📚 Cambridge AS Psychology Formula & Key Theorem Quick Reference | 剑桥AS心理学公式定理速查手册
This quick reference handbook compiles all essential statistical formulas, decision rules, and key quantitative concepts you will encounter in Cambridge AS Psychology (9990). It is designed for Year 12 students who need a rapid revision tool to consolidate research methods and data analysis skills before the exam. Each formula is presented with its typical usage in core studies and a worked-style explanation to reinforce your understanding of inferential testing.
本速查手册汇集了剑桥AS心理学（9990）中你将遇到的所有关键统计公式、决策规则和重要量化概念。它专为Year 12学生设计，可作为考前快速回顾研究方法和数据分析技能的备考工具。每个公式均配有在核心研究中的典型应用场景，并通过示例式解释帮助你巩固对推断性检验的理解。
1. Mean and Median | 平均数和中位数
The mean (x̄) is the arithmetic average of a data set, calculated by summing all scores and dividing by the total number of scores (n). It is used with interval/ratio data in normally distributed samples, for example when reporting the average number of dreams recalled in Dement and Kleitman (1957).
平均数（x̄）是数据集的算术平均值，计算方法是将所有分数相加后除以总个数（n）。当数据为等距/等比变量且呈正态分布时使用平均数，例如在Dement和Kleitman（1957）的研究中报告梦境回忆的平均数量。
x̄ = Σx / n
The median (Mdn) is the middle value in an ordered list. It is reported when the data set contains extreme scores or is ordinal, as the median is not distorted by outliers. For example, reaction times in a cognitive task might be summarised by the median.
中位数（Mdn）是有序列表的中间值。当数据集中包含极端值或属于顺序变量时报告中位数，因为中位数不受异常值影响。例如，认知任务中的反应时通常用中位数汇总。
2. Standard Deviation and Variance | 标准差与方差
Standard deviation (s) measures the dispersion of scores around the mean. A larger standard deviation indicates greater variability in the data. It is essential for evaluating consistency, such as variations in anxiety scores across conditions. Variance is simply the square of the standard deviation, but standard deviation is preferred because it is expressed in the original units.
标准差（s）衡量分数围绕平均数的离散程度。标准差越大表示数据变异性越大。它对于评估一致性非常关键，例如测量不同实验条件下焦虑分数的变动。方差是标准差的平方，但标准差因与原数据单位一致而更常用。
s = √[ Σ(x − x̄)² / (n − 1) ]
The denominator (n−1) is used for a sample, as it provides an unbiased estimate of the population standard deviation. In the core study by Hassett et al. (2008) on monkey toy preferences, standard deviation could be computed for the duration of interactions with each toy type.
分母使用（n−1）来计算样本标准差，因为它给出总体标准差的无偏估计。在Hassett等人（2008）关于猴子玩具偏好的核心研究中，可以计算与各类玩具互动时长的标准差。
3. Spearman’s Rank Correlation Coefficient | 斯皮尔曼秩相关系数
Spearman’s rho (rₛ) is a non-parametric test of correlation used when at least one variable is ordinal or when ranked data are used. It tells us the strength and direction of a relationship between two co-variables, with a value ranging from −1 (perfect negative) to +1 (perfect positive).
斯皮尔曼秩相关系数（rₛ）是一种非参数相关检验，适用于至少有一个变量为顺序变量或使用排名数据时。它说明两个共变量关系的强度和方向，取值从−1（完全负相关）到+1（完全正相关）。
rₛ = 1 − (6 Σd²) / (n(n² − 1))
Where d is the difference between the ranks assigned to each participant’s two scores, and n is the number of paired observations. For instance, you might use rₛ to examine the relationship between self-reported stress rank and hours of sleep rank in Bandura et al. (1961) data.
其中d是每个参与者两个分数所分配等级的差值，n是配对观测值数量。例如，你可以用rₛ来检验Bandura等人（1961）数据中自我报告的压力等级与睡眠时长等级之间的关系。
4. Chi-Square Test for Independence | 独立性卡方检验
The chi-square (χ²) test determines whether there is a significant association between two categorical variables (nominal data). It compares observed frequencies (O) with the frequencies expected (E) if the null hypothesis were true. In AS Psychology, it is used in studies with independent groups design and frequency data, such as choice of doll in a social learning experiment.
