📚 A-Level WJEC Psychology: Formula & Theorem Quick Reference Handbook | WJEC 心理学公式定理速查手册
This handbook brings together every key formula, statistical test equation, and classic psychophysical law required for the WJEC A-Level Psychology specification. Whether you are computing a standard deviation for a research methods question, running a Spearman’s rho on correlational data, or applying Weber’s Law in a nominal sensory scenario, you can find the concise mathematical expression, its usage conditions, and a worked glance of meaning here. Use it as a revision checklist, a top-up reminder before your examination, or a quick-fix reference when tackling past papers.
本手册汇总了 WJEC A-Level 心理学考试中必需的所有核心公式、统计检验方程和经典心理物理学定律。无论您是在研究方法题中计算标准差、对相关分析数据执行斯皮尔曼等级相关系数，还是在知觉情境中应用韦伯定律，都能在此找到简洁的数学表达式、适用条件和使用要点。您可以将其用作复习清单、考前强化或刷真题时的速查工具。
1. Mean, Variance and Standard Deviation | 平均值、方差与标准差
The arithmetic mean (M) is the sum of all scores divided by the number of scores. For a sample, M = ΣX / n, where ΣX is the sum of raw scores and n is the sample size. In WJEC psychology, you must be able to calculate the mean from raw data and understand its sensitivity to outliers.
算术平均数 (M) 是所有分数之和除以分数的个数。对于样本，M = ΣX / n，其中 ΣX 为原始分数总和，n 为样本量。在 WJEC 心理学中，您必须能够根据原始数据计算平均数，并理解其对异常值的敏感性。
Variance and standard deviation describe how spread out scores are around the mean. Sample variance is s² = Σ(X − M)² / (n − 1). Taking the square root gives the standard deviation:
方差与标准差用于描述分数围绕均值的离散程度。样本方差公式为 s² = Σ(X − M)² / (n − 1)。取其平方根即得标准差：
s = √[ Σ(X − M)² / (n − 1) ]
A larger standard deviation indicates greater variability in the data set, which can affect the choice of inferential test.
标准差越大，表明数据集的变异性越大，这会影响到推断统计检验的选择。
2. Z-Scores (Standard Scores) | Z 分数（标准分数）
A z-score expresses how many standard deviations a raw score lies above or below the mean. It is fundamental for comparing scores from different distributions, locating a value on the normal curve, and checking for outliers.
Z 分数表示一个原始分数距离均值有多少个标准差。它是在正态曲线上定位数值、比较来自不同分布的分数以及检验异常值的基础工具。
z = (X − μ) / σ
For a sample, μ is replaced by M and σ by s. In WJEC assessment, you may be given a table of z-scores and asked to determine the percentage of the population falling above or below a certain point, typically applying the 68‑95‑99.7 empirical rule for normally distributed data.
对于样本，μ 用 M 替代，σ 用 s 替代。在 WJEC 评估中，您可能会拿到 Z 分数表，并被要求确定总体中落在某一点之上或之下的百分比，通常会应用到正态分布数据的 68‑95‑99.7 经验法则。
3. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s rho (rs) is a non-parametric test of correlation used when data are ordinal or when the relationship is monotonic but not necessarily linear. It is frequently examined in WJEC Unit 2 alongside the concepts of correlational design and alternative hypotheses.
斯皮尔曼等级相关系数 (rs) 是一种非参数相关检验，适用于顺序数据或者关系是单调但未必线性的情况。在 WJEC 第二单元中经常考到，常与相关研究设计和备择假设等概念一同出现。
rs = 1 − (6 Σd²) / [n(n² − 1)]
Where d is the difference between the ranks of each pair of scores, and n is the number of pairs. The obtained rs is compared with a critical value table; if |rs| is greater than the critical value, the null hypothesis is rejected.
式中 d 为每对分数的秩次差，n 为对数。计算出的 rs 与临界值表比较；若 |rs| 大于临界值，则拒绝零假设。
4. Pearson’s Product-Moment Correlation Coefficient | 皮尔逊积差相关系数
Pearson’s r measures the strength and direction of a linear relationship between two interval/ratio variables. Although WJEC highlights Spearman’s rho more prominently, Pearson’s r may appear when data meet parametric assumptions.
