Year 12 CAIE Psychology: Quick Reference to Formulas and Key Principles | Year 12 CAIE 心理学：公式定理速查手册

📚 Year 12 CAIE Psychology: Quick Reference to Formulas and Key Principles | Year 12 CAIE 心理学：公式定理速查手册

This handbook provides a concise, exam-focused revision of the essential formulas, statistical tests, and key theoretical principles required for Year 12 CAIE Psychology (9990). It covers the descriptive and inferential statistics you must calculate and interpret, alongside the foundational laws and concepts that underpin the core studies. Use it to check your understanding before assessments, and to build confidence in applying numerical and conceptual knowledge.

本手册为 Year 12 CAIE 心理学 (9990) 提供简明、紧扣考点的复习内容，涵盖必修的描述与推断统计公式，以及支撑核心研究的基本定律和概念。你需要在考试中计算并解释这些统计量，同时理解相关的理论原理。请在评估前用它检查掌握程度，增强应用数值与概念知识的信心。

1. Measures of Central Tendency | 集中趋势测量

The three main measures summarise a typical value in a data set. The mean is the arithmetic average, the median is the middle score when data are ordered, and the mode is the most frequently occurring score. In CAIE Psychology, you must be able to identify which measure is most appropriate for different levels of measurement, and whether the distribution is skewed.

三种主要测量指标概括数据中的典型值。算术平均数（mean）是算术平均；中位数（median）是排序后中间的值；众数（mode）是出现频率最高的值。在 CAIE 心理学中，你需要能够判断对于不同测量层次以及分布是否偏斜，哪种测量最为合适。

Mean: sum of all scores divided by the number of scores. Formula: x̄ = Σx / n. Used with interval/ratio data, but sensitive to extreme values.

平均数：所有数值之和除以数值个数。公式：x̄ = Σx / n。用于间隔/比率数据，但对极端值敏感。

Median: middle value in an ordered list. For an even number, average the two middle values. Best for ordinal data or skewed distributions.

中位数：有序列表中的中间值。偶数个时取中间两个值的平均值。最适用于顺序数据或偏斜分布。

Mode: most common value. Useful for nominal data. There can be more than one mode.

众数：最常见的值。适用于名义数据。可能存在多个众数。

2. Measures of Dispersion | 离散程度测量

Dispersion tells us how spread out the scores are around the central tendency. The range and standard deviation are the two measures examined at this level. You must be able to calculate and comment on the variation within a sample, as well as consider its impact on the reliability of findings.

离散程度反映数据在集中趋势周围的分散状况。极差（range）和标准差（standard deviation）是这一阶段考查的两种测量。你需要能够计算并评价样本内的变异，并考虑它对研究结果可靠性的影响。

Range: highest score minus lowest score. Indicates the spread but is strongly affected by outliers.

极差：最高分减去最低分。反映离散情况，但极易受异常值影响。

Standard Deviation (SD): average distance of each score from the mean. A larger SD means greater variability. Formula: SD = √[Σ(x − x̄)²/(n−1)] for a sample. You do not need to compute this manually in CAIE if given summary data, but should be able to interpret it.

标准差 (SD)：各数据与平均数之间的平均距离。SD 越大，变异程度越大。样本标准差公式：SD = √[Σ(x − x̄)²/(n−1)]。CAIE 中若给出汇总数据，通常不要求手动开方计算，但需能解读。

3. Inferential Statistics: Core Concepts | 推断统计核心概念

Inferential tests determine whether the results obtained are likely due to the manipulation of the independent variable or due to chance. Key terms include probability (p-value), level of significance, null hypothesis, alternative hypothesis, and Type I/II errors. CAIE requires you to select the correct test based on study design, level of data, and whether the design is independent or repeated measures.

推断检验旨在判断结果更可能由操纵自变量引起，还是由偶然因素引起。关键术语包括概率（p 值）、显著性水平、零假设、备择假设以及第一类/第二类错误。CAIE 要求你根据研究设计、数据层次以及是独立设计还是重复测量设计来选择合适的检验。

The conventional significance level p ≤ 0.05 means there is a 5% or less probability that the observed difference/correlation is due to chance. If the calculated statistic is equal to or more extreme than the critical value, we reject the null hypothesis. Always link this decision to the actual study context.

