📚 Pre-U WJEC Psychology: Formula & Theorem Quick Reference Handbook | Pre-U WJEC 心理学：公式定理速查手册
This handbook is designed as a compact reference for Pre-U WJEC Psychology students, bringing together the essential statistical formulas, psychophysical laws, and key theoretical equations required across research methods and core topic areas. Each entry is presented with clear notation, a concise explanation, and a paired Chinese translation, making it ideal for rapid revision and in-class application.
本手册专为 Pre-U WJEC 心理学学生设计，是一本浓缩的速查工具，汇集了研究方法与核心专题领域所必备的描述统计、推断检验及重要心理学定律的公式。每个条目均配有清晰的符号标记、简明解释及配对的中文翻译，非常适合快速复习与课堂应用。
1. Descriptive Statistics: Mean, Median, Mode | 描述统计：平均数、中位数、众数
The arithmetic mean for a set of scores X is calculated as the sum of all scores divided by the number of scores, n. It represents the central point of a distribution but is sensitive to extreme values.
算术平均数是将所有分数 X 的总和除以数据个数 n。它表示分布的中心点，但对极端值较为敏感。
M or X̄ = ΣX / n
The median is the middle value when scores are arranged in numerical order. For an odd number of scores, it is the central score; for an even number, it is the mean of the two middle scores. The mode is simply the most frequently occurring score in a dataset.
中位数是将数据按大小排序后居于中间位置的数值。当数据个数为奇数时，中位数为正中间的数值；当数据个数为偶数时，则为中间两个数值的平均数。众数则是数据集中出现次数最多的数值。
2. Variance and Standard Deviation | 方差与标准差
Variance (s²) measures the average squared deviation of each score from the mean, while standard deviation (s) is its square root and restores the original units. For a sample, we use n−1 in the denominator to provide an unbiased estimate of the population variance.
方差（s²）衡量的是每个数值与平均数之差的平方的平均数，而标准差（s）是方差的平方根，恢复了原始单位。对于样本数据，分母使用 n−1 以得到总体方差的无偏估计。
s² = Σ(X − X̄)² / (n − 1)
s = √[Σ(X − X̄)² / (n − 1)]
A smaller standard deviation indicates that scores cluster closely around the mean, whereas a larger one signals greater spread. Understanding variability is crucial when comparing groups in experimental designs.
较小的标准差表明分数紧密围绕在平均数周围，而较大的标准差则意味着数据分布更广。在实验设计中比较不同组别时，理解变异性至关重要。
3. Z-Scores and the Normal Distribution | Z 分数与正态分布
A z-score expresses how many standard deviations a raw score lies above or below the mean. It is fundamental for standardising and comparing scores from different distributions.
Z 分数表示一个原始分数距离平均数有多少个标准差。它对来自不同分布的分数进行标准化和比较起着基础性作用。
z = (X − μ) / σ
In a standard normal distribution, approximately 68% of values fall within ±1σ, 95% within ±1.96σ, and 99.7% within ±3σ. Psychologists use z-tables to determine the probability of obtaining a particular score under the null hypothesis.
在标准正态分布中，约 68% 的数值落在 ±1σ 内，95% 落在 ±1.96σ 内，99.7% 落在 ±3σ 内。心理学家使用 Z 表来确定在原假设下获得某个特定分数的概率。
4. Independent Samples t-test | 独立样本 t 检验
The independent t-test evaluates whether the means of two unrelated groups differ significantly. It pools variance estimates to calculate a t statistic under the assumption of homogeneity of variance.
独立样本 t 检验用于评估两个无关组别的平均数是否存在显著差异。它在假定方差齐性的前提下，合并方差估计值来计算 t 统计量。
t = (X̄₁ − X̄₂) / √[sₚ²(1/n₁ + 1/n₂)]
Here, sₚ² is the pooled variance, computed by weighting each group’s variance by its degrees of freedom. The obtained t-value is compared against critical values with df = n₁ + n₂ − 2. If the absolute t exceeds the critical value, the null hypothesis is rejected.
其中 sₚ² 为合并方差，通过各组的自由度对其方差进行加权计算得出。将得到的 t 值与自由度为 n₁ + n₂ − 2 的临界值进行比较。若 t 的绝对值大于临界值，则拒绝原假设。
5. Chi-Square Test for Independence | 卡方独立性检验
The chi-square (χ²) test examines whether there is a significant association between two categorical variables. Observed frequencies are compared with expected frequencies derived from the marginal totals.
卡方（χ²）检验用于考察两个类别变量之间是否存在显著关联。将观察频数与由边际总和得出的期望频数进行比较。
χ² = Σ[(O − E)² / E]
For each cell in a contingency table, O is the observed count and E is the expected count. The degrees of freedom are (rows − 1) × (columns − 1). A statistically significant χ² suggests that the variables are not independent, but it does not indicate the strength of the relationship – Cramér’s V or phi coefficient can be used for that purpose.
