📚 Formula & Theorem Quick Reference Handbook for Year 13 CAIE Psychology | A2 心理学公式定理速查手册
This handbook collates the essential statistical formulas, decision rules, and classic quantitative laws most relevant to the Year 13 CAIE Psychology syllabus. From measures of central tendency and dispersion to the inferential tests you must calculate by hand (Wilcoxon, Mann–Whitney U, Chi-squared, Spearman’s rank, and the Binomial Sign test), every entry is presented with its computational formula, conditions of use, and interpretation of outcomes. A dedicated section on psychophysical laws rounds off the collection, linking numeric models to perceptual experience.
本手册汇编了与 CAIE 心理学 A2 阶段最为相关的统计公式、决策规则和经典定量定律。从集中趋势与离散量，到需要手算的推断检验（威尔科克森符号秩检验、曼-惠特尼 U 检验、卡方检验、斯皮尔曼等级相关和二项符号检验），每个条目都列出了计算公式、使用条件和结果解读。专设的心理物理学定律一节则将数值模型与感知体验联系起来，使手册更完整。
1. Descriptive Statistics: Central Tendency & Dispersion | 描述统计：集中趋势与离散量
The sample mean (x̄) is the sum of all scores divided by the number of observations n: x̄ when data are ordered, and the mode is the most frequent score. For interval/ratio data, the mean is usually the most informative measure.
样本均值 (x̄) 是所有分数之和除以观测数量 n：x̄ = Σx / n。中位数是数据排序后位于中间的数值，众数是出现频率最高的数值。对于等距/比率数据，均值通常提供最多信息。
Dispersion is captured by the range (highest – and standard deviation (s). The standard deviation formula for a sample is s = √[ Σ(x - x̄)² / (n - 1) ]. A larger standard deviation indicates greater spread around the mean.
离散程度可用全距（最大值减最小值）、方差 (s²) 和标准差 (s) 来描述。样本标准差公式为 s = √[ Σ(x - x̄)² / (n - 1) ]。标准差越大，数据在均值周围的散布越广。
s = √[ Σ(x - x̄)² ÷ (n - 1) ]
2. The Normal Distribution & z-Scores | 正态分布与 z 分数
Many psychological variables (e.g., IQ scores) approximate a normal distribution – a bell-shaped, symmetrical curve where the mean, median, and mode coincide. About 68% of values lie within ±1 SD, 95% within ±1.96 SD, and 99.7% within ±3 SD from the mean.
许多心理学变量（如智商分数）近似服从正态分布——这是一种钟形对称曲线，均值、中位数和众数重合。约 68% 的数值落在均值 ±1 SD 内，95% 落在 ±1.96 SD 内，99.7% 落在 ±3 SD 内。
A z-score expresses a raw score’s distance from the mean in standard deviation units. It is used to compare performances across different distributions.
z 分数以标准差为单位表示原始分数与均值的距离，可用于跨分布比较。
z = (x - μ) ÷ σ
3. Hypothesis Testing & p-Values | 假设检验与 p 值
We begin with a null hypothesis (H₀) stating no effect or no difference, and an alternative hypothesis (H₁) predicting a directional (one-tailed) or non-directional (two-tailed) effect. The significance level α (commonly 0.05) is the probability threshold for rejecting H₀.
我们首先设立零假设 (H₀)，即无效应或无差异；以及备择假设 (H₁)，预测有方向（单尾）或无方向（双尾）的效应。显著性水平 α（通常为 0.05）是拒绝 H₀ 的概率阈值。
the result is declared statistically significant; we reject H₀. When p > α, we retain H₀. A significant result does not necessarily imply a large or important effect – it simply suggests the outcome is unlikely to be due to chance alone.
p 值是在 H₀ 为真时获得当前结果（或更极端结果）的概率。当 p ≤ α 时，结果具有统计显著性，我们拒绝 H₀；当 p > α 时，保留 H₀。显著结果并不必然意味着效应量很大或很重要——它只说明该结果不太可能仅由偶然因素造成。
4. The Binomial Sign Test | 二项符号检验
Used for repeated-measures or matched-pairs designs when the DV yields nominal data (e.g., improvement vs. no improvement). The test counts the number of pluses (+) and minuses (−) for pairs of scores, ignoring ties. The observed value (S) is the smaller of the two counts.
