0.05）下的临界值进行比较。若ρ大于或等于临界值，则相关性具有统计显著性。
A significant positive Spearman’s rho indicates that as one variable rises, so does the other; a significant negative value shows an inverse relationship. Always check that data meet the assumptions: pairs of scores from the same participants and a monotonic relationship.
显著的斯皮尔曼正相关表明一个变量上升时另一个也上升；显著的负值则表明反向关系。务必检查数据满足假设：成对分数来自同一批被试，且呈单调关系。
3. Chi-Square Test for Independence | 卡方独立性检验
The chi‑square (χ²) test determines whether there is a significant association between two categorical variables. It compares observed frequencies (O) with expected frequencies (E) under the null hypothesis of no association.
卡方（χ²）检验用于判断两个分类变量之间是否存在显著关联。它将观察频次（O）与零假设（无关联）下的期望频次（E）进行比较。
χ² = Σ((O − E)² / E)
Expected frequencies are calculated for each cell of a contingency table: E = (row total × column total) / grand total. The degrees of freedom (df) equal (number of rows − 1) × (number of columns − 1).
期望频次针对列联表每个单元格计算：E =（行合计×列合计）/ 总计数值。自由度df = (行数−1)×(列数−1)。
After computing χ², consult a chi‑square distribution table with the correct df and chosen α level (commonly 0.05). If the calculated χ² exceeds the critical value, reject the null hypothesis – there is a significant association.
计算出χ²后，查阅相应自由度和所选α水平（通常为0.05）的卡方分布表。若计算值大于临界值，则拒绝零假设——即存在显著关联。
Key assumptions: data must be frequencies (not percentages), each observation should be independent, and expected frequencies should not be too small (typically at least 5 in each cell). The test is widely used in CIE scenarios, such as surveying gender differences in opinion.
关键假设：数据必须是频次（非百分比），每次观察应独立，且期望频次不应过小（通常每个单元格至少为5）。该检验在CIE情境中应用广泛，例如调查观点上的性别差异。
4. Mann-Whitney U Test | 曼‑惠特尼U检验
The Mann‑Whitney U test is a non‑parametric alternative to the independent samples t‑test. It compares two independent groups to determine whether their distributions differ. All scores are ranked together from 1 to n, and the sum of ranks for the smaller group is often denoted R₁.
曼‑惠特尼U检验是独立样本t检验的非参数替代方法。它比较两个独立组，以判断其分布是否不同。所有分数合并后从1到n统一排序，较小群体的秩和通常记作R₁。
U = n₁n₂ + n₁(n₁ + 1) / 2 − R₁
Here n₁ and n₂ are the sample sizes of the two groups, and R₁ is the sum of ranks for the group with n₁ participants. The other U value (U’) can be found via n₁n₂ − U, and the smaller of the two is used for comparison with the critical value.
其中n₁和n₂为两组的样本量，R₁为样本量为n₁的那个组的秩和。另一个U值（U’）可通过n₁n₂−U得到，取两者中较小者与临界值比较。
If the observed U (or smaller U) is less than or equal to the critical value from the Mann‑Whitney table, the difference between groups is statistically significant. Remember, this test applies to ordinal or interval data that are not normally distributed, using an independent groups design.
若观察到的U（较小的U）小于或等于曼‑惠特尼临界值表中的值，则组间差异具有统计显著性。谨记，该检验适用于非正态分布的顺序或等距数据，且采用独立组设计。
5. Wilcoxon Signed-Rank Test | 威尔科克森符号秩检验
The Wilcoxon signed‑rank test is a non‑parametric counterpart to the paired samples t‑test. It is used for repeated measures or matched pairs designs. The test subtracts one condition score from the other for each participant, records the sign of the difference, ranks the absolute differences, and then sums the ranks of the less frequent sign – this sum is the test statistic T.
威尔科克森符号秩检验是配对样本t检验的非参数版本，适用于重复测量或配对设计。该检验逐名被试计算两种条件下的分数差，记录差值的符号，对绝对差值排序，然后将出现较少的符号的秩求和——此和即为检验统计量T。
T = Σ ranks of the less frequent sign
Ignore any differences of zero – they reduce the effective sample size. The smaller the T value, the more likely the difference is significant. Compare T to the critical value from the Wilcoxon table for the relevant n and α.
忽略任何差值为零的数据——这会使有效样本量减少。T值越小，差异越可能显著。将T与对应n和α的威尔科克森临界值表比较。
If T is less than or equal to the critical value, reject the null hypothesis. This test is common in CIE questions that involve before‑and‑after designs or twin studies, as it requires only that the differences are ordinal and symmetrically distributed.
若T小于或等于临界值，则拒绝零假设。该检验在涉及前后测设计或双胞胎研究的CIE试题中十分常见，因为它仅要求差值具有顺序性并呈对称分布。
6. Signal Detection Theory | 信号检测论
Signal Detection Theory (SDT) provides a framework for measuring decision‑making under uncertainty. It separates sensitivity (d’) from response bias (criterion). The key formula expresses d’ as the difference between the z‑scores of the hit rate (H) and the false alarm rate (FA).
信号检测论提供了一个在不确定条件下测量决策的框架，它将敏感度（d’）与反应偏向（标准）区分开来。其核心公式将d’表示为击中率（H）与虚报率（FA）的z分数之差。
d’ = z(H) − z(FA)
A higher d’ indicates better sensitivity – the participant can more easily distinguish signal from noise. Hit rate is the proportion of ‘yes’ responses when a signal is present, while false alarm rate is the proportion of ‘yes’ responses when only noise is present.
d’值越高，表明敏感度越好——被试能更容易区分信号与噪音。击中率是当信号出现时回答“是”的比例，而虚报率是仅存在噪音时回答“是”的比例。
SDT is often used in memory or perception studies. Although you may not calculate z‑scores manually in the exam, understanding d’ helps explain why some findings are due to sensitivity shifts rather than simple response biases.
