📚 Cambridge Year 12 Psychology: Formula & Theorem Quick Reference Guide | 剑桥Year 12心理学：公式定理速查手册
This quick reference guide summarises the essential formulas, statistical tests, and psychological laws required for the Cambridge International AS Level Psychology (9990). Understanding when and how to apply these equations is vital for Research Methods questions and for interpreting data in core studies. The handbook covers measures of central tendency, dispersion, correlation, inferential tests, as well as classic psychophysical and behavioural laws.
本速查手册总结了剑桥国际AS阶段心理学（9990）所必需的基本公式、统计检验和心理定律。掌握何时以及如何应用这些方程式对于研究方法题目以及解释核心研究中的数据至关重要。本手册涵盖集中趋势测量、离散程度、相关性、推断检验，以及经典的心理物理学和行为定律。
1. Measures of Central Tendency | 集中趋势测量
The mean (M) is the arithmetic average of a set of scores. It is calculated by summing all values and dividing by the number of values.
均值（M）是一组分数的算术平均值，计算方法是将所有数值相加再除以数值的个数。
Mean = ΣX / N
where ΣX is the sum of all scores and N is the total number of scores. The median is the middle score when data are arranged in ascending order; for an even number of scores it is the average of the two central values. The mode is the most frequently occurring score.
其中 ΣX 为所有分数之和，N 为分数的总个数。中位数是将数据按升序排列后位于中间的数值；若个数为偶数，则取中间两个值的平均数。众数是出现频率最高的分数。
2. Measures of Dispersion (Variability) | 离散程度（变异性）测量
The range is the simplest measure of spread, computed as the difference between the maximum and minimum scores.
极差是最简单的离散指标，即最大值与最小值之差。
Range = Xmax – Xmin
Variance and standard deviation describe how far scores deviate from the mean. For sample data, the variance (s²) uses n–1 in the denominator to provide an unbiased estimate of the population variance.
方差和标准差描述分数偏离均值的程度。对于样本数据，方差（s²）的分母为 n–1，以提供总体方差的无偏估计。
Variance (s²) = Σ(X – X̄)² / (n – 1)
Standard Deviation (s) = √[ Σ(X – X̄)² / (n – 1) ]
Here X̄ is the sample mean, X represents each individual score, and n is the sample size. Always report the mean alongside the standard deviation when summarising data.
其中 X̄ 是样本均值，X 代表每个个体分数，n 为样本容量。总结数据时，应同时报告均值和标准差。
3. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s rho (rₛ) measures the strength and direction of association between two ranked (ordinal) variables. It is a non‑parametric test suitable when data do not meet the assumptions of Pearson’s r.
斯皮尔曼等级相关系数（rₛ）测量两个定序变量之间关联的强度和方向。它是一种非参数检验，适用于数据不满足皮尔逊积差相关假设的情形。
rₛ = 1 – [ 6 ΣD² / (n (n² – 1)) ]
Where D is the difference between the ranks of each pair, and n is the number of paired observations. After computing rₛ, compare it with the critical value from Spearman’s table for the given sample size. If rₛ is greater than or equal to the critical value, reject the null hypothesis of no correlation.
其中 D 是每对数据的等级差，n 是成对观察值的数量。计算 rₛ 后，将其与给定样本量下斯皮尔曼表的临界值进行比较。若 rₛ 大于或等于临界值，则拒绝无相关的零假设。
4. Chi‑Square Test (χ²) | 卡方检验
The chi‑square test is used with categorical (nominal) data to determine whether there is a significant difference between observed and expected frequencies. Two common forms are the goodness‑of‑fit test and the test of association (independence).
卡方检验用于类别（称名）数据，检验观测频数与期望频数之间是否存在显著差异。常见形式有拟合优度检验和独立性（关联性）检验。
χ² = Σ [ (O – E)² / E ]
O is the observed frequency, E is the expected frequency. Expected frequencies are calculated as:
O 为观测频数，E 为期望频数。期望频数的计算方式如下：
E = (row total × column total) / grand total
For a goodness‑of‑fit test, degrees of freedom (df) = number of categories – 1. For a test of independence, df = (number of rows – 1) × (number of columns – 1). Compare the calculated χ² with the critical value from the chi‑square distribution table at the chosen significance level.
对于拟合优度检验，自由度（df）= 类别数 – 1。对于独立性检验，自由度 = (行数 – 1) × (列数 – 1)。将计算出的 χ² 值与所选显著性水平下的卡方分布表临界值进行比较。
5. 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 the central tendency of two unrelated groups using ordinal (ranked) data.
曼‑惠特尼U检验是独立样本t检验的非参数替代方法，使用定序（等级）数据比较两个独立组的集中趋势。
U₁ = n₁n₂ + [ n₁(n₁+1) / 2 ] – R₁
U₂ = n₁n₂ + [ n₂(n₂+1) / 2 ] – R₂
n₁ and n₂ are the sample sizes of the two groups; R₁ and R₂ are the sums of the ranks for group 1 and group 2 respectively, after all scores from both groups are combined and ranked together. The test statistic U is the smaller of U₁ and U₂, compared against critical values in the Mann‑Whitney U table.
n₁ 和 n₂ 是两组的样本量；R₁ 和 R₂ 分别是将两组所有分数合并排序后，第一组和第二组的等级和。检验统计量 U 取 U₁ 和 U₂ 中的较小值，并与曼‑惠特尼U检验表中的临界值进行比较。
6. Wilcoxon Signed‑Rank Test | 威尔科克森符号秩检验
This test is a non‑parametric equivalent to the paired‑samples t‑test. It evaluates whether there is a significant difference between two related conditions or matched pairs using ordinal data.