适用于重复测量或配对设计，且因变量产生称名数据（如改善/未改善）。检验计算每对分数的正号 (+) 与负号 (−) 数量，忽略平局。观测值 (S) 为两者中较小的计数。
and significance level. If S ≤ critical value, the result is significant; H₀ is rejected.
将 S 与二项检验表中对应 n（非零差异对数）和显著性水平的临界值比较。若 S ≤ 临界值，结果显著，拒绝 H₀。
No additional computation formula is required; the test uses binomial probabilities tabulated in standard tables.
无需额外计算公式；该检验直接使用标准表格中的二项分布概率。
5. Wilcoxon Signed-Ranks Test | 威尔科克森符号秩检验
Applied to repeated-measures (or matched-pairs) designs with at least ordinal data. The test considers both the direction and the magnitude of differences. Steps: (1) calculate the difference score for each pair; (2) rank the absolute differences (ignoring zeros), assigning average ranks to ties; (3) sum the ranks of the positive differences (R₊) and negative differences (R₋). The test statistic T is the smaller of these two sums.
适用于至少有顺序数据的重复测量（或配对）设计。检验同时考虑差异的方向与大小。步骤：(1) 计算每对数据的差异值；(2) 对差异的绝对值排序（忽略零），平局赋予平均秩次；(3) 分别计算正差异秩和 (R₊) 与负差异秩和 (R₋)。检验统计量 T 取两者中较小值。
T = min(R₊, R₋)
ro difference pairs), if T ≤ critical value, the result is significant and H₀ is rejected.
将 T 与威尔科克森表临界值比较。对于给定的 n（非零差异对数），若 T ≤ 临界值，结果显著，拒绝 H₀。
6. Mann–Whitney U Test | 曼-惠特尼 U 检验
The Mann–Whitney U test compares two independent groups when the DV is at least ordinal. It is the non-parametric equivalent of the independent t-test. After ranking all scores together, the sum of ranks for each group (R₁ and R₂) is computed, and U is calculated for each group. The smaller U is taken as the test statistic.
曼-惠特尼 U 检验用于比较两个独立组，因变量至少为顺序数据，是独立 t 检验的非参数替代。将所有分数统一排序后，计算每组的秩和 (R₁, R₂)，再分别计算每组 U 值，取较小 U 为检验统计量。
U₁ = R₁ - n₁(n₁ + 1) ÷ 2
U₂ = R₂ - n₂(n₂ + 1) ÷ 2
U = min(U₁, U₂)
Compare U with the Mann– mple sizes n₁ and n₂, if U ≤ critical value, the result is significant and H₀ is rejected.
将 U 与曼-惠特尼临界值表比较。对于样本量 n₁ 和 n₂，若 U ≤ 临界值，结果显著，拒绝 H₀。
7. Chi-Squared Tests (Goodness of Fit & Association) | 卡方检验（拟合度与关联性）
Chi-squared tests work with frequency (count) data. The Goodness of Fit test compares observed frequencies in one categorical variable against expected theoretical frequencies. The test for Association (independence) examines whether two categorical variables are related. The same formula is used; only the expected frequencies are computed differently.
卡方检验用于频次数据。拟合度检验比较单一类别变量的观测频次与理论期望频次；独立性（关联性）检验考察两个类别变量是否相关。两者使用同一公式，仅期望频次的计算方法不同。
χ² = Σ (O - E)² ÷ E
tal) ÷ grand total. Degrees of freedom (df) = (number of rows - 1) × (number of columns - 1) for association; for goodness of fit, df = number of categories - 1.
