📚 Year 13 CIE Psychology: Quick Reference Handbook of Formulas & Theorems | Year 13 CIE 心理学：公式定理速查手册
For CIE A Level Psychology (9990), success in research methods, biopsychology, and cognitive papers hinges on a firm grasp of the key statistical formulas, psychophysical laws, and influential theoretical models. This concise handbook compiles the essential equations, decision rules, and named principles you are expected to apply, interpret, or evaluate across Paper 2, Paper 3, and Paper 4. Keep it at hand while revising independent measures design, signal detection theory, or the Yerkes–Dodson curve, and use it as a last-minute checker before the exam.
在 CIE A Level 心理学（9990）考试中，研究方法、生物心理学和认知卷的成败往往取决于对关键统计公式、心理物理学定律以及重要理论模型的牢固掌握。这本速查手册汇集了你需要在 Paper 2、Paper 3 和 Paper 4 中应用、解释或评价的核心方程、决策规则和命名原理。无论你正在复习独立组设计、信号检测论，还是耶克斯–多德森曲线，都可以将它放在手边，作为考前最后的检查清单。
1. Measures of Central Tendency: Mean, Median, Mode | 集中趋势量数：平均数、中位数、众数
The mean (M) is the arithmetic average of a data set: M = Σx / N, where Σx is the sum of all scores and N is the number of scores. Use it for interval/ratio data that are normally distributed without extreme outliers.
平均数（M）是一组数据的算术平均值：M = Σx / N，其中 Σx 是所有分数的总和，N 为分数个数。适用于无极端异常值的正态分布等距/等比数据。
The median is the middle score when data are ordered. It is the preferred measure for ordinal data or when the distribution is skewed, as it is unaffected by outliers.
中位数是将数据排序后位于中间的值。当数据为顺序尺度或分布偏态时优先选用，因为中位数不受异常值影响。
The mode is the most frequently occurring value and can be used with nominal data. A distribution may be unimodal, bimodal, or multimodal.
众数是出现频率最高的数值，可用于称名数据。分布可以是单峰、双峰或多峰的。
Mean = Σx ÷ N
2. Measures of Dispersion: Range and Standard Deviation | 离散量数：全距与标准差
The range is the simplest measure: Range = highest score – lowest score. It only considers the two extreme values, so it can be misleading when outliers are present.
全距是最简单的量数：全距 = 最高分 – 最低分。它只考虑两个极值，因此当存在异常值时可能产生误导。
Standard deviation (SD) describes the average distance of each score from the mean. For a sample, the formula is:
标准差（SD）描述每个分数与平均数的平均距离。样本标准差的公式为：
s = √[ Σ(x – M)² / (n – 1) ]
Where x is each individual score, M is the sample mean, and n is the sample size. Dividing by (n – 1) gives an unbiased estimate of the population SD. A larger SD indicates greater spread.
其中 x 为每个个体的分数，M 为样本平均数，n 为样本容量。分母用 (n – 1) 可得到总体标准差的无偏估计。SD 越大，数据的离散程度越高。
3. Correlation: Pearson's r and Spearman's rho | 相关：皮尔逊 r 与斯皮尔曼 rho
Pearson's product-moment correlation coefficient (r) measures the strength and direction of a linear relationship between two interval/ratio variables. It ranges from –1 to +1. The computational formula is:
皮尔逊积差相关系数（r）衡量两个等距/等比变量之间线性关系的强度和方向，取值范围为 –1 至 +1。计算公式为：
r = [ nΣxy – (Σx)(Σy) ] / √[ (nΣx² – (Σx)²)(nΣy² – (Σy)²) ]
Where n is the number of paired scores, x and y are the two variables. A score of +0.8 indicates a strong positive correlation; –0.3 indicates a weak negative correlation.
其中 n 为配对数据的数目，x 和 y 为两个变量。+0.8 表示强正相关，–0.3 表示弱负相关。
Spearman's rank correlation coefficient (rₛ) is used when at least one variable is ordinal or when the relationship is monotonic but not linear. Rank the data, apply the formula:
斯皮尔曼等级相关系数（rₛ）用于至少有一个顺序变量或关系是单调但非线性的情况。对数据排序后，使用公式：
rₛ = 1 – [ 6Σd² / (n(n² – 1)) ]
Where d is the difference between ranks for each pair. Tied ranks require a correction factor; the simplified formula above holds when there are few or no ties.
