📚 Pre-U Cambridge Psychology: Key Formulas & Theorems Quick Handbook | Pre-U 剑桥心理学：核心公式定理速查手册
This quick-reference handbook brings together the quantitative laws, statistical formulas, and fundamental theorems that Pre-U Cambridge Psychology students must master. From the earliest psychophysical discoveries to the inferential tests used in modern research methods, each entry is presented with clear notation, worked meanings, and exam-focused commentary. Use this guide alongside your notes and past papers to secure the numerical and conceptual reasoning required at Pre-U level.
本手册汇总 Pre-U 剑桥心理学学生必须掌握的核心定量定律、统计公式与基础定理。从最早的心理物理学发现到现代研究方法中的推论检验，每个条目都以清晰符号、释义和应考要点呈现。请将本手册与你的笔记和历年真题配合使用，巩固 Pre-U 阶段所需的数值推导与概念推理能力。
1. Weber’s Law | 韦伯定律
Weber’s Law states that the just noticeable difference (JND) between two stimuli is a constant proportion of the original stimulus intensity. It is expressed as:
韦伯定律指出，两个刺激之间的最小可觉差（JND）与原始刺激强度的比例是一个常数。其表达式为：
ΔI / I = k
where ΔI is the increment in stimulus intensity required to produce the JND, I is the baseline intensity, and k is the Weber fraction. For example, if the Weber fraction for lifted weights is 0.02, a person holding a 100 g weight would just notice a 2 g increment. This law holds well in the mid-range of sensory magnitudes but often breaks down at extremely low or high intensities. In the Pre-U paper, you may need to calculate the JND, identify k from a data set, or evaluate the law’s limitations across different sensory modalities.
其中 ΔI 代表产生最小可觉差所需的刺激强度增量，I 为基础强度，k 为韦伯分数。例如，若提重实验中韦伯分数为 0.02，手持 100 g 重物的人恰好能觉察到 2 g 的重量变化。该定律在感觉强度的中间范围适用性较好，但在极低或极高强度时常会失效。在 Pre-U 考试中，你可能需要计算 JND、从数据集中识别 k 值，或评价该定律在不同感觉通道中的局限性。
2. Fechner’s Law | 费希纳定律
Fechner built on Weber’s work to propose a logarithmic relationship between physical stimulus intensity and the perceived sensation magnitude. Fechner’s Law is written as:
费希纳在韦伯研究的基础上提出，物理刺激强度与感觉量之间存在对数关系。费希纳定律的表达式为：
S = k log I
Here S stands for the subjective sensation, I is the physical intensity, and k is a modality‑specific constant. The law implies that equal ratios of physical change produce equal increments in sensation. This principle explains why a candle in a dark room appears dramatically brighter, whereas adding the same candle to an already bright hall barely alters perceived brightness. You should be ready to interpret graphs of S against log I and to contrast Fechner’s approach with Stevens’ direct scaling methods.
式中 S 代表主观感觉量，I 为物理强度，k 为取决于感觉通道的常数。该定律意味着，物理变化的相等比率会产生相等的感觉增量。这一原理可以解释，为什么在黑暗房间中点燃一支蜡烛会显得格外明亮，而在本就明亮的厅堂中增加同一支蜡烛几乎不会改变主观亮度。你应当做好解读 S – log I 关系图，以及对比费希纳思路与史蒂文斯直接量表法的准备。
3. Stevens’ Power Law | 史蒂文斯幂定律
Stevens argued that sensation magnitude relates to stimulus intensity via a power function, not a logarithmic one:
史蒂文斯主张感觉量与刺激强度之间呈幂函数关系，而非对数关系：
S = k Iⁿ
where n is an exponent specific to the sensory modality. For brightness, n is typically less than 1 (about 0.33 for a small spot of light), meaning sensation grows more slowly than physical intensity; for electric shock, n can be greater than 1 (around 3.5), indicating a rapid escalation of perceived intensity. The Pre-U syllabus expects you to understand how magnitude estimation experiments support this law and how the exponent changes with the nature of the stimulus.