对于列联表中的每个单元格，O 为观察频数，E 为期望频数。自由度为（行数 − 1）×（列数 − 1）。统计上显著的 χ² 表明变量间并非独立，但并不能指示关系的强弱——为此可以使用 Cramér’s V 或 φ 系数。
6. Pearson’s r and Spearman’s Rho | 皮尔逊 r 与斯皮尔曼 ρ
Pearson’s product-moment correlation coefficient (r) measures the strength and direction of a linear relationship between two continuous variables. It ranges from −1 (perfect negative) to +1 (perfect positive), with 0 indicating no linear correlation.
皮尔逊积差相关系数（r）用于衡量两个连续变量之间线性关系的强度和方向。其取值范围从 −1（完全负相关）到 +1（完全正相关），0 表示不存在线性相关。
r = Σ[(X − X̄)(Y − Ȳ)] / √[Σ(X − X̄)² Σ(Y − Ȳ)²]
Spearman’s rank correlation coefficient (ρ or rₛ) is used when data are ordinal or when assumptions of normality are violated. It operates on ranked data rather than raw scores.
斯皮尔曼等级相关系数（ρ 或 rₛ）适用于顺序数据或当正态性假设被违反时。它基于数据的排序等级而非原始分数进行计算。
ρ = 1 − (6Σd²) / [n(n² − 1)]
Here, d is the difference between the ranks of each pair, and n is the number of paired observations. Both coefficients require cautious interpretation: correlation does not imply causation, and outliers can distort Pearson’s r considerably.
此处 d 为每对数据排序等级之差，n 为配对观测值的数量。两种系数都需谨慎解读：相关关系并不意味因果关系，且异常值可能严重扭曲皮尔逊 r。
7. Mann-Whitney U Test | 曼-惠特尼 U 检验
The Mann-Whitney U test is a non-parametric alternative to the independent t-test. It compares the ranks of two independent samples when the assumptions of normality or equal variance are not met.
曼-惠特尼 U 检验是独立样本 t 检验的非参数替代方法。当数据不满足正态性或方差齐性假设时，它通过比较两个独立样本的排序等级来进行检验。
U₁ = n₁n₂ + n₁(n₁ + 1)/2 − R₁
U₂ = n₁n₂ − U₁
R₁ denotes the sum of ranks for group 1. The smaller U value is used as the test statistic. For larger sample sizes (typically n > 20), the sampling distribution of U approximates normality, and a z-score can be computed to assess significance.
R₁ 表示组 1 的等级总和。以较小的 U 值作为检验统计量。对于较大的样本量（通常 n > 20），U 的抽样分布近似正态，此时可计算 Z 分数来判断显著性。
8. Weber’s Law and Fechner’s Law | 韦伯定律与费希纳定律
Weber’s Law states that the just noticeable difference (JND) between two stimuli is proportional to the magnitude of the standard stimulus. The constant of proportionality, k, is known as the Weber fraction.
韦伯定律指出，两个刺激之间可被觉察的最小差异（最小可觉差）与标准刺激的强度成正比。该比例常数 k 称为韦伯分数。
ΔI / I = k
Fechner built upon Weber’s work by assuming that each JND produces an equal increment in sensation. This yields a logarithmic relationship between perceived intensity S and physical intensity I.
费希纳以韦伯的研究为基础，假设每一个最小可觉差产生相等的感觉增量。由此导出了知觉强度 S 与物理强度 I 之间的对数关系。
S = k log I
Fechner’s law implies that as stimulus intensity grows, increasingly larger physical changes are required to produce a constant perceived change. This underpins early psychophysical scaling and remains relevant in sensory psychology today.
费希纳定律意味着随着刺激强度增加，需要越来越大的物理变化才能引起恒定的知觉变化。这奠定了早期心理物理量表的理论基础，在当今的感觉心理学中仍具有重要价值。
9. Stevens’ Power Law | 斯蒂文斯幂定律
Stevens proposed that the relationship between stimulus intensity and perceived magnitude is better described by a power function. The exponent n varies across sensory modalities and determines whether perception compresses or expands the physical range.
斯蒂文斯提出，刺激强度与知觉大小之间的关系更适合用幂函数来描述。指数 n 因感觉通道而异，并决定了知觉对物理范围的压缩或扩展程度。
S = k Iⁿ
For example, the exponent for brightness is approximately 0.33, indicating response compression, while for electric shock it is around 3.5, showing extreme expansion. This law is central to the techniques of magnitude estimation used in psychophysics.
