📚 AS Cambridge Psychology: Formula & Theorem Quick Reference | AS 剑桥心理学：公式定理速查手册
This quick reference guide consolidates the essential formulas, statistical tests and key theorems you need for the Cambridge AS Psychology (9990) examination. It covers the entire research methods toolkit alongside threshold laws and cognitive principles that appear across biological, cognitive and learning approaches. Use this handbook to check your calculations, understand when to apply each test and reinforce the laws that underpin psychological phenomena.
这本速查手册汇集了剑桥 AS 心理学 (9990) 考试所需的关键公式、统计检验和重要定理。内容覆盖整个研究方法工具箱，以及出现在生物心理学、认知心理学和学习心理学中的阈限定律和认知原则。你可以借助本手册核对计算、理解每种检验的适用条件，并巩固支撑心理现象的定律。
1. Measures of Central Tendency | 集中趋势的度量
Central tendency tells you where the bulk of your data lies. The three main measures are the mean (arithmetic average), the median (middle value in an ordered list) and the mode (most frequent value). In AS Psychology, you will often calculate the mean for interval/ratio data and the median or mode when data are ordinal or skewed.
集中趋势反映数据的主体位置。三种主要度量是平均数（算术平均值）、中位数（有序序列的中间值）和众数（出现频率最高的值）。在 AS 心理学中，常对等距/比率数据计算平均数，而对顺序数据或存在偏态时则用中位数或众数。
The mean is sensitive to extreme scores; the median resists outliers; the mode is the only choice for truly categorical data. Reporting all three can give a fuller picture of your data distribution.
平均数对极端值敏感；中位数不受离群值影响；众数则是分类数据的唯一选择。同时报告这三个统计量能更全面地展示数据分布。
2. Measures of Dispersion | 离散趋势的度量
Dispersion describes the spread of scores. The range (maximum – minimum) gives a quick but crude measure. The interquartile range (IQR = Q3 – Q1) trims the top and bottom 25 %, resisting outliers. These are essential starting points before choosing an inferential test.
离散趋势描述分数的散布情况。全距（最大值 – 最小值）提供快速但粗略的度量。四分位距（IQR = Q3 – Q1）切掉顶端和底端各 25%，能抵抗离群值。这些是选择推断检验之前的基础步骤。
Always inspect dispersion before interpreting a mean. A small IQR suggests tight agreement, whereas a huge range signals variability that might hide important subgroups.
在解释平均数之前一定要检查离散程度。较小的 IQR 表明数据集中，而很大的全距则暗示变异性高，可能掩盖重要的子群体。
3. Standard Deviation Formula | 标准差公式
Standard deviation (SD) quantifies the average distance of each data point from the mean. In Cambridge AS, you may encounter both population and sample versions. The population SD divides by N; the sample SD (used for estimation) divides by (n − 1).
标准差（SD）量化每个数据点与平均数的平均距离。剑桥 AS 考试中可能同时出现总体和样本版本。总体标准差除以 N；用于估计的样本标准差除以 (n − 1)。
Population SD formula:
σ = √[ Σ(X − μ)² / N ]
总体标准差公式：
σ = √[ Σ(X − μ)² / N ]
Sample SD formula (the one most commonly used in inferential work):
s = √[ Σ(X − X̄)² / (n − 1) ]
样本标准差公式（推断工作中最常用）：
s = √[ Σ(X − X̄)² / (n − 1) ]
A larger SD means greater variability. When you compute SD for a data set, link it back to the shape of the distribution and the likelihood of extreme values.
SD 越大表示变异性越高。计算完数据集的 SD 后，要将其与分布形状和出现极端值的可能性联系起来解释。
4. 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 a monotonic relationship between two ordinal (or ranked interval) variables. It is a non-parametric alternative to Pearson’s r and is examined heavily in Paper 2 research methods.
斯皮尔曼等级相关系数（rs）衡量两个顺序变量（或排序后的等距变量）之间单调关系的强度和方向。它是皮尔逊 r 的非参数替代方法，在卷二研究方法部分考查频繁。
The formula:
rs = 1 − (6 Σ d²) / [n (n² − 1)]
其中 d = each paired rank difference，n = number of pairs.
公式：
rs = 1 − (6 Σ d²) / [n (n² − 1)]
其中 d = 每对秩次差值，n = 配对数目。
rs ranges from −1 (perfect negative) to +1 (perfect positive). You must compare your obtained rs against a critical value table at p ≤ 0.05 with degrees of freedom = n. A significant correlation implies that as one variable increases, the other tends to change in the same direction (positive) or opposite direction (negative).
rs 取值从 −1（完全负相关）到 +1（完全正相关）。必须将计算得到的 rs 与临界值表比较（p ≤ 0.05，自由度 = n）。显著的相关意味着当一变量增大时，另一变量倾向于同方向（正相关）或反方向（负相关）变化。
5. Chi-Square Test | 卡方检验
The chi-square test of association (χ² test) is used on independent groups design with nominal (category) data. It checks whether the observed frequencies in categories differ significantly from those expected by chance. The test of independence is especially common in social and cognitive studies.