卡方（χ²）检验用于判断两个分类变量（称名数据）之间是否存在显著关联。它比较观测频数（O）与在零假设成立时预期的频数（E）。在AS心理学中，它用于独立组设计和频数数据，例如社会学习实验中的娃娃选择。
χ² = Σ (O − E)² / E
Degrees of freedom (df) for a two-way contingency table is calculated as (number of rows − 1) × (number of columns − 1). The computed χ² value is compared against a critical value from the chi-square distribution table at a chosen significance level, typically p ≤ 0.05.
双向列联表的自由度（df）计算公式为（行数−1）×（列数−1）。算出的χ²值要与选定的显著性水平（通常p ≤ 0.05）下的卡方分布临界值表进行比较。
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 t-test, used to compare two independent groups when the dependent variable is at least ordinal or the data violate parametric assumptions. It is frequently applied in AS core studies like Canli et al. (2000) when comparing emotional intensity ratings between two participant groups.
曼–惠特尼U检验是独立样本t检验的非参数替代方法，用于在因变量至少为顺序变量或数据违背参数假设时比较两个独立组。它常用于AS核心研究，例如Canli等人（2000）中比较两组参与者的情绪强度评分。
U₁ = n₁n₂ + n₁(n₁+1)/2 − R₁
U₂ = n₁n₂ − U₁
Here n₁ and n₂ are the sample sizes, and R₁ is the sum of ranks for group 1. The smaller of U₁ and U₂ is the test statistic U. If the calculated U is less than or equal to the critical value from the Mann–Whitney table, the result is significant.
此处n₁与n₂为样本量，R₁为组1的秩和。U₁和U₂中较小的值即为检验统计量U。若算出的U小于或等于曼–惠特尼表中的临界值，则结果显著。
6. Wilcoxon Signed-Rank Test | 威尔科克森符号秩检验
The Wilcoxon signed-rank test is a non-parametric test for repeated measures or matched pairs designs. It examines whether there is a significant difference between two related conditions by considering both the direction and magnitude of difference scores. It is appropriate when the level of measurement is at least ordinal.
威尔科克森符号秩检验是一种用于重复测量或配对组设计的非参数检验。它通过考量差异分数的方向与大小，检验两个相关条件是否存在显著差异。该检验适用于至少顺序变量的测量层次。
T = sum of ranks for the less frequent sign
To compute T, calculate the difference between each pair of scores, ignore zero differences, rank the absolute differences from smallest to largest, then separately sum the ranks for positive and negative signs. The smaller of these two sums is the T statistic. In Laney et al. (2008), the test could compare food preference ratings before and after a false memory implantation.
计算T时，先求出每对分数的差值，忽略零差值，将差值的绝对值从最小到最大编秩，然后分别求出正秩和与负秩和。两个秩和中较小者即为T统计量。在Laney等人（2008）的研究中，可用此检验比较虚假记忆植入前后食物偏好评分的差异。
7. Understanding p-values and Significance Levels | 理解p值与显著性水平
A p-value indicates the probability of obtaining the observed results, or more extreme, if the null hypothesis is true. In AS Psychology, the conventional significance level is α = 0.05. If p ≤ 0.05, the result is judged statistically significant, and the null hypothesis is rejected. This threshold corresponds to a 5% risk of a Type I error.
p值表示在零假设为真的条件下，获得当前结果或更极端结果的概率。在AS心理学中，常用显著性水平为α = 0.05。若p ≤ 0.05，则结果被认为具有统计显著性，并拒绝零假设。该阈值对应5%的第一类错误风险。
The p-value is compared against the chosen alpha level; it is not a direct measure of effect size or practical importance. Researchers also look at whether the test is one-tailed (directional hypothesis) or two-tailed (non-directional hypothesis), which affects the location of the critical region.
p值与选定的α水平进行比较；它并非效应量或实际重要性的直接度量。研究者还需确定检验是单尾（方向性假设）还是双尾（非方向性假设），这会影响临界区域的位置。
8. Type I and Type II Errors | 第一类错误与第二类错误
A Type I error (false positive) occurs when a null hypothesis is wrongly rejected, concluding there is an effect when in fact there is none. The probability of making a Type I error is equal to the significance level (α). A Type II error (false negative) happens when the null hypothesis is falsely retained, missing a real effect. Its probability is denoted by β.