其中 d 为每对数据的等级差。出现相同等级时需校正因子；在没有或极少相同等级时上述简化公式适用。
4. Inferential Statistics: Mann–Whitney U Test | 推断统计：曼–惠特尼 U 检验
The Mann–Whitney U test is the non-parametric alternative to the independent t-test, used with an independent groups design and ordinal (or interval) data. Calculate U for both groups:
曼–惠特尼 U 检验是独立样本 t 检验的非参数替代方案，用于独立组设计且有顺序（或等距）数据。分别计算两组的 U 值：
U₁ = n₁n₂ + n₁(n₁ + 1)/2 – R₁
U₂ = n₁n₂ – U₁
l value table. If Uobserved ≤ Ucritical, the result is significant (p < 0.05, two-tailed).
其中 n₁、n₂ 为两个样本的容量，R₁ 为第一组的等级和。将较小的 U 值与临界值表比较。若 Uobserved ≤ Ucritical，则结果显著（p < 0.05，双尾检验）。
When reporting, state: U = [value], p < 0.05 (or > 0.05), and the direction of difference if significant.
报告时需写出：U = [数值]，p < 0.05（或 > 0.05），若显著则说明差异方向。
5. Inferential Statistics: Wilcoxon Signed-Ranks Test | 推断统计：威尔科克森符号秩检验
Use the Wilcoxon test for a repeated measures or matched pairs design with ordinal (or interval) data. It is the non-parametric equivalent of the paired t-test. Compute difference scores (D), rank the absolute values of D (ignoring zeros), assign the sign of the original D to each rank, and then sum the positive ranks (T+) and negative ranks (T–).
威尔科克森检验用于重复测量或匹配对设计，数据为顺序（或等距）尺度。它是配对 t 检验的非参数等价。计算差值 D，将 D 的绝对值排序（忽略零），给每个等级赋予原始差值的符号，然后分别求正等级和（T+）与负等级和（T–）。
The test statistic T is the smaller of T+ and T– Compare T with the critical value table: if T ≤ critical value, the result is significant.
检验统计量 T 取 T+ 与 T– 中较小的那个。将 T 与临界值表比较：若 T ≤ 临界值，结果显著。
Report as T = [value], p < 0.05 and note the direction of change (e.g., 'scores increased significantly after the intervention').
报告格式为 T = [数值]，p < 0.05，并注明变化方向（例如“干预后分数显著提高”）。
6. Inferential Statistics: Chi-Squared Test for Association or Goodness of Fit | 推断统计：卡方检验——关联性或拟合优度
The chi-squared (χ²) test is used with nominal (categorical) data in an independent groups design. The general formula is:
卡方（χ²）检验用于称名（类别）数据的独立组设计。通用公式为：
χ² = Σ [ (O – E)² / E ]
E is the expected frequency. For a 2×2 contingency table, degrees of freedom (df) = (rows – 1) × (columns – 1).
其中 O 为观察频数，E 为期望频数。对于 2×2 列联表，自由度（df）=（行数 – 1）×（列数 – 1）。
l value: if χ²calc ≥ χ²crit, reject the null hypothesis. Always check that no more than 20% of expected frequencies are below 5, and none below 1.
将计算得到的 χ² 值与临界值比较：若 χ²calc ≥ χ²crit，则拒绝零假设。需确保不超过 20% 的期望频数低于 5，且无期望频数低于 1。
7. Effect Size: Cohen's d | 效应量：科恩 d
Cohen's d quantifies the size of the difference between two independent group means in standard deviation units. It is essential to report alongside a significant t-test result.
科恩 d 以标准差为单位量化两个独立组平均数之间的差异大小，与显著的 t 检验结果一同报告至关重要。
d = (M₁ – M₂) / Spooled
d standard deviation. A rough guide: d = 0.2 is a small effect, 0.5 medium, and 0.8 large.
其中 M₁ 和 M₂ 为两组平均数，Spooled 为合并标准差。粗略指南：d = 0.2 为小效应，0.5 为中等，0.8 为大效应。
8. Weber's Law | 韦伯定律
Weber's law describes the just noticeable difference (JND) in sensory perception. It states that the change in a stimulus that will be just noticeable is a constant ratio of the original stimulus intensity:
韦伯定律描述感觉知觉中的最小可觉差（JND）。该定律指出，能引起刚好注意到的刺激变化量与原始刺激强度之比是一个常数：
ΔI / I = k
Where ΔI is the increment in intensity required to produce a JND, I is the initial intensity, and k is the Weber fraction that varies across senses (e.g., ~0.08 for light intensity, ~0.02 for sound). For lifted weights, k is about 0.05–0.08.
其中 ΔI 为产生最小可觉差所需的强度增量，I 为初始强度，k 为韦伯分数，不同感觉通道的 k 值不同（例如光强约为 0.08，声音约为 0.02）。对于举重，k 约为 0.05–0.08。
Weber's law holds well for mid-range intensities but breaks down at very low and very high intensities.