信号检测论常用于记忆或知觉研究。尽管考试中你可能无需手动计算z分数，但理解d’有助于解释为何某些结果源于敏感度的改变而非单纯的反应偏向。
7. Weber’s Law | 韦伯定律
Weber’s Law quantifies the just noticeable difference (JND) – the smallest change in a stimulus that can be detected. The law states that the JND (ΔI) is a constant proportion of the original stimulus intensity (I).
韦伯定律量化了最小可觉差（JND）——即能被察觉到的最小刺激变化量。该定律指出，最小可觉差ΔI与原刺激强度I呈恒定的比例关系。
ΔI / I = k
Here k is the Weber fraction, which varies across sensory modalities. For example, the Weber fraction for lifted weights is approximately 0.02, meaning a 100 g weight must be increased by around 2 g to be noticed as heavier.
此处k为韦伯分数，因感觉通道不同而异。例如，举重感觉的韦伯分数约为0.02，意味着一个100克的重量大约需要增加2克才能被察觉为更重。
Weber’s Law helps explain why we notice small sound changes in quiet environments but require larger changes in noisy settings. It underpins the concept of relative sensitivity and is a precursor to Fechner’s logarithmic scale. In CIE, you may be asked to describe the law and apply its formula to predict perception thresholds.
韦伯定律有助于解释为何安静环境中我们能注意到微小声音变化，而在嘈杂环境中则需要较大的变化。它奠定了相对敏感度的概念，也是费希纳对数定律的先导。在CIE考试中，你可能需要描述该定律并用其公式预测知觉阈限。
8. Yerkes‑Dodson Law | 耶克斯‑多德森定律
The Yerkes‑Dodson Law describes the empirical relationship between arousal and performance. It is typically depicted as an inverted U‑shaped curve. Performance increases with moderate arousal up to an optimal point, after which further arousal leads to a decline.
耶克斯‑多德森定律描述了唤醒水平与表现之间的经验关系，通常呈现为倒U形曲线。随着唤醒水平升高，表现最初上升，直至某一最佳点，之后进一步升高的唤醒则导致表现下降。
No single formula captures the law, but it can be stated as: Optimal performance = f(moderate arousal). The location of the optimal point shift with task difficulty – simpler or well‑learned tasks tolerate higher arousal, while complex or novel tasks require lower arousal for peak performance.
该定律没有单一公式，但可表述为：最佳表现 = f(中等唤醒)。最佳点的位置随任务难度而移动——简单或熟练掌握的任务能耐受较高唤醒，而复杂或新颖的任务则需较低唤醒才能达到最佳表现。
This principle is widely applied in sports psychology and exam preparation. Understanding it helps explain why a little anxiety can boost test performance but too much can cause choking. CIE questions often require you to draw and interpret the inverted‑U and discuss implications for real‑world behaviour.
这一原则广泛应用于运动心理学和备考情境。理解它有助于解释为何适度焦虑能提升考试成绩，但过度焦虑会导致发挥失常。CIE试题常要求你绘制并解读倒U形曲线，并讨论其对现实行为的意义。
9. Statistical Significance, p‑value & Decision Errors | 统计显著性、p值与决策错误
Statistical significance tells us whether an observed result is likely due to chance. The p‑value is the probability of obtaining the observed test statistic (or one more extreme), assuming the null hypothesis is true. In psychology, the common significance level (α) is 0.05.
统计显著性告诉我们所观察到的结果是否很可能源于偶然。p值是在零假设为真时，获得等于或超过当前检验统计量的概率。心理学中常用的显著性水平α为0.05。
If p ≤ α, reject H₀ (result is significant).
When making decisions based on p‑values, two types of errors can occur. A Type I error (false positive) occurs when the null hypothesis is rejected even though it is true. The probability of a Type I error equals α.
基于p值做决策时，可能出现两类错误。I类错误（假阳性）是指零假设为真却被拒绝的情况，其发生概率等于α。
A Type II error (false negative) happens when the null hypothesis is not rejected despite being false. Its probability is denoted β. Power, which is 1 − β, reflects the test’s ability to detect a real effect.
II类错误（假阴性）则指零假设为假却未被拒绝的情况，其概率记作β。统计检验力等于1−β，反映检验检测真实效应的能力。
Increasing sample size can reduce the risk of both error types and improve power. In CIE answer scripts, always state the chosen α, report whether p < 0.05, and link significance to the research hypothesis without overclaiming causation.
增加样本量可以降低两类错误的风险，并提高检验力。在CIE答卷中，务必说明选定的α水平，报告p是否小于0.05，并将显著性与研究假设相联系，但避免过度声称因果关系。
10. Choosing the Right Inferential Test | 正确选择推断性检验
Selecting an appropriate statistical test is a core CIE skill. The choice depends on the experimental design, the level of measurement, and whether parametric assumptions are met. The table below summarizes the most common tests you will encounter.
选择合适的统计检验是CIE的核心技能之一。选择取决于实验设计、测量层次以及是否满足参数假设。下表汇总了你将遇到的最常用的检验。
| Design / Data Type | Non‑parametric Test | Parametric Alternative |
|---|---|---|
| Independent groups (ordinal or non‑normal) | Mann‑Whitney U | Independent t‑test |
| Repeated /
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