常规显著性水平 p ≤ 0.05 表示观察到的差异/相关由偶然引起的概率不大于 5%。如果计算出的统计量等于或比临界值更极端，我们就拒绝零假设。必须始终将这一判断与实际研究背景相联系。

Null hypothesis (H₀): no difference/relationship. | 零假设 (H₀)：无差异/无关系。
Alternative hypothesis (H₁): there is a difference/relationship. | 备择假设 (H₁)：存在差异/关系。
Type I error: rejecting a true null. | 第一类错误：拒绝一个真实的零假设。
Type II error: failing to reject a false null. | 第二类错误：未能拒绝一个错误的零假设。

4. Chi-Square (χ²) Test | 卡方检验

The Chi-square test for independence is used when the data are nominal (categories) and the design is independent measures. It tests whether there is an association between two categorical variables. In CAIE Psychology, you might use it to analyse questionnaire data or frequency counts observed in studies such as Andrade’s doodling experiment if re-coded.

独立性的卡方检验用于名义（分类）数据且设计为独立测量时。它检验两个类别变量之间是否存在关联。在 CAIE 心理学中，你可能将它用于分析问卷数据或类似 Andrade 涂鸦实验中重新编码后的频率计数。

χ² = Σ (O − E)² / E

O = observed frequency, E = expected frequency. Expected frequencies are calculated under the assumption of no association. Degrees of freedom (df) = (number of rows − 1) × (number of columns − 1). Compare the obtained χ² with the critical value from the table. If calculated χ² ≥ critical value, the result is significant at the chosen p-value.

O = 观察频次，E = 期望频次。期望频次是在无关联假设下计算出的。自由度 (df) = (行数 − 1) × (列数 − 1)。将得到的 χ² 值与表格中的临界值比较。若计算得到的 χ² ≥ 临界值，则在所选 p 值下结果显著。

5. Mann–Whitney U Test | 曼–惠特尼 U 检验

Mann–Whitney is an ordinal-level test for independent measures designs. It compares two sets of scores to see if one tends to be significantly higher than the other. The test is regularly used in CAIE to analyse data from studies like Canli et al.’s emotional memory experiment or Laney et al.’s false memory study if the data are treated as ordinal.

曼–惠特尼 U 检验是一种用于独立测量设计的顺序数据检验。它比较两组分数，以判断是否一组倾向于显著高于另一组。该检验在 CAIE 中常用于分析如 Canli 等人的情绪记忆实验或 Laney 等人的虚假记忆研究的数据（当数据被视为顺序尺度时）。

Steps: Rank all scores together ignoring group membership; sum the ranks for each group (R₁ and R₂); compute U for each group.

步骤：将所有分数混合排序，忽略组别；计算每组的秩和 (R₁ 和 R₂)；对每一组计算 U。

U₁ = n₁n₂ + n₁(n₁+1)/2 − R₁
U₂ = n₁n₂ + n₂(n₂+1)/2 − R₂

The smaller U is taken as the test statistic. Compare this with the critical value for the given n₁ and n₂. For significance at 0.05, the obtained U must be equal to or less than the critical value.

取较小的 U 作为检验统计量。将其与给定 n₁、n₂ 的临界值比较。要在 0.05 水平上显著，得到的 U 必须等于或小于临界值。

6. Wilcoxon Signed-Rank Test | 威尔科克森符号秩检验

Wilcoxon T test is used when the data are at least ordinal and the design is repeated measures (or matched pairs). It examines whether there is a significant difference between two conditions for the same participants. In core studies, this test fits designs where participants are tested before and after an intervention, or give two ratings under different conditions.

威尔科克森符号秩检验（T 检验）用于数据至少是顺序尺度且为重复测量（或配对）设计时。它检验同一参与者在两种条件下是否有显著差异。在核心研究中，此检验适用于参与者在干预前后接受测试，或在不同条件下给出两种评价的设计。

Procedure: Calculate the difference between each pair of scores; ignore pairs with zero difference; rank the absolute differences; apply the sign of the difference to the ranks; sum the positive ranks and the negative ranks separately. The test statistic T is the smaller of the two sums. Compare T with the critical value: for significance at 0.05, T must be equal to or less than the table value.

步骤：计算每对分数之差；忽略差值为零的对；对绝对差值进行排序；将差的符号赋予秩；分别求出正秩和与负秩和。检验统计量 T 为两者中较小的和。将 T 与临界值比较：在 0.05 水平上显著要求 T 等于或小于表值。

CAIE often asks you to state that the Wilcoxon test reduces the impact of extreme scores compared with a parametric test because it works with ranks, not actual values.