第一类错误（错误的肯定）指错误地拒绝了零假设，即得出结论认为存在效应而实际上没有。犯第一类错误的概率等于显著性水平（α）。第二类错误（错误的否定）指错误地保留了零假设，错过了真实的效应，其概率用β表示。
Increasing sample size, using a less stringent alpha, or employing a more powerful statistical test can reduce the likelihood of a Type II error. Balancing these errors is a crucial part of experimental design, especially when evaluating studies like Milgram (1963), where ethical cost of false conclusions is high.
增加样本量、使用较宽松的α水平或采用统计效力更高的检验可降低第二类错误发生的可能性。平衡这两类错误是实验设计的关键部分，尤其是在评估像Milgram（1963）这样得出错误结论的伦理代价很高的研究时。
9. Normal Distribution and Standard Scores | 正态分布与标准分数
In a perfectly normal distribution, approximately 68% of scores fall within 1 standard deviation of the mean, 95% within 1.96 SDs, and 99.7% within 3 SDs. Although AS Psychology mainly uses non-parametric tests, understanding normality helps in deciding the appropriate test and interpreting standardised data.
在一个完美的正态分布中，约68%的分数落在均值的1个标准差范围内，95%落在1.96个标准差内，99.7%落在3个标准差内。尽管AS心理学主要使用非参数检验，了解正态性有助于选择合适的检验以及解释标准化数据。
Standard scores (z-scores) express how many standard deviations a raw score is from the mean: z = (x − x̄) / s. They are used when comparing scores from different distributions, though direct computation is less common in AS practicals.
标准分数（z分数）表示原始分数距离均值几个标准差：z = (x − x̄) / s。在比较不同分布的数据时用到，但在AS操作考试中直接计算较少见。
10. Critical Values and Degrees of Freedom | 临界值与自由度
When performing an inferential test, the computed statistic is compared with a critical value found in a statistical table. For the chi-square test, degrees of freedom (df) must be determined first. For the chi-square test of independence, df = (r − 1)(c − 1), where r and c are the number of rows and columns. For Spearman’s rho and Mann–Whitney U tests, the table entry usually corresponds directly to sample size (N or n) rather than a separate df value.
进行推断性检验时，需要将计算出的统计量与统计表中查得的临界值进行比较。对于卡方检验，必须首先确定自由度（df）。独立性卡方检验的自由度为df = (r − 1)(c − 1)，其中r与c分别为行数与列数。对于斯皮尔曼秩相关与曼–惠特尼U检验，通常直接根据样本量（N或n）查表，而不依赖单独的自由度值。
Observed value must be more extreme than the critical value to reject the null hypothesis. For correlation, the absolute calculated rho must be greater than or equal to the tabled value; for U and T, the computed statistic must be equal to or smaller than the critical value. Always check whether the hypothesis is one-tailed or two-tailed before selecting the correct column.
观测值必须比临界值更极端（更远离零假设区域）才能拒绝零假设。对于相关分析，计算出的rho绝对值必须大于或等于表中的临界值；对于U和T检验，计算出的统计量必须等于或小于临界值。在选择正确的列之前，务必确认假设是单尾还是双尾。
11. Miller’s Magic Number (7 ± 2) | 米勒的神奇数字（7±2）
George Miller (1956) proposed that the capacity of short-term memory is approximately 7 items (plus or minus 2). This ‘magic number’ refers to the number of discrete chunks of information that can be held in working memory at once. It is not a statistical formula but a key cognitive principle referred to when evaluating memory studies such as Andrade (2010) on doodling and concentration.
乔治·米勒（1956）提出短时记忆的容量大约为7个项目（上下浮动2个）。这个“神奇数字”指工作记忆中一次能容纳的离散信息组块数量。它虽非统计公式，却是评价记忆研究（如Andrade（2010）关于涂鸦与注意力的实验）时的核心认知原理。
When applying this principle, remember that chunking—grouping bits of information into meaningful units—can expand functional memory capacity. This helps explain why participants in the Baron-Cohen et al. (1997) eyes test might rely on quick pattern recognition rather than holding each eye region as a separate item in memory.
运用这一原理时需记住，组块化（将信息片段组合成有意义单元）可以扩大功能性记忆容量。这有助于解释为何Baron-Cohen等人（1997）眼神测试中的参与者可能依赖快速的模式识别，而非将每个眼区作为独立项目保持在记忆中。
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