皮尔逊积差相关系数 r 用于衡量两个间距/比率变量之间线性关系的强度和方向。尽管 WJEC 更突出斯皮尔曼等级相关，但当数据满足参数检验假定时，皮尔逊 r 也有可能出现。
r = [ Σ(X − Mₓ)(Y − My) ] / [ √( Σ(X − Mₓ)² × Σ(Y − My)² ) ]
Values of r range from −1 (perfect negative) to +1 (perfect positive). A result close to zero suggests no linear relationship. The significance is checked against df = n − 2.
r 的取值范围从 −1（完全负相关）到 +1（完全正相关）。结果接近零意味着不存在线性关系。显著性检验的自由度为 n − 2。
5. Chi-Squared Test for Independence | 独立性卡方检验
The chi-squared (χ²) test is a non-parametric test of association used with nominal (categorical) data in an independent groups design. It tells you whether the observed frequencies differ significantly from the expected frequencies.
卡方 (χ²) 检验是一种适用于称名（分类）数据和独立组设计的非参数关联检验，用以判断观测频次与期望频次是否存在显著差异。
χ² = Σ [ (O − E)² / E ]
O represents the observed frequency in each cell, E the expected frequency (row total × column total / grand total). Degrees of freedom = (number of rows − 1) × (number of columns − 1). The calculated χ² is compared to a critical value; if it exceeds the critical value, the result is significant.
O 代表各单元格的观测频次，E 代表期望频次（行合计 × 列合计 / 总计）。自由度 = (行数 − 1) × (列数 − 1)。计算出的 χ² 值须与临界值比较；若大于临界值，则表示结果显著。
6. The Sign Test | 符号检验
The sign test is one of the simplest non-parametric tests for a repeated measures design. It is used when data are at least ordinal and you want to know whether there is a significant difference in a specific direction between two conditions. WJEC frequently includes it as a test of difference for related data.
符号检验是最简单的非参数检验之一，适用于重复测量设计。当数据至少为顺序水平、且您希望了解两个条件下的某一特定方向上是否存在显著差异时，可使用该检验。WJEC 常将其作为相关样本差异检验来考查。
s = number of less frequent sign (+ or -)
Count the number of positive and negative differences between pairs; the test statistic s is the smaller of the two counts. Compare s to a critical value for a given n (number of non-zero differences). If s is equal to or less than the critical value, the null hypothesis can be rejected.
先计算每对数据的正、负差值个数；检验统计量 s 是两者中较小的数值。将 s 与给定 n（非零差异的对数）下的临界值进行比较。若 s 小于或等于临界值，则可以拒绝零假设。
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 tests whether two independent groups differ on an ordinal measure and is regularly required in WJEC Unit 2 research methods questions.
曼–惠特尼 U 检验是独立样本 t 检验的非参数替代方案，用于检验两个独立组在顺序测量上是否存在差异，在 WJEC 第二单元研究方法题中频繁出现。
U1 = n1n2 + n1(n1+1)/2 − R1
U2 = n1n2 + n2(n2+1)/2 − R2
n1 and n2 are the sample sizes, R1 and R2 the sum of ranks for each group. The smaller U value is compared to a critical table. If U is smaller than or equal to the critical value, the result is significant, indicating a real difference between groups.
n1 和 n2 为两组样本量，R1 和 R2 是每组的秩和。选择较小的 U 值与临界值表比较。若 U 小于或等于临界值，则结果显著，表明两组间存在真实差异。
8. Wilcoxon Signed-Ranks Test | 威尔科克森符号秩检验
The Wilcoxon test is a non-parametric test for a related (repeated measures) design, using ordinal data. It considers both the direction and the magnitude of differences between paired scores, making it more powerful than the simple sign test.
威尔科克森检验是一种适用于相关（重复测量）设计和非参数数据的检验方法，同时考虑配对分数差异的方向和大小，因此比单纯的符号检验更敏感。
T = sum of ranks for the less frequent sign
Ignore zero differences, rank the remaining absolute differences from smallest to largest, and then sum the ranks of the positive differences and the negative differences separately. The test statistic T is the smaller of the two sums. Compare T to a critical Wilcoxon table; if T is less than or equal to the critical value, you can reject the null hypothesis.