独立性卡方检验（χ² 检验）适用于独立组设计且数据为称名量表（类别数据）的情境。它判断各类别的观察频次是否与随机期望频次显著不同。独立性检验在社会心理学和认知研究中尤为常见。
The formula:
χ² = Σ [ (O − E)² / E ]
Where O = observed frequency, E = expected frequency (row total × column total / grand total).
公式：
χ² = Σ [ (O − E)² / E ]
其中 O = 观察频次，E = 期望频次（行总计 × 列总计 / 总样本量）。
Degrees of freedom: (rows − 1) × (columns − 1). When any expected frequency is below 5, a Yates’ correction may be required. Compare calculated χ² against critical χ² values; if calculated ≥ critical, reject the null hypothesis of no association.
自由度 = (行数 − 1) × (列数 − 1)。当任一期望频次低于 5 时，可能需要使用耶茨校正。将计算得到的 χ² 与临界 χ² 比较；若计算值 ≥ 临界值，拒绝无关联的零假设。
6. Mann-Whitney U Test | 曼-惠特尼 U 检验
The Mann-Whitney U test is a non-parametric test for an independent groups (unrelated) design when the dependent variable yields at least ordinal data. It asks whether the ranks from one condition are significantly higher or lower than those from the other.
曼-惠特尼 U 检验是一种用于独立组（无关）设计的非参数检验，适用于因变量至少产生顺序数据的情境。它检验一种条件下的秩次是否显著高于或低于另一种条件。
The formulas (two U values are calculated and the smaller is taken):
U1 = n1n2 + n1(n1+1)/2 − R1
U2 = n1n2 + n2(n2+1)/2 − R2
Where R1 is the sum of ranks for group 1, R2 for group 2. U = min(U1, U2).
公式（计算两个 U 值，取较小的那个）：
U1 = n1n2 + n1(n1+1)/2 − R1
U2 = n1n2 + n2(n2+1)/2 − R2
其中 R1 为组 1 的秩和，R2 为组 2 的秩和。U 取 U1 和 U2 中较小者。
Compare U to the critical value (one-tailed or two-tailed). If obtained U is less than or equal to the critical value, the difference is significant. Always report sample sizes and explain what the rank sums suggest about direction.
将 U 与临界值（单尾或双尾）比较。若得到的 U ≤ 临界值，差异显著。报告时需写明样本量并解释秩和大小暗示的方向性。
7. Wilcoxon Signed-Rank Test | 威尔科克森符号秩检验
The Wilcoxon signed-rank test is the related-design (e.g., repeated measures or matched pairs) counterpart of the Mann-Whitney U. It works with ordinal data from pairs of scores and tests whether the median difference score is significantly different from zero.
威尔科克森符号秩检验是与曼-惠特尼 U 对应的相关设计（如重复测量或配对设计）检验。它处理配对分数的顺序数据，检验差值的中位数是否显著偏离零。
The formula for T (the test statistic) is the smaller of the sums of positive ranks or negative ranks:
T = smaller of Σ R+ and Σ R−
(where ranks of absolute differences are summed separately for positive and negative directions)
T（检验统计量）是正秩和与负秩和中较小的那个：
T = Σ R+ 与 Σ R− 中较小者
（其中绝对差值的秩按正负方向分别求和）
T is compared against a critical T table for n pairs (ignore pairs where difference = 0). A small T (≤ critical value) indicates a significant difference. Explain the direction by noting which sign has the larger sum of ranks.
将 T 与 n 对配对（忽略差值为 0 的配对）对应的临界 T 表比较。T 越小（≤ 临界值），差异越显著。通过比较正负秩和的大小来解释方向。
8. Weber’s Law | 韦伯定律
Weber’s Law underpins the concept of the just-noticeable difference (jnd) in psychophysics. It states that the increase in intensity needed to detect a change is a constant proportion of the original stimulus intensity.
韦伯定律奠定了心理物理学中最小可觉差 (jnd) 的概念。它指出，为察觉变化所需的刺激强度增量是原强度的恒定比例。
Weber’s fraction:
ΔI / I = k
Where ΔI = increment needed for jnd, I = original intensity, k = Weber constant.
韦伯分数：
ΔI / I = k
其中 ΔI = 产生 jnd 所需的增量，I = 原始强度，k = 韦伯常数。
Different sensory modalities have different Weber constants (e.g., brightness ~0.08, lifted weights ~0.02). Weber’s law breaks down at very low and very high intensities, but it framed early experimental psychology by showing a lawful relationship between physical and psychological intensity.