其中 n 取决于感觉通道的指数。对于亮度，n 通常小于 1（小光点约 0.33），意味着感觉增长速度慢于物理强度；对于电击，n 可能大于 1（约 3.5），反映主观强度的迅速攀升。Pre-U 大纲要求你理解数量估计实验如何支持该定律，以及指数如何随刺激性质而变化。
4. Signal Detection Theory: Sensitivity and Bias | 信号检测论：敏感性与反应偏向
Signal detection theory (SDT) separates an observer’s sensory sensitivity from their decision criterion. The key sensitivity measure is d’ (d-prime):
信号检测论（SDT）将观察者的感觉敏感性与判断标准分离开来。关键的敏感性指标是 d’（d-prime）：
d’ = z(Hit) – z(False Alarm)
where z(Hit) is the z-score of the hit rate and z(False Alarm) is the z-score of the false alarm rate. A higher d’ means the signal and noise distributions are better separated. The decision criterion β (beta) can be expressed as the ratio of the likelihood of signal-plus-noise to noise alone at the decision point. A liberal criterion (β < 1) increases hits but also false alarms; a conservative criterion (β > 1) reduces false alarms at the cost of misses. In part (b) exam questions you might be asked to compute d’ from a confusion matrix or to explain how criterion shifts affect detection in eyewitness testimony or medical screening.
式中 z(Hit) 为击中率的 z 分数，z(False Alarm) 为虚报率的 z 分数。d’ 越大，表明信号与噪声分布的分离程度越好。判断标准 β（beta）可表示为判断点处信号加噪声的似然性与噪声似然性之比。宽松标准（β < 1）既提高击中率，也增加虚报率；保守标准（β > 1）则以漏报为代价降低虚报。在第 (b) 部分考题中，你可能需要根据混淆矩阵计算 d’，或解释标准变化如何在目击证词或医学筛查中影响检测表现。
5. Rescorla-Wagner Model | 雷斯克拉-瓦格纳模型
The Rescorla-Wagner model describes classical conditioning as a process where learning is driven by prediction error. The change in associative strength (ΔV) on a trial is given by:
雷斯克拉-瓦格纳模型将经典条件作用描述为由预测误差驱动的学习过程。一次试验中联想强度的变化量（ΔV）由下式表示：
ΔV = αβ (λ – ∑V)
where α is the salience of the conditioned stimulus, β is the learning rate parameter linked to the unconditioned stimulus, λ is the maximum associative strength the US can support, and ∑V is the sum of current associative strengths of all cues present. The term (λ – ∑V) is the prediction error. For example, if a bell (CS) has already acquired some strength, a subsequent compound of bell and light would show weaker conditioning to the light because the total V is already close to λ, illustrating the blocking effect. This model features in explanations of phobia acquisition and extinction and requires you to apply the equation to novel scenarios.
式中 α 为条件刺激的显著性，β 为与非条件刺激相关联的学习率参数，λ 为非条件刺激所能支持的最大联想强度，∑V 为当前所有呈现线索的联想强度总和。(λ – ∑V) 即为预测误差。例如，若铃声（CS）已获得一定强度，随后的“铃声 + 灯光”复合刺激对灯光的条件化就会较弱，因为此时总 V 已接近 λ，这正是“阻断效应”。该模型在恐惧症习得与消退的解释中出现，并需要你将方程式应用于新情境。
6. Yerkes-Dodson Law | 耶克斯-多德森定律
The Yerkes-Dodson Law states that performance increases with physiological or mental arousal up to an optimal point, after which further increases in arousal cause a decline in performance. The relationship follows an inverted-U curve and can be described by the heuristic equation:
耶克斯-多德森定律指出，表现随生理或心理唤醒水平提高而上升，直至一个最佳点，此后唤醒的进一步升高反而导致表现下降。该关系遵循倒 U 形曲线，并可用如下启发式方程描述：
Performance = E₀ – b (Arousal – Aopt)²
where E₀ is the maximum performance achievable at the optimal arousal level Aopt, and b is a positive constant reflecting task complexity. For simple or well‑learned tasks, Aopt is higher; for complex or novel tasks, Aopt is lower. In examination essays, you might be asked to draw and label the curve, explain why high anxiety harms exam performance, or discuss the law’s relevance to sports and eyewitness recall.
其中 E₀ 为最佳唤醒水平 Aopt 时能达到的最高表现，b 为反映任务复杂性的正常数。对于简单或熟练任务，Aopt 较高；对于复杂或新任务，Aopt 较低。在论文题中，你可能需要绘制并标注曲线，解释为什么高焦虑会损害考试表现，或讨论该定律在体育竞赛和目击回忆中的意义。
7. Descriptive Statistics: Mean, Variance, and Standard Deviation | 描述统计：均值、方差与标准差
In any empirical report, you must be able to calculate and interpret measures of central tendency and dispersion.
