📚 Year 12 CAIE Psychology Formula & Theorem Quick Reference | Year 12 CAIE 心理学公式定理速查手册
Psychology at Year 12 under the CAIE syllabus blends scientific methodology with statistical reasoning. While the subject is known for case studies and theories, quantitative competence is crucial. This handbook covers the essential formulas, statistical tests, and mathematical relationships underpinning key psychological concepts—from measures of central tendency and dispersion to inferential tests like the chi-squared, Mann–Whitney U, and Spearman’s rank correlation. Each entry is presented in a clear, exam-oriented format with both English and Chinese explanations, so you can quickly locate and revise the mathematical tools you need for Papers 1–4.
Year 12 CAIE 心理学将科学方法与统计推理相结合。虽然这门学科以案例研究和理论闻名，但定量能力同样至关重要。本手册涵盖了核心心理学概念背后的基本公式、统计检验和数学关系——从集中趋势和离散程度的测量到推理性检验，如卡方检验、曼-惠特尼U检验和斯皮尔曼等级相关。每个条目都以清晰的、面向考试的形式呈现，配有中英文解释，让你可以快速定位和复习论文1–4所需的数学工具。
1. Measures of Central Tendency | 集中趋势的测量
The mean is the arithmetic average, calculated by summing all scores and dividing by the number of scores. It uses all data but is sensitive to extreme outliers. The formula is:
均值是算术平均数，通过将所有分数相加再除以分数个数计算得出。它使用了所有数据，但对极端异常值敏感。公式为：
x̄ = Σx / N
where Σx is the sum of all scores and N is the number of scores.
其中 Σx 是所有分数的总和，N 是分数的个数。
The median is the middle value when data are ordered. For an odd number of scores, it is the central score; for an even number, it is the mean of the two middle scores. The median is robust against outliers, making it better for skewed data.
中位数是数据排序后位于中间的值。当分数个数为奇数时，它是正中间的分数；为偶数时，它是中间两个分数的均值。中位数对异常值不敏感，因此更适合偏态数据。
The mode is the most frequently occurring score in a dataset. A dataset can have no mode, one mode (unimodal), or multiple modes (bimodal/multimodal). The mode is the only measure suitable for nominal (categorical) data.
众数是数据集中出现频率最高的分数。一个数据集可能没有众数、只有一个众数（单峰）或多个众数（双峰/多峰）。众数是唯一适用于名义（分类）数据的集中趋势测量。
2. Measures of Dispersion | 离散程度的测量
The range is the difference between the highest and lowest scores: Range = Xmax – Xmin. It gives a quick sense of spread but ignores the distribution’s shape and can be inflated by outliers.
极差是最高分与最低分之间的差值：极差 = Xmax – Xmin。它能快速反映数据跨度，但忽略了分布形态，且可能被异常值夸大。
Variance measures the average squared deviation from the mean. The formula for sample variance (s²) is:
方差测量的是各数据与均值之差的平方的平均数。样本方差（s²）的公式为：
s² = Σ(xᵢ – x̄)² / (N – 1)
where xᵢ is each individual score, x̄ is the sample mean, and N is the sample size. We divide by N–1 to give an unbiased estimate of the population variance.
其中 xᵢ 为每个个体分数，x̄ 为样本均值，N 为样本容量。我们除以 N–1 以得到总体方差的无偏估计。
Standard deviation (s) is the square root of variance: s = √s². It is expressed in the original measurement units and is the most widely reported measure of spread in psychological research.
标准差（s）是方差的平方根：s = √s²。它以原始测量单位表示，是心理学研究中最常报告的离散程度指标。
3. Normal Distribution and Standard Scores | 正态分布与标准分数
In a normal distribution, 68% of observations fall within ±1 standard deviation of the mean, 95% within ±1.96 SD, and 99.7% within ±3 SD. This property is used to identify unusual scores and to perform parametric tests.