该检验是配对样本t检验的非参数版本，使用定序数据评估两个相关条件或配对组之间是否存在显著差异。
Steps: (1) compute the difference score (D) for each pair; (2) ignore zero differences and rank the absolute values of the differences from 1 to n (smallest to largest); (3) sum the ranks of positive differences (T₊) and the ranks of negative differences (T₋); (4) the test statistic T is the smaller of T₊ and T₋.
步骤：(1) 计算每对数据的差值 (D)；(2) 忽略差值为零的配对，将差值的绝对值从1到n进行排序（由小到大）；(3) 分别求正差值的等级和 (T₊) 与负差值的等级和 (T₋)；(4) 检验统计量 T 取 T₊ 与 T₋ 中的较小值。
Compare T with the critical value for N (number of non‑zero difference pairs). If T is equal to or less than the critical value, the result is significant.
将 T 与 N（非零差值对的数目）对应的临界值比较。若 T 等于或小于临界值，则结果显著。
7. Sign Test | 符号检验
The sign test is the simplest non‑parametric test for a repeated measures or matched‑pairs design. It only considers the direction of the difference (+, –, or tie) rather than the magnitude of the difference.
符号检验是用于重复测量或配对设计的最简单非参数检验，它仅考虑差值的符号（+、–或平局），而不考虑差值的大小。
Procedure: for each pair, record whether the score in condition B is greater than (+) or less than (–) the score in condition A. Ties are omitted. Count the number of pluses and minuses. The test statistic S is the number of occurrences of the less frequent sign. Compare S with the critical value from the binomial distribution table for N non‑tied pairs.
步骤：对每对数据，记录条件B的分数是否大于(+)或小于(–)条件A的分数，平局忽略。计算正号和负号的数量。检验统计量 S 取出现次数较少的符号的数目。将 S 与 N（非平局配对数量）对应的二项分布临界值表进行比较。
8. Weber–Fechner Laws (Psychophysics) | 韦伯‑费希纳定律（心理物理学）
Weber’s Law states that the size of the just noticeable difference (JND) between two stimuli is a constant proportion of the original stimulus intensity. It highlights the relative nature of sensory discrimination.
韦伯定律指出，两个刺激之间的最小可觉差（JND）与原始刺激强度成正比。它揭示了感觉分辨的相对性质。
ΔI / I = k
ΔI is the increment in intensity needed to produce a JND, I is the initial stimulus intensity, and k is the Weber fraction (constant for a given sensory modality). Fechner extended this by proposing a logarithmic relationship between physical intensity and perceived sensation:
ΔI 是产生最小可觉差所需的强度增量，I 是初始刺激强度，k 是韦伯分数（对特定感觉通道为常数）。费希纳在此基础上提出物理强度与心理感觉之间存在对数关系：
S = k log I
S is the perceived sensation magnitude. These laws help explain why, for example, you can easily detect a small added weight to a light object but not to a heavy one.
S 是感觉量的大小。这些定律可以解释为什么例如在轻物上增加一点重量很容易察觉，而在重物上则不易察觉。
9. Yerkes‑Dodson Law | 耶克斯‑多德森定律
The Yerkes‑Dodson law describes the relationship between arousal and performance as an inverted U‑shaped curve. Performance improves with moderate arousal but declines when arousal becomes too high or too low.
耶克斯‑多德森定律将唤醒水平与表现之间的关系描述为一条倒U形曲线。中等程度的唤醒使表现最佳，而唤醒过高或过低均会使表现下降。
The optimal level of arousal also depends on task complexity: simple or well‑learned tasks benefit from higher arousal, whereas complex or novel tasks require lower arousal for peak performance. This principle is widely referenced in explanations of anxiety, motivation, and memory.
最佳唤醒水平还取决于任务复杂度：简单或熟练的任务在较高唤醒下表现更佳，而复杂或新颖的任务则需较低唤醒才能达到最佳表现。这一原理在解释焦虑、动机和记忆时被广泛引用。
10. Signal Detection Theory (Key Metrics) | 信号检测论（关键指标）
Signal detection theory (SDT) provides a framework for separating sensitivity from response bias when a person must decide whether a stimulus (signal) is present or absent amidst noise.
信号检测论（SDT）提供了一个将感受性与反应偏差分开考察的框架，应用于个体在噪音背景中判断刺激（信号）是否存在的情形。
Sensitivity (d′) measures how easily the signal can be discriminated from noise. It is defined as the difference between the z‑scores of the hit rate (H) and the false alarm rate (F).
感受性指标 d′ 衡量将信号从噪音中区分出来的能力，定义为击中率 (H) 与虚报率 (F) 的 Z 分数之差。
d′ = z(H) – z(F)
A higher d′ indicates better discriminability. Response bias (criterion c) reflects the observer’s general tendency to respond ‘yes’ or ‘no’, independent of stimulus strength.
d′ 越高，分辨能力越强。反应偏向（判断标准 c）反映观察者应答“是”或“否”的总体倾向，与刺激强度无关。
c = –½ [ z(H) + z(F) ]
In these formulas, z(H) and z(F) are the z‑scores corresponding to the proportion of hits and false alarms, obtained from the standard normal distribution table. SDT is used in studies of attention, memory recognition, and clinical decision‑making.
在这些公式中，z(H) 和 z(F) 分别是根据标准正态分布表查出的击中率和虚报率所对应的 Z 值。信号检测论常被应用于注意、记忆再认以及临床决策等研究领域。
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