忽略零差异，将剩余差异的绝对值从小到大排序，然后分别计算正、负差值的秩和，取其中较小的作为检验统计量 T。将 T 与威尔科克森临界值表比较；若 T 小于或等于临界值，则可拒绝零假设。
9. Independent and Related t-Tests | 独立样本与相关样本 t 检验
When data are interval/ratio and normally distributed, the independent t‑test compares the means of two unrelated groups, whereas the related t‑test compares two means from the same participants. WJEC expects you to recognise the parametric assumptions and be able to quote the general formula.
当数据为间距/比率水平且呈正态分布时，独立样本 t 检验比较两个不相关组的均值，而相关样本 t 检验比较同一批被试的两个均值。WJEC 期望您能识别参数检验的假定条件并引用基本公式。
Independent t‑test (assuming equal variances):
独立样本 t 检验（假设方差相等）：
t = (M₁ − M₂) / √( s₁²/n₁ + s₂²/n₂ )
Related t‑test: t = (MD) / (sD/√n), where MD is the mean of the difference scores and sD their standard deviation. The calculated t is compared with a critical value for the appropriate degrees of freedom.
相关样本 t 检验：t = (MD) / (sD/√n)，其中 MD 为差异分数均值，sD 为其标准差。计算出的 t 值与相应自由度下的临界值进行比较。
10. Effect Size: Cohen’s d | 效应量：科恩 d
Cohen’s d quantifies the size of the difference between two group means in standard deviation units, independent of sample size. In WJEC coursework and interpretation questions, reporting an effect size strengthens the conclusion by indicating practical significance.
科恩 d 以标准差为单位量化两组均值间差异的大小，且不受样本量影响。在 WJEC 课程作业与解读题目中，报告效应量可以通过展示实际显著性来强化结论。
d = (M₁ − M₂) / spooled
Where spooled = √[ (s₁² + s₂²) / 2 ]. Guidelines: 0.2 = small, 0.5 = medium, 0.8 = large effect. For the related t‑test, d = MD / sD.
式中 spooled = √[ (s₁² + s₂²) / 2 ]。经验准则：0.2 为小效应，0.5 为中效应，0.8 为大效应。对于相关样本 t 检验，d = MD / sD。
11. 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 a constant proportion of the original stimulus intensity. This classical psychophysical principle is often referenced in WJEC when discussing sensory processes and experimental psychology.
韦伯定律指出，两种刺激之间的最小可觉差（JND）与原刺激强度的比例是恒定的。这条经典的心理物理学原理在 WJEC 讨论感觉过程和实验心理学时经常被引用。
ΔI / I = k
ΔI is the increment in intensity needed to detect a difference, I the initial intensity, and k the Weber fraction (e.g., ~0.02 for visual brightness).
ΔI 为产生差异知觉所需的增量，I 为初始强度，k 为韦伯分数（例如视觉亮度 k 约 0.02）。
Fechner’s Law extends this by proposing that sensation magnitude (S) grows logarithmically with physical intensity: S = k log I. Though not always numerically tested, the law helps explain why equal physical steps produce diminishing perceived changes.
费希纳定律将此扩展为感觉量 (S) 随物理强度呈对数增长：S = k log I。虽然不常进行定量考查，但该定律有助于解释为何等量的物理变化会引起逐渐减小的知觉变化。
12. Probability, P‑Values and the Null Hypothesis | 概率、P 值与零假设
In inferential statistics, the p‑value represents the probability of obtaining the observed result, or one more extreme, if the null hypothesis (H0) is true. The conventional significance level in WJEC psychology is p ≤ 0.05, though more stringent levels (p ≤ 0.01) may be applied.
在推断统计中，p 值表示在零假设 (H0) 真的情况下，获得当前观测结果或更极端结果的概率。WJEC 心理学中常规的显著性水平为 p ≤ 0.05，有时也会使用更严格的标准 (p ≤ 0.01)。
Type I error (α) is rejecting a true H0; Type II error (β) is failing to reject a false H0. Understanding these concepts is essential for evaluating the validity of psychological research on the WJEC exam.
第一类错误 (α) 是错误拒绝了真的 H0；第二类错误 (β) 是未能拒绝假的 H0。理解这些概念对于在 WJEC 考试中评价心理学研究的效度至关重要。
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