在正态分布中，68% 的观测值落在均值±1个标准差之内，95%落在±1.96个标准差之内，99.7%落在±3个标准差之内。这一特性用于识别异常值并进行参数检验。
A z-score represents how many standard deviations a raw score is from the mean: z = (x – x̄) / s. It allows comparison across different distributions. Standardisation is essential when combining scores from different tests.
z分数表示一个原始分数距离均值多少个标准差：z = (x – x̄) / s。它允许在不同分布之间进行比较。在合并不同测试的分数时，标准化是必不可少的。
However, CAIE candidates are rarely asked to compute z-scores for large data sets by hand. Understanding the principle is more important: a z-score of +1.65 corresponds to the top 5% of the distribution (one-tailed).
然而，CAIE 考生很少被要求手工计算大数据集的 z 分数。理解原理更为重要：z 分数为+1.65 对应分布中前 5% 的位置（单尾）。
4. Correlation: Spearman’s Rank | 相关：斯皮尔曼等级相关
Spearman’s rho (ρ) is a non-parametric test for correlation between two sets of ranks. It is used when data are ordinal or when assumptions for Pearson’s r are violated. The formula is:
斯皮尔曼等级相关系数（ρ）是一种用于两组等级数据之间相关性的非参数检验。当数据为顺序数据或违反皮尔逊 r 的假设时使用。公式为：
ρ = 1 – (6 Σd²) / (N(N² – 1))
where d is the difference between the ranks of each pair, and N is the number of paired observations. The value falls between –1 (perfect negative correlation) and +1 (perfect positive correlation).
其中 d 是每对数据的等级差值，N 是配对观测的个数。该值介于 –1（完全负相关）和 +1（完全正相关）之间。
After calculating ρ, you compare it against a critical value table with N in the left column and your significance level (usually p < 0.05). If ρ (ignoring sign) is equal to or greater than the critical value, the correlation is statistically significant.
计算出 ρ 后，将其与临界值表进行比较，左边一列为 N，并选择显著性水平（通常 p < 0.05）。若 ρ 的绝对值（忽略符号）等于或大于临界值，则相关性具有统计学显著性。
5. Chi-Squared Test | 卡方检验
The chi-squared (χ²) test is used with independent groups and nominal (category) data to examine whether there is a significant association between two variables, or a difference between observed and expected frequencies. The formula is:
卡方（χ²）检验用于独立组别和名义（分类）数据，以检验两个变量之间是否存在显著关联，或观测频数与期望频数之间的差异。公式为：
χ² = Σ (O – E)² / E
where O is the observed frequency and E is the expected frequency for each cell. Expected frequency is calculated as (row total × column total) / grand total.
其中 O 为每个单元格的观测频数，E 为期望频数。期望频数计算公式为（行合计 × 列合计）/ 总计。
Degrees of freedom (df) for a contingency table is (number of rows – 1) × (number of columns – 1). Compare the calculated χ² with the critical value from the chi-squared distribution table. If χ² ≥ critical value, reject the null hypothesis.
列联表的自由度（df）为（行数 – 1）×（列数 – 1）。将计算出的 χ² 与卡方分布表中的临界值进行比较。若 χ² ≥ 临界值，则拒绝零假设。
Chi-squared applies only to frequency data, not percentages or scores. Each expected frequency should be at least 5 for the test to be valid.
卡方检验仅适用于频数数据，而非百分比或分数。每个期望频数应至少为 5，检验才有效。
6. 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 when data are at least ordinal and the design is independent measures. It tests whether two independent groups come from the same population. The formula involves two U values:
曼-惠特尼U检验是独立样本 t 检验的非参数替代方法，适用于数据至少为顺序数据且设计为独立测量时。它用于检验两个独立组是否来自同一总体。公式涉及两个 U 值：
U₁ = N₁N₂ + (N₁(N₁ + 1))/2 – R₁
U₂ = N₁N₂ – U₁
where 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.
其中 N₁ 和 N₂ 为样本容量，R₁ 为组 1 的秩和。U₁ 和 U₂ 中较小的值即为检验统计量 U。
You compare U against a critical value table for given sample sizes. If U is equal to or less than the critical value, the difference is significant (p < 0.05). Note: unlike many tests, a smaller U indicates a greater difference.