CAIE 经常要求你说明，与参数检验相比，威尔科克森检验减少极端值的影响，因为它基于秩而不是实际数值。

7. Sign Test | 符号检验

The sign test is the simplest non-parametric test for repeated measures design with nominal or paired data. It is used when you can only record the direction of change (e.g., more/less, yes/no) rather than the magnitude. It can be applied to data from studies like Schachter and Singer’s emotion study if responses were coded categorically.

符号检验是最简单的非参数检验，适用于名义或配对数据的重复测量设计。当你仅能记录变化的方向（如更多/更少、是/否）而非变化的程度时使用。若反应被编为类别数据，它可应用于如 Schachter 和 Singer 的情绪研究的数据。

Count the number of plus and minus signs. The test statistic S is the number of occurrences of the less frequent sign. N = total number of non-zero differences. Compare S with the critical value in the binomial sign test table: for significance, S must be equal to or less than the critical value. Always check whether the hypothesis is one-tailed or two-tailed.

统计正负符号的个数。检验统计量 S 为出现较少的符号的数目。N = 非零差异总数。将 S 与二项符号检验表中的临界值比较：要达到显著性，S 必须等于或小于临界值。务必检查假设是单尾还是双尾。

8. Spearman’s Rank Correlation (rₛ) | 斯皮尔曼等级相关

Spearman’s Rho is used to assess the strength and direction of a relationship between two ordinal variables. It does not assume a linear relationship, and is therefore appropriate for studies that look at associations, such as the correlation between age and memory performance in Canli et al. or between questionnaire scores.

斯皮尔曼等级相关系数用于评估两个顺序变量之间关系的强度和方向。它不假定线性关系，因此适用于探索关联的研究，例如 Canli 等人研究中年龄与记忆表现的关联，或问卷分数之间的关联。

rₛ = 1 − [6 Σd² / n(n²−1)]

d = difference between the ranks of each pair, n = number of pairs. The value of rₛ ranges from −1 (perfect negative correlation) to +1 (perfect positive correlation), with 0 indicating no correlation. Check critical value table: the obtained rₛ must be equal to or greater than the critical value to be significant.

d = 每对数据秩的差值，n = 对子数。rₛ 的取值范围为 −1（完全负相关）到 +1（完全正相关），0 表示无相关。查临界值表：要达到显著，得到的 rₛ 必须等于或大于临界值。

9. Type I and Type II Errors: Decision Matrix | 第一类与第二类错误：决策矩阵

Understanding errors is essential for evaluating the validity of psychological conclusions. A Type I error is a false positive, while a Type II error is a false negative. When you comment on significance in the exam, you should acknowledge that the 5% level still carries a risk of error, and that sample size can affect the probability of a Type II error.

理解错误对于评估心理学结论的有效性至关重要。第一类错误是假阳性，第二类错误是假阴性。在考试中评论显著性时，你应当意识到 5% 水平仍然存在错误风险，且样本量会影响第二类错误的概率。

Decision	H₀ is true	H₀ is false
Reject H₀	Type I error	Correct decision
Fail to reject H₀	Correct decision	Type II error

When the level of significance is made more stringent (e.g., p ≤ 0.01), the risk of a Type I error decreases, but the risk of a Type II error increases. You may refer to this trade-off when discussing why psychology uses p ≤ 0.05 as a convention.

当显著性水平更为严格时（如 p ≤ 0.01），第一类错误的风险降低，但第二类错误的风险增加。你可以在讨论为什么心理学通常采用 p ≤ 0.05 这一惯例时提及这一权衡。

10. Key Principles: Weber’s Law and the Yerkes–Dodson Law | 关键原理：韦伯定律与耶克斯–多德森定律

Two psychological laws that frequently appear in CAIE core studies and exam questions are Weber’s Law (perception) and the Yerkes–Dodson Law (arousal and performance). Although there are no calculations involved, knowing these principles aids in designing and evaluating studies.

两个常在 CAIE 核心研究和考题中出现的心理学定律是韦伯定律（知觉）和耶克斯–多德森定律（唤醒与表现）。虽然不涉及计算，但掌握这些原理有助于设计和评价研究。

Weber’s Law: The just noticeable difference (JND) between two stimuli is a constant proportion of the original stimulus. It helps explain detection of change in psychophysics, and can be applied to studies on subliminal perception or change blindness.