例如，亮度的指数大约为 0.33，表明反应压缩；而电击的指数约为 3.5，表现出极强的扩展性。该定律在心理物理学中使用的数量估计技术中居于核心地位。
10. Rescorla-Wagner Model of Classical Conditioning | 经典条件反射的 Rescorla-Wagner 模型
The Rescorla-Wagner model explains how the associative strength between a conditioned stimulus (CS) and an unconditioned stimulus (US) develops across trials. Central to the model is the idea that learning is driven by prediction error.
Rescorla-Wagner 模型解释了条件刺激（CS）与非条件刺激（US）之间的联结强度如何在试验过程中发展。该模型的核心思想是，学习由预测误差所驱动。
ΔV = αβ(λ − ΣV)
ΔV is the change in associative strength on a given trial; α is the salience of the CS; β is the learning rate determined by US magnitude; λ represents the maximum associative strength that the US can support; and ΣV is the total associative strength of all CSs present. When (λ − ΣV) is large, learning is rapid; when it approaches zero, learning asymptotes.
ΔV 是某次试验中联结强度的变化量；α 为 CS 的显著性；β 是由 US 强度决定的学习率；λ 表示 US 所能产生的最大联结强度；ΣV 则是所有在场 CS 的联结强度总和。当 (λ − ΣV) 较大时，学习速度很快；当其接近于零时，学习趋于渐近线。
This model elegantly accounts for blocking, overshadowing, and conditioned inhibition, making it one of the most influential quantitative theories of associative learning.
该模型巧妙地解释了阻断效应、遮掩效应以及条件性抑制，因而成为最具影响力的联结学习量化理论之一。
11. Hull’s Drive Theory Formula | 赫尔驱力理论公式
Clark Hull’s drive theory proposes that the tendency to make a response, or excitatory potential (sEᵣ), is a multiplicative function of habit strength (sHᵣ), drive (D), and incentive motivation (K).
克拉克·赫尔的驱力理论认为，作出某种反应的趋势或称兴奋势能（sEᵣ）是习惯强度（sHᵣ）、驱力（D）与诱因动机（K）的乘积函数。
sEᵣ = sHᵣ × D × K
Habit strength (sHᵣ) builds through reinforced pairings of stimulus and response; drive (D) refers to internal states such as hunger or thirst that energise behaviour; and incentive motivation (K) captures the anticipated reward magnitude. If any factor is zero, the excitatory potential collapses, theoretically precluding behaviour.
习惯强度（sHᵣ）通过刺激与反应之间的强化配对而建立；驱力（D）指饥饿或口渴等为行为提供能量的内部状态；诱因动机（K）则反映了对奖赏大小的预期。若任一因子为零，兴奋势能将消失，从理论上排除行为的产生。
This formula was later refined by Spence to account for incentive learning, leading to the Hull-Spence model frequently cited in theoretical accounts of motivation and avoidance learning.
这一公式后来由斯彭斯加以完善以解释诱因学习，形成了在动机与回避学习理论解释中常被引用的 Hull-Spence 模型。
12. Signal Detection Theory: d′ and Criterion | 信号检测论：d′ 与判断标准
Signal detection theory (SDT) separates an observer’s sensitivity from their response bias. The sensitivity index d′ (d-prime) quantifies the distance between the signal and noise distributions in standard deviation units.
信号检测论（SDT）将观察者的感受性与反应偏向区分开来。感受性指标 d′（d-prime）量化了信号分布与噪音分布之间以标准差为单位的距离。
d′ = z(Hit) − z(FA)
Here, z(Hit) is the z-score corresponding to the proportion of hits (correctly detecting a signal) and z(FA) is the z-score for false alarms (responding ‘yes’ when only noise is present). A higher d′ indicates better sensitivity, independent of the participant’s willingness to say ‘yes’.
此处 z(Hit) 是与击中率（正确检测到信号）对应的 Z 分数，z(FA) 是虚报率（仅在噪音出现时回答“是”）的 Z 分数。d′ 越高表示感受性越好，这与被试说“是”的意愿无关。
The response criterion c, or alternatively β, indicates the threshold at which the observer decides a signal is present. A conservative criterion reduces both hits and false alarms, assessed by:
反应标准 c（或另一种指标 β）标示了观察者判断信号是否出现的临界点。较为保守的标准会同时降低击中与虚报，可通过以下公式进行评估：
c = −½[z(Hit) + z(FA)]
SDT is widely applied in memory research, medical diagnosis, and eyewitness testimony evaluation, providing a robust framework for disentangling accuracy from bias.
信号检测论广泛应用于记忆研究、医学诊断及目击证人证词评估，为分离准确性与偏向提供了强有力的框架。
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