将 U 与给定样本容量的临界值表进行比较。若 U 等于或小于临界值，则差异显著（p < 0.05）。注意：与许多检验不同，U 值越小表示差异越大。
Common CAIE exam trick: ensure you rank all data from both groups together, assigning rank 1 to the lowest score. Use mean ranks for tied scores.
CAIE 考试常见技巧：确保将两组所有数据放在一起排序，最低分给秩 1。并列分数使用平均秩。
7. Wilcoxon Signed-Ranks Test | 威尔科克森符号秩检验
This test is used for a repeated measures or matched-pairs design when the data are at least ordinal. It assesses whether there is a significant difference between two related conditions. The formula for the test statistic T is:
该检验用于重复测量或配对设计，数据至少为顺序数据。它评估两个相关条件之间是否存在显著差异。检验统计量 T 的计算步骤为：
Calculate the difference between each pair, ignore zero differences, rank the absolute differences (ignoring sign), then sum the ranks of the less frequent sign. That sum is T.
计算每对数据之间的差值，忽略零差值，对差值的绝对值进行排序（忽略符号），然后将较少见的符号的秩相加。这个总和就是 T。
T is compared with a critical value table. For significance, the calculated T must be less than or equal to the critical T. This is the opposite of most test statistics (smaller T means a more significant difference).
将 T 与临界值表进行比较。要达到显著性，计算出的 T 必须小于或等于临界 T。这与大多数检验统计量相反（T 越小表示差异越显著）。
Always remember to state N as the number of non-zero difference scores. This adjusted N is used for looking up critical values.
始终记住将 N 报告为非零差值分数的个数。这个调整后的 N 用于查阅临界值。
8. Standard Error and Confidence Intervals | 标准误和置信区间
The standard error of the mean (SEM) estimates the variability of sample means around the population mean. It is calculated as:
均值的标准误（SEM）估计样本均值围绕总体均值的变异性。计算公式为：
SEM = s / √N
where s is the sample standard deviation and N is sample size. SEM decreases as sample size increases, reflecting greater precision.
其中 s 为样本标准差，N 为样本容量。随着样本容量的增加，SEM 减小，反映出更高的精确度。
A 95% confidence interval for the mean is typically given by:
均值的 95% 置信区间通常由下式给出：
CI = x̄ ± (z × SEM)
For a 95% CI, the z-value is 1.96 (if population standard deviation is known) or use the t-distribution value when N is small.
对于 95% 置信区间，z 值为 1.96（当总体标准差已知时），或在 N 较小时使用 t 分布值。
Understanding SEM and CIs is crucial for evaluating the trustworthiness of sample data. Wider intervals suggest less precision and greater uncertainty.
理解 SEM 和置信区间对于评估样本数据的可信度至关重要。较宽的区间表明精度较低且不确定性更大。
9. Effect Size: Cohen’s d | 效应量：科恩 d 值
Effect size quantifies the magnitude of a difference, independent of sample size. Cohen’s d for an independent groups design is:
效应量量化差异的大小，与样本容量无关。独立组设计的科恩 d 值计算公式为：
d = (M₁ – M₂) / s_pooled
where M₁ and M₂ are the group means, and s_pooled is the pooled standard deviation. In simple terms, it expresses the difference in standard deviation units.
其中 M₁ 和 M₂ 是两组的均值，s_pooled 是合并标准差。简而言之，它以标准差为单位表示差异。
Cohen’s guidelines: d = 0.2 small, 0.5 medium, 0.8 large. Reporting effect size alongside p-values is becoming standard practice in psychology to avoid over-reliance on significance testing.
科恩的参考标准：d = 0.2 为小效应，0.5 为中等效应，0.8 为大效应。在心理学中，同时报告效应量和 p 值正在成为标准做法，以避免过度依赖显著性检验。
Although CAIE does not always require computation of d, knowing the concept helps in evaluating research findings and answering questions about practical significance.
尽管 CAIE 并不总是要求计算 d 值，但了解这一概念有助于评估研究发现，并回答关于实际显著性的问题。
10. Weber–Fechner Law | 韦伯-费希纳定律
Ernst Weber’s law states that the just noticeable difference (JND) between two stimuli is a constant proportion of the original stimulus intensity. It is expressed as:
恩斯特·韦伯定律指出，两个刺激之间的最小可觉差（JND）是原始刺激强度的恒定比例。其表达式为：
k = ΔI / I
where I is the initial stimulus intensity, ΔI is the increment needed for a JND, and k is the Weber fraction. For example, k for lifted weights is about 0.02.
