📚 Year 12 Edexcel Psychology: Formula & Theorem Quick Reference Guide | Year 12 Edexcel 心理学公式定理速查手册
Welcome to your quick-reference companion for the Edexcel AS Psychology specification. This guide brings together the key statistical formulas, decision rules and classical psychophysical laws you will need throughout Year 12. It covers descriptive statistics, inferential tests, probability theory and the handful of elegant mathematical relationships that underpin sensory psychology. Use this as your go-to revision sheet when tackling research methods questions or interpreting data in the exam.
欢迎使用这本为 Edexcel AS 心理学考试大纲量身定制的速查手册。本手册汇集了 Year 12 课程中必备的核心统计公式、决策准则和经典心理物理学定律，涵盖描述统计、推论检验、概率理论以及支撑感觉心理学的几个优美数学关系。在复习研究方法或解读考试数据时，它就是你的随身锦囊。
1. Descriptive Statistics: Measures of Central Tendency & Dispersion | 描述统计：集中趋势与离散量数
Three measures summarise the centre of a data set, while two measures capture its spread. For a sample of size N and raw scores X, the arithmetic mean is the sum of all scores divided by N. The median is the middle score when data are ordered; the mode is the most frequent score. The range is simply the difference between the highest and lowest values, whereas the standard deviation reflects how far scores deviate from the mean on average. The formulae below use sample notation (n-1) for an unbiased estimate.
三种量数概括数据集的中心，两种量数捕捉其离散程度。对于样本容量 N 和原始分数 X，算术平均数为所有分数之和除以 N。中位数是排序后居中的那个分数；众数是出现频率最高的分数。全距直接是最大值与最小值之差，而标准差反映分数平均偏离均值的程度。以下公式采用样本符号 (n-1) 以求得无偏估计。
Mean (Mean): M = ΣX / N
Variance (样本方差): s² = Σ (X – M)² / (n – 1)
Standard Deviation (标准差): s = √ [ Σ (X – M)² / (n – 1) ]
The variance and standard deviation are squared and square-root expressions of the same averaged squared deviation. Psychologists report the standard deviation because it is in the original units of measurement.
方差和标准差是基于平均离差平方的平方与开方表达。心理学家报告标准差，因为它与原始测量单位一致。
2. Probability, p-Values and Decision Errors | 概率、p 值与决策错误
In null-hypothesis significance testing (NHST), the p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. The conventional alpha level in psychology is 0.05. If p < 0.05, the result is declared statistically significant and the null hypothesis is rejected. However, this decision carries two possible errors: a Type I error (false positive) occurs when a true null is wrongly rejected; a Type II error (false negative) occurs when a false null is not rejected. Power (1 - beta) is the probability of correctly rejecting a false null.
在虚无假设显著性检验 (NHST) 中，p 值是在虚无假设为真的前提下，得到至少与观测值一样极端的检验统计量的概率。心理学的惯例 alpha 水平为 0.05。若 p < 0.05，结果被宣布为统计显著，虚无假设被拒绝。然而，这一决策伴随两种可能的错误：I 型错误（假阳性）指错误地拒绝了真实的虚无假设；II 型错误（假阴性）指未能拒绝错误的虚无假设。统计功效 (1 - β) 是正确拒绝错误虚无假设的概率。
Students must be able to state whether p < 0.05 or p > 0.05 when given a critical value table and an observed test statistic. Remember: a non-significant result does not prove the null; it simply means there is insufficient evidence against it.
学生必须学会在给定临界值表和观测检验统计量时，判断 p < 0.05 还是 p > 0.05。请记住：不显著的结果并不能证明虚无假设为真，只是意味着反对它的证据不足。
3. The Normal Distribution and Its Key Properties | 正态分布及其关键性质
Many psychological variables are roughly normally distributed in the population. The normal curve is bell-shaped, symmetric, and unimodal. In a perfect normal distribution, the mean, median and mode are identical. The distribution is fully described by its mean (mu) and standard deviation (sigma). A beautiful empirical rule states that approximately 68% of values lie within 1 standard deviation of the mean, 95% within 2 standard deviations and 99.7% within 3 standard deviations. These percentages are critical when assessing how unusual an observation is.
许多心理变量在总体中近似呈正态分布。正态曲线呈钟形，对称且单峰。在完美的正态分布中，平均数、中位数和众数完全重合。分布由其均值 (μ) 和标准差 (σ) 完整描述。一条优美的经验规则指出：约 68% 的值落在均值 ±1 个标准差内，95% 落在 ±2 个标准差内，99.7% 落在 ±3 个标准差内。这些百分比在评估观测值的异常程度时至关重要。
In Edexcel AS, you may be asked to interpret data against the normal distribution but are not required to calculate z-scores unless given the formula. However, the concept of standardising scores often appears in the context of deviation and percentile ranks.