在任何实证报告中，你都必须会计算并能解释集中趋势与离散程度的指标。
| Statistic | Formula (Unicode) | Meaning |
| Mean (x̄) | x̄ = (∑x) / n | Arithmetic average of all scores. |
| Variance (s²) | s² = ∑(x – x̄)² / (n – 1) | Average squared deviation from the mean; uses n – 1 for a sample. |
| Standard Deviation (s) | s = √[∑(x – x̄)² / (n – 1)] | Square root of variance, in the original units. |
Using n – 1 gives an unbiased estimate of the population variance. Always check whether your data come from a population or a sample; Pre-U papers usually supply sample data. Be careful with units and round appropriately in calculations.
使用 n – 1 能够得到总体方差的无偏估计。务必确认数据来自总体还是样本；Pre-U 试卷通常提供样本数据。计算时需注意单位并合理取整。
8. Standard Error and the t-Test | 标准误与 t 检验
The standard error of the mean (SE) measures how much sample means scatter around the population mean:
平均数的标准误（SE）衡量样本均值在总体均值周围的散布程度：
SE = s / √n
For an independent-measures t-test, the test statistic is:
对于独立测量 t 检验，检验统计量为：
t = (x̄₁ – x̄₂) / SEpooled
where SEpooled = √[sp²(1/n₁ + 1/n₂)] and sp² is the pooled variance. The degrees of freedom (df) equal n₁ + n₂ – 2. You compare the obtained t with the critical t from a table at your chosen significance level (usually 0.05). A significant t (> critical) indicates a low probability that the observed difference occurred by sampling error, allowing you to reject the null hypothesis. Pre-U questions often require you to complete a partially filled t-table, state whether results are significant, and write a conclusion in the context of the study.
其中 SEpooled = √[sp²(1/n₁ + 1/n₂)]，sp² 为合并方差。自由度 (df) 等于 n₁ + n₂ – 2。你需要将计算得到的 t 值与在选定显著性水平（通常为 0.05）下查表所得的临界 t 进行比较。若 t 显著（大于临界值），表明观察到的差异由抽样误差造成的概率很低，从而可拒绝虚无假设。Pre-U 试题常要求你完善一个不完整的 t 表格，说明结果是否显著，并在研究背景下写出结论。
9. Chi-Square Test | 卡方检验
The chi-square (χ²) test evaluates whether observed frequencies differ from expected frequencies under the null hypothesis of no association. The basic formula is:
卡方 (χ²) 检验用于评估在“无关联”的虚无假设下，观察频次是否与期望频次存在差异。其基本公式为：
χ² = ∑ [(O – E)² / E]
where O is the observed frequency and E is the expected frequency for each cell. For a contingency table, E is calculated as (row total × column total) / grand total. Degrees of freedom = (number of rows – 1) × (number of columns – 1). The obtained χ² is compared with the critical χ² from the distribution table. A significant χ² indicates that the variables are related beyond what would be expected by chance. In part (b) questions you may be given a table and asked to compute χ² step by step, or to discuss the test’s suitability over a t-test when data are nominal.
式中 O 为各单元格的观察频次，E 为期望频次。对于列联表，E 通过 (行合计 × 列合计) / 总计 计算。自由度 = (行数 – 1) × (列数 – 1)。将得到的 χ² 值与分布表中的临界 χ² 进行比较。显著的 χ² 表明变量间存在超出随机预期的关联。在第 (b) 部分中，你可能会得到一个表格并要求逐步计算 χ²，或讨论在名义数据条件下该检验相较 t 检验的适切性。
10. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
When data are ordinal or do not meet parametric assumptions, Spearman’s rho (ρ) measures the strength and direction of association between two ranked variables:
当数据为定序或不符合参数检验假设时，斯皮尔曼等级相关系数 (ρ) 可测量两个排序变量之间的关联强度与方向：
ρ = 1 – (6 ∑ d²) / [n(n² – 1)]
where d is the difference between the ranks of each paired observation and n is the number of pairs. ρ ranges from –1 (perfect negative correlation) to +1 (perfect positive correlation), with 0 indicating no association. To use this formula, you first rank each set of scores separately, then compute d for each pair, square those differences, and sum them. Pre-U tasks typically involve a small dataset; you might also be asked to interpret the obtained ρ in terms of effect size or to suggest a follow‑up test if a significant correlation implies causality.
式中 d 为每对观察值等级之差，n 为配对数量。ρ 取值范围从 –1（完全负相关）到 +1（完全正相关），0 表示无关联。使用该公式时，你需要先分别对每组分数排秩，然后计算每对数据的等级差 d，平方后求和。Pre-U 任务通常给出较小的数据集；你还可能被要求从效应量的角度解释所得的 ρ，或当显著相关暗示因果关系时建议后续检验。
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