O = 观测频次，E = 期望频次。在 2×2（或 r×c）列联表的关联性检验中，每个单元格的 E =（行合计×列合计）÷ 总计。关联性检验的自由度 df =（行数 - 1）×（列数 - 1）；拟合度检验的自由度 df = 类别数 - 1。
tical value from χ² tables at the appropriate df and α. If χ² ≥ critical value, reject H₀.
将计算所得 χ² 与卡方表中对应 df 和 α 的临界值比较。若 χ² ≥ 临界值，拒绝 H₀。
8. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数
Spearman’s coefficient (rₛ or ρ) measures the strength and direction of association between two ordinal variables or between two variables that can be ranked. It takes values from −1 (perfect negative correlation) to +1 (perfect positive correlation), with 0 indicating no monotonic relationship.
斯皮尔曼系数 (rₛ 或 ρ) 衡量两个顺序变量或可排序变量之间的关联强度与方向。取值范围从 −1（完全负相关）到 +1（完全正相关），0 表示无单调关系。
fference d for each pair of ranks; (3) square each d and sum them. Use the formula below. If ties exist, assign average ranks. Significance is determined by comparing rₛ to a critical value table.
步骤：(1) 分别对每个变量排序（1 = 最低）；(2) 计算每对秩次的差值 d；(3) 求各 d 的平方与总和。若存在平局，赋予平均秩次。显著性通过将 rₛ 与临界值表比较来确定。
rₛ = 1 - (6Σd²) ÷ [n(n² - 1)]
9. Type I & Type II Errors | 第一类与第二类错误
A Type I error occurs when the null hypothesis is rejected when it is actually true – a false positive. The probability of a Type I error is set by the researcher’s chosen significance level α (e.g., 5%). A Type II error is the failure to reject a false null hypothesis – a false negative. Its probability is denoted β.
第一类错误发生在 H₀ 为真却被拒绝时，即假阳性。其概率由研究者选定的显著性水平 α（如 5%）决定。第二类错误是未能拒绝一个错误的 H₀，即假阴性，其概率记为 β。
Statistical power (1 - β) is the probability of correctly detecting a genuine effect. Larger sample sizes, larger effect sizes, and one-tailed tests all increase power and reduce the chance of a Type II error. There is a trade-off: lowering α to guard against Type I errors raises the risk of Type II errors unless sample size is increased.
统计功效 (1 - β) 是正确检测出真实效应的概率。增大样本量、增大效应量、使用单尾检验都能提高功效、降低第二类错误风险。存在权衡：为了防范第一类错误而降低 α，会增加第二类错误的风险，除非同时扩大样本量。
10. Psychophysical Laws: Weber, Fechner & Stevens | 心理物理学定律：韦伯、费希纳与斯蒂文斯
Weber’s Law states that the just noticeable difference (JND) between two stimuli is a constant proportion (k) of the initial stimulus intensity. This proportion, the Weber fraction, varies across sensory modalities.
韦伯定律指出，两个刺激的最小可觉差 (JND) 是初始刺激强度的一个恒定比例 (k)。该比例（韦伯分数）因感觉通道而异。
ΔI = k × I
Fechner’s Law extends Weber’s insight by proposing that sensation magnitude (S) grows logarithmically with physical intensity (I). This explains why it takes ever larger changes in stimulation to produce equal steps in perceived intensity (e.g., doubling the physical brightness produces a smaller subjective increase).
费希纳定律延伸了韦伯的发现，提出感觉量 (S) 与物理强度 (I) 呈对数增长。这解释了为何需要越来越大的刺激变化才能产生相等的感知增量（例如，物理亮度加倍时，主观亮度增量较小）。
S = k × log I
exponent differs across sensory continua. For brightness, n ≈ 0.33; for electric shock, n ≈ 3.5, producing a steeply accelerating function.
斯蒂文斯幂律随后修正了这一关系，显示感觉量与物理强度的某次幂 (n) 成正比，指数 n 因感觉连续体而异。例如，明度 n ≈ 0.33，而电击 n ≈ 3.5，形成陡峭的加速函数。
S = k × Iⁿ
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