在 Edexcel AS 考试中，你可能需要对照正态分布解读数据，但不一定需要计算 z 分数（除非提供公式）。不过，分数标准化的概念经常在离差和百分等级的情境中出现。
4. Chi-Square Test: Formulas for Association and Goodness of Fit | 卡方检验：关联性与拟合优度公式
The chi-square test works with frequency (count) data. The test statistic compares observed frequencies (O) with expected frequencies (E) under the null hypothesis. The basic formula is the same for both the test of association (independence) and the goodness-of-fit test:
卡方检验处理频数（计数）数据。检验统计量将虚无假设下的观察频数 (O) 与期望频数 (E) 进行比较。独立性检验和拟合优度检验共用相同的基本公式：
χ² = Σ [ (O – E)² / E ]
For a test of association on a contingency table, the expected frequency for each cell is calculated as (row total × column total) ÷ grand total. The degrees of freedom (df) = (number of rows – 1) × (number of columns – 1). For a goodness-of-fit test with a single categorical variable, the expected values are normally based on chance or a theoretical ratio, and df = number of categories – 1. The larger the chi-square value relative to the critical value, the more likely it is to be significant.
对于列联表的独立性检验，每个单元格的期望频数 = (行总和 × 列总和) ÷ 总频数。自由度 (df) = (行数 – 1) × (列数 – 1)。对于单一分类变量的拟合优度检验，期望值通常基于概率或理论比率，df = 类别数 – 1。卡方值相对于临界值越大，越可能显著。
5. Sign Test: A Simple Test for Repeated-Measures Data | 符号检验：适用于重复测量数据的简易检验
The sign test is the only non-parametric test used in AS psychology that analyses nominal data derived from a repeated-measures or matched-pairs design. It simply records the direction of change between two conditions: a ‘+’ for increase, a ‘-‘ for decrease, and ties are discarded. The test statistic S is the count of the less frequent sign. You compare this S value to the critical values in the sign test table for the number of participants (N, excluding ties). If S is equal to or smaller than the critical value, the result is significant at that level.
符号检验是 AS 心理学中唯一分析名义数据的非参数检验，用于重复测量或配对设计。它仅记录两种条件间的变化方向：增加记为 ‘+’，减少记为 ‘-‘，平局剔除不计。检验统计量 S 为出现次数较少的符号的计数。你将此 S 值与符号检验表中对应于参与者数 (N，剔除平局后) 的临界值进行比较。若 S 等于或小于临界值，则在该水平上显著。
No complex formula is needed beyond S = number of signs of the less common sign. The logic is probabilistic: under the null hypothesis both signs are equally likely, so a large imbalance suggests a real difference.
除了 S = 出现较少符号的个数外，这里无需复杂公式。其逻辑基于概率：虚无假设下两种符号出现概率相等，因此严重的失衡暗示存在真实差异。
6. Mann-Whitney U Test: Comparing Two Independent Groups | 曼-惠特尼 U 检验：比较两个独立组
When an experiment uses an independent-groups design and the data are at least ordinal, the Mann-Whitney U test can be used instead of the sign test. It ranks all the scores from both groups together, ignoring group membership. The formulas for the U statistic are:
当实验采用独立组设计且数据至少为顺序水平时，可选用曼-惠特尼 U 检验代替符号检验。它将两组所有分数混合后统一排序，忽略分组。U 统计量的公式为：
U₁ = n₁n₂ + [n₁(n₁ + 1) / 2] – R₁
U₂ = n₁n₂ – U₁
U = the smaller of U₁ and U₂
Here, n₁ and n₂ are the sample sizes of the two groups, and R₁ is the sum of the ranks for group 1. The obtained U is compared against the critical values for the Mann-Whitney table at the appropriate n₁, n₂ and alpha level. A significant result means the two groups have different central tendencies.
此处，n₁ 与 n₂ 为两组的样本容量，R₁ 为第一组的秩和。将计算所得的 U 与曼-惠特尼表格中对应 n₁、n₂ 和 alpha 水平的临界值比较。小U值若等于或小于临界值，表示显著，意味着两组集中趋势不同。
7. Wilcoxon Signed-Rank Test: For Pairs of Related Observations | 威尔科克森符号秩检验：适用于成对相关观测
The Wilcoxon signed-rank test is the ordinal counterpart to the sign test for repeated-measures or matched-pairs designs, using the magnitude of differences as well as their direction. After calculating the difference score for each pair, ignore zero differences, rank the absolute differences from smallest to largest (assigning average ranks for ties), and then attach the original sign to each rank. The test statistic T (or W) is the smaller of the sum of positive ranks and the sum of negative ranks.
威尔科克森符号秩检验是符号检验的顺序水平对应版，适用于重复测量或配对设计，既利用变化方向也利用变化幅度。计算每对数据的差值后，忽略零差异，将差值的绝对值从小到大排秩（遇平局分配平均秩），再给每个秩附上原来的正负号。检验统计量 T (或 W) 取正秩和与负秩和中较小的那一个。
No single formula is written here because the computation follows a procedure rather than a plug-in equation, but the test statistic is simply the smaller rank sum. Compare it with the critical Wilcoxon T values for the relevant N (number of non-zero pairs). If the observed T is equal to or less than the critical value, the difference is significant.