韦伯定律在中段强度范围内成立，但在极低和极高强度下失效。
9. Fechner's Law | 费希纳定律
Fechner extended Weber's law to relate physical intensity to perceived sensation. He proposed that sensation magnitude (S) grows as the logarithm of physical intensity (I):
费希纳将韦伯定律延伸至物理强度与知觉感受之间的关系。他提出感觉量（S）随物理强度（I）的对数增长：
S = k · log(I / I₀)
Where k is a constant that depends on the modality, and I₀ is the absolute threshold (the minimum detectable intensity). This logarithmic relationship explains why a doubling of light intensity does not produce a doubling of perceived brightness.
其中 k 为依赖于感觉通道的常数，I₀ 为绝对阈限（最小可觉察强度）。该对数关系解释了为何光强翻倍并不会使主观亮度翻倍。
Although superseded by Stevens' power law in many contexts, Fechner's law remains a cornerstone of psychophysics and is often cited when discussing logarithmic coding in sensory neurons.
尽管在许多情境下已被史蒂文斯幂定律取代，费希纳定律仍是心理物理学的基石，在讨论感觉神经元对数编码时常被引用。
10. Signal Detection Theory: d-prime (d') | 信号检测论：d'（d-prime）
Signal detection theory (SDT) separates an observer's sensitivity from their response bias. Sensitivity is indexed by d' (d-prime), which measures the distance between the noise and signal+noise distributions:
信号检测论（SDT）将观察者的感受性与反应偏向分离开来。感受性指标为 d'（d-prime），衡量噪音分布与信号+噪音分布之间的距离：
d' = z(hit rate) – z(false alarm rate)
e of 0.95 corresponds to a z-score of +1.64; a false alarm rate of 0.10 corresponds to z = –1.28, making d' = 1.64 – (–1.28) = 2.92, indicating high sensitivity.
其中 z 为正态分布函数的逆函数。击中率 0.95 对应的 z 分数为 +1.64；虚报率 0.10 对应的 z 为 –1.28，因此 d' = 1.64 – (–1.28) = 2.92，表示高感受性。
Response bias (criterion C) can be computed as C = –0.5 [ z(hit rate) + z(false alarm rate) ]. A negative C indicates a liberal bias (saying 'yes' more often); a positive C indicates a conservative bias.
反应偏向（判定标准 C）可计算为 C = –0.5 [ z(击中率) + z(虚报率) ]。负 C 表示宽松偏向（更倾向于说“是”）；正 C 表示保守偏向。
11. Yerkes–Dodson Law | 耶克斯–多德森定律
Though not a mathematical formula, the Yerkes–Dodson law is a fundamental principle of arousal and performance. It states that performance increases with physiological or mental arousal to an optimal point, after which further arousal leads to a decline in performance. The relationship follows an inverted-U curve.
耶克斯–多德森定律虽非数学公式，却是关于唤醒与绩效的基本原理。它表明随着生理或心理唤醒水平升高，绩效会提高到某个最优点，之后继续增加的唤醒反而导致绩效下降。该关系呈倒 U 形曲线。
Optimal arousal is task-dependent: complex or difficult tasks require lower arousal for peak performance; simple tasks may benefit from higher arousal. This law is regularly invoked to evaluate studies on anxiety, memory, and eyewitness testimony.
最佳唤醒水平取决于任务：复杂或困难任务需要较低唤醒才能达到最佳绩效；简单任务则可受益于较高唤醒。在评价有关焦虑、记忆和目击证词的研究时，这一定律经常被援引。
12. Long-Term Potentiation: The Hebbian Learning Rule (Summary) | 长时程增强：赫布学习规则（概述）
Often paraphrased as 'neurons that fire together wire together', Donald Hebb's postulate provides a cellular basis for associative learning. While not a single equation in exam answers, the principle that simultaneous activation of pre- and post-synaptic neurons strengthens their synaptic connection underpins LTP and biological explanations of memory.
唐纳德·赫布的信条常被概括为“一起发放的神经元连接在一起”，为联想学习提供了细胞基础。虽然考试答案中不要求单一方程，但突触前与突触后神经元同时激活会增强其突触连接这一原理，支撑了 LTP 和记忆的生物学解释。
In neurobiology, LTP is measured as an increase in excitatory post-synaptic potentials (EPSPs) following high-frequency stimulation. The formula for percent potentiation is:
在神经生物学中，LTP 被测量为高频刺激后兴奋性突触后电位（EPSP）的增强。增强百分比的公式为：
% Potentiation = [(post-stimulus EPSP – baseline EPSP) / baseline EPSP] × 100
This quantitative measure allows researchers to compare the degree of synaptic strengthening across conditions, e.g., in studies of hippocampal slices.
这一量化指标使研究者能够比较不同条件下的突触强化程度，例如在海马脑片研究中。
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