韦伯定律：两个刺激之间的最小可觉差 (JND) 是原始刺激的一个恒定比例。它有助于解释心理物理学中对变化的检测，可应用于潜意识知觉或变化盲视研究。

Yerkes–Dodson Law: Performance increases with physiological or mental arousal, but only up to a point. When arousal becomes too high, performance decreases. The optimal level of arousal is lower for difficult tasks and higher for simple tasks. It is useful when evaluating studies like the Dement and Kleitman dream research or any study involving stress and memory.

耶克斯–多德森定律：表现随生理或心理唤醒水平的提高而提高，但只到一定程度。当唤醒水平过高时，表现下降。困难任务的最佳唤醒水平较低，简单任务则较高。它在评价如 Dement 和 Kleitman 的梦境研究或任何涉及压力与记忆的研究时十分有用。

11. Research Design and Sampling Quick Reference | 研究设计与抽样速查

In the CAIE exam, you are often asked to identify or justify a design: independent measures, repeated measures, or matched pairs. Each has strengths and weaknesses regarding participant variables, order effects, and demand characteristics. Sampling methods such as opportunity, random, volunteer, and stratified sampling must also be linked to representativeness and bias.

在 CAIE 考试中，常要求你识别或论证某一种设计：独立测量、重复测量或配对组设计。每种设计在参与者变量、顺序效应和需求特征方面各有优劣。抽样方法如机会抽样、随机抽样、志愿者抽样和分层抽样也需与代表性和偏差相联系。

Independent measures: different participants in each condition; avoids order effects, but may need more people and can have participant variables. | 独立测量：每种条件下使用不同的参与者；避免顺序效应，但可能需要更多人且存在参与者变量。
Repeated measures: same participants in all conditions; controls participant variables, but order effects must be controlled (e.g., counterbalancing). | 重复测量：所有条件下都是相同的参与者；控制参与者变量，但必须控制顺序效应（如平衡法）。
Matched pairs: pairs of participants matched on a relevant trait then split; reduces participant variables and avoids order effects, but matching is imperfect. | 配对组：根据相关特质将参与者配对，再分入不同条件；减少参与者变量且避免顺序效应，但匹配并不完美。

When writing about sampling, always consider whether the sample can be generalised to the target population. In core studies, many samples are opportunity samples from a limited demographic, so representativeness is often low.

在描述抽样时，始终思考样本是否能推广到目标人群。在核心研究中，许多样本是来自有限人群的机会样本，因此代表性往往较低。

12. Ethical and Validity Considerations | 伦理与效度考量

All core studies must be evaluated against ethical guidelines: informed consent, right to withdraw, confidentiality, protection from harm, debriefing, and deception. When a study involves deception (e.g., Milgram, Laney et al.), you must consider whether the deception was justified and whether adequate debriefing was provided.

所有核心研究都必须对照伦理准则加以评价：知情同意、退出权、保密、免受伤害、事后解释以及欺骗。当研究涉及欺骗时（如 Milgram、Laney 等人的研究），你必须考虑欺骗是否合理，以及是否提供了充分的事后解释。

Validity refers to whether a study measures what it intends to measure. Internal validity concerns whether the IV truly caused the change in the DV (control of confounds). External validity, including ecological and population validity, asks whether findings can be generalised. Use these terms precisely in your answers, and link them to specific features of the study design.

效度指研究是否测量了它想要测量的东西。内部效度关注是否确实是自变量导致了因变量的变化（混淆变量的控制）。外部效度，包括生态效度和人群效度，是指研究结果能否被推广。在你的答案中要精确使用这些术语，并将它们与研究设计的具体特征相联系。

Reliability is consistency. If a measurement is repeated under the same conditions, it should produce similar results. In CAIE, you can discuss inter-rater reliability for observational studies, and standardisation of procedures to improve experimental reliability.

信度即一致性。如果在同样条件下重复测量，应该得到类似的结果。在 CAIE 中，你可以讨论观察研究中的评分者间信度，以及通过程序标准化来提高实验信度。

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Year 12 CAIE Psychology: Quick Reference to Formulas and Key Principles | Year 12 CAIE 心理学：公式定理速查手册