此处无需给出单一公式，因为计算遵循一套步骤而非代入方程，但检验统计量就是较小的那个秩和。将其与相应 N（非零对数）的 Wilcoxon T 临界值比较。若观测 T 等于或小于临界值，差异显著。
8. Spearman’s Rank Correlation Coefficient | 斯皮尔曼秩相关系数
Spearman’s rho (rₛ) measures the strength and direction of association between two ordinal variables. Both sets of scores are ranked separately, and the difference (d) between ranks for each participant is squared. The formula is elegantly straightforward:
斯皮尔曼等级相关系数 (rₛ) 测量两个顺序变量之间关联的强度和方向。两组分数各自独立排秩，计算每位参与者的秩差 d 后平方。公式简洁优美：
rₛ = 1 – [6 Σ d² / n (n² – 1)]
Here, n is the number of paired observations. The coefficient ranges from -1 (perfect negative correlation) through 0 (no correlation) to +1 (perfect positive correlation). To determine significance, compare the calculated rₛ to the critical values for Spearman’s table. If the absolute rₛ is equal to or greater than the critical value, the correlation is significant.
其中 n 为配对观测的数量。系数取值范围从 -1（完全负相关）到 0（无相关）再到 +1（完全正相关）。判断显著性时，将计算所得的 rₛ 与斯皮尔曼临界值表对照。若 rₛ 的绝对值等于或大于临界值，则相关显著。
9. Effect Size: Cohen’s d and Family | 效应量：Cohen’s d 及其族系
Statistical significance tells you whether an effect probably exists; effect size tells you how large it is. The most common measure for experiments comparing two independent means is Cohen’s d:
统计显著性告诉你效应是否存在，效应量则告诉你效应有多大。比较两个独立组均值时最常用的量数为 Cohen’s d：
d = (M₁ – M₂) / s_pooled
where M₁ and M₂ are the group means, and s_pooled is the weighted pooled standard deviation. By convention, d = 0.2 is a small effect, d = 0.5 a medium effect, and d = 0.8 a large effect. For correlational studies, the effect size r (Pearson or Spearman) itself is a direct measure, with 0.1 small, 0.3 medium and 0.5 large. Edexcel questions often ask you to interpret how meaningful a result is, not just whether it is significant.
其中 M₁ 和 M₂ 为组均值，s_pooled 为加权合并标准差。依惯例，d = 0.2 为小效应，d = 0.5 为中等效应，d = 0.8 为大效应。对于相关研究，效应量 r（皮尔逊或斯皮尔曼）本身就是直接量尺，0.1 为小，0.3 为中，0.5 为大。Edexcel 考题常要求你解读结果的实际意义，而不只是看它是否显著。
10. Classical Psychological Laws: Weber, Fechner and Stevens | 经典心理学定律：韦伯、费希纳与史蒂文斯
Psychophysics, a branch that studies the relationship between physical stimuli and perceptual experience, has yielded three classical ‘laws’ that appear across the A-level syllabus. Weber’s Law states that the just noticeable difference (JND) between two stimuli is a constant proportion of the original stimulus intensity:
心理物理学研究物理刺激与知觉体验之间的关系，产生了三条贯穿 A-level 课程的经典“定律”。韦伯定律指出，两个刺激的最小可觉差 (JND) 与原始刺激强度成正比：
ΔI / I = k (Weber fraction)
Fechner expanded on Weber’s work and proposed that subjective sensation (S) grows logarithmically with physical intensity (I):
费希纳在韦伯工作基础上提出，主观感觉量 (S) 随物理强度 (I) 呈对数增长：
S = k log I (Fechner’s Law)
Later, S.S. Stevens argued for a power-law relationship, which works better for many modalities:
后来，S.S. 史蒂文斯主张幂函数关系，对许多感觉通道更为适用：
S = k Iⁿ (Stevens’ Power Law)
In these equations, k and the exponent n are constants that depend on the sensory modality. For example, for brightness n ≈ 0.33, whereas for electric shock n ≈ 3.5. These psychophysical functions bridge mind and matter, and they are elegant examples of how mathematics can model mental experience. The Edexcel specification expects you to know the basic concepts, not to compute the constants, but being able to recognise the forms adds depth to your answers on perception.
在这些方程中，k 和指数 n 是取决于感觉通道的常数。例如，亮度的 n 约为 0.33，而电击的 n 约为 3.5。这些心理物理函数沟通了心与物，是用数学建模心理体验的优美示例。Edexcel 大纲要求你了解基本概念而非计算常数，但能识别这些形式将为你关于知觉的答案增加深度。
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