不同感觉通道有不同的韦伯常数（如亮度约 0.08，提升重物约 0.02）。韦伯定律在极低和极高强度下失效，但它通过揭示物理强度与心理强度之间的规律性关系，奠定了早期实验心理学的基础。
9. Fechner’s Law | 费希纳定律
Gustav Fechner built on Weber’s work to propose a logarithmic relationship between physical intensity and perceived sensation. Fechner’s law describes how sensation magnitude (S) grows as the logarithm of the stimulus intensity (I).
古斯塔夫·费希纳在韦伯工作的基础上提出物理强度与感知感觉之间存在对数关系。费希纳定律描述感觉大小 (S) 随刺激强度 (I) 的对数增长。
Fechner’s equation:
S = k log I
Where k is a modality-specific constant.
费希纳方程：
S = k log I
其中 k 是特定感觉通道的常数。
The implication is that equal ratios of physical increase produce equal arithmetic steps in sensation. This is historically vital for linking psychophysics with introspection-based measurement. In the AS context, both Weber’s and Fechner’s laws appear in the biological approach when discussing sensory thresholds and the nature of perception.
这表示物理强度按等比增加时，感觉按等差增加。这一原理在历史上对于将心理物理学与内省测量联系起来至关重要。在 AS 情境中，韦伯定律和费希纳定律都出现在生物心理学视角中，用于讨论感觉阈限和知觉本质。
10. Signal Detection Theory | 信号检测论
Signal Detection Theory (SDT) separates sensitivity (d’) from response bias (β or c) when an observer must decide whether a faint signal is present amid noise. The key formula for sensitivity is the distance between the means of the noise and signal-plus-noise distributions.
信号检测论将感受性 (d’) 与反应偏向 (β 或 c) 分离，适用于观察者需在噪声中判断微弱信号是否出现的情境。感受性的关键公式是噪声分布与信号加噪声分布均值之间的距离。
Sensitivity index:
d’ = z(Hit) − z(False Alarm)
Where z refers to the inverse of the normal distribution (z-score). A larger d’ indicates better discriminability. Response bias is measured by the criterion location; a liberal criterion yields many hits and many false alarms, while a conservative one reduces both.
感受性指数：
d’ = z(击中率) − z(虚报率)
其中 z 指正态分布的反函数（z 分数）。d’ 越大，分辨力越强。反应偏向由判定标准的位置衡量；宽松标准导致高击中率也伴有高虚报率，保守标准则两者都低。
SDT is frequently cited in studies of attention, memory recognition and eyewitness testimony. The AS syllabus expects you to explain how these two independent processes influence decision-making, not simply the calculation.
信号检测论常被引用于注意、记忆再认和目击证词的研究。AS 考纲要求你解释这两个独立过程如何影响决策，而不仅仅是计算。
11. Miller’s Law (Magic Number 7±2) | 米勒定律（神奇数字 7±2）
George Miller’s classic 1956 paper “The Magical Number Seven, Plus or Minus Two” proposed that the capacity of short-term memory (STM) for the immediate recall of information is approximately 7 ± 2 items or chunks.
乔治·米勒 1956 年的经典论文“神奇数字 7±2”提出，短时记忆 (STM) 对即时回忆的信息容量大约为 7 ± 2 个项目或组块。
This is a descriptive law rather than a mathematical formula, but it has driven decades of cognitive research. Chunking – grouping individual bits into meaningful units – can expand functional capacity. Miller’s law is central to the multi-store model of memory and is used to explain serial position effects and limits of attention.
这是一条描述性定律而非数学公式，却推动了数十年的认知研究。组块化——将单个信息联合为有意义的单元——可以扩展功能容量。米勒定律是记忆多贮存模型的核心，也用于解释序列位置效应和注意力的局限。
12. All-or-None Law (Neuronal) | 全或无定律（神经元）
The all-or-none law states that a neurone either fires an action potential at full strength or does not fire at all. Once the threshold of excitation is reached, the resulting spike is of constant amplitude and does not vary with stimulus strength.
全或无定律指出，神经元要么以全幅发放动作电位，要么完全不发放。一旦达到兴奋阈限，所产生的锋电位幅度恒定，不随刺激强度变化。
This is a fundamental principle in the biological approach. Information about stimulus intensity is coded by the frequency of firing, not the size of the action potential. The law links directly to synaptic transmission and research on neurotransmitters examined in AS Psychology.
这是生物心理学视角中的一条基本原则。关于刺激强度的信息由发放频率编码，而非动作电位的大小。这一定律直接与突触传递及 AS 心理学中考查的神经递质研究相联系。
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