其中 I 为初始刺激强度，ΔI 为产生最小可觉差所需的增量，k 为韦伯分数。例如，提举重物的 k 值约为 0.02。
Fechner extended this into a logarithmic relationship: perceived sensation (S) = k log(I). This means that equal ratios of physical intensity produce equal intervals of perceived intensity, explaining why adding one candle to a brightly lit room is less noticeable than to a dark room.
费希纳将其扩展为对数关系：感觉量（S）= k log(I)。这意味着物理强度的等比例变化会产生等间隔的感知强度变化，这解释了为什么在明亮的房间里加一支蜡烛不如在黑暗的房间里加一支蜡烛那么明显。
These equations underpin the psychophysical study of perception. While not heavily formula-based in exams, the proportional and logarithmic principles are often examined in the context of thresholds and sensory adaptation.
这些方程是心理物理学对知觉研究的基础。虽然考试中不会大量涉及公式，但比例和对数原理经常在阈限和感觉适应的情境中被考查。
11. Signal Detection Theory: Key Equations | 信号检测论：关键方程
Signal detection theory (SDT) separates a perceiver’s sensitivity from their response bias. The sensitivity index d’ (d-prime) is calculated as:
信号检测论（SDT）将感知者的感受性与反应偏差分离开来。敏感性指标 d’（d 素数）的计算公式为：
d’ = z(hit) – z(false alarm)
where z(hit) is the z-score for the hit rate (correctly identifying a signal) and z(false alarm) is the z-score for the false alarm rate (saying ‘yes’ when no signal is present). Larger d’ indicates better sensitivity.
其中 z(hit) 是击中率（正确识别出信号）的 z 分数，z(false alarm) 是虚报率（没有信号却说‘是’）的 z 分数。d’ 越大表示敏感性越好。
The response criterion (c) is given by c = –0.5[z(hit) + z(false alarm)]. It measures bias: negative c indicates a liberal strategy (saying ‘yes’ easily), positive c indicates a conservative strategy.
反应标准（c）的公式为 c = –0.5[z(hit) + z(false alarm)]。它测量偏差：负的 c 表示宽松策略（轻易说‘是’），正的 c 表示保守策略。
While students do not need to compute d’ in CAIE exams, understanding that performance can be broken into two independent components—sensitivity and bias—is key for evaluating eyewitness testimony studies and vigilance tasks.
虽然 CAIE 考试不要求学生计算 d’，但理解表现可以分解为两个独立成分——敏感性和偏差——对于评估目击者证词研究和警戒任务的研究至关重要。
12. Percentages and Probability in Research | 研究中的百分比和概率
Psychological studies frequently report percentages to describe participant demographics or behavioural outcomes. The formula for a percentage is (part / whole) × 100. For instance, if 15 out of 60 participants recalled a word, the recall rate is (15/60)×100 = 25%.
心理学研究经常用百分比来描述参与者的人口统计学特征或行为结果。百分比的公式是（部分 / 整体）× 100。例如，如果 60 名参与者中有 15 人回忆出一个词，则回忆率为 (15/60)×100 = 25%。
Probability (p) is central to inferential statistics. A p-value is the probability of obtaining the observed results (or more extreme) if the null hypothesis were true. The conventional significance threshold is p ≤ 0.05, meaning fewer than 5 chances in 100 that the result is a fluke.
概率（p）是推断统计学的核心。p 值是在零假设为真的前提下，获得观测结果（或更极端结果）的概率。传统的显著性阈值是 p ≤ 0.05，即结果纯属偶然的可能性低于 5/100。
Understanding that p is a conditional probability—it does not indicate the probability that the null hypothesis is true—is a critical nuance. The Type I error rate is exactly the significance level α, while Type II error (β) relates to statistical power (1 – β).
理解 p 是一个条件概率——它并不表示零假设为真的概率——这一点至关重要。I 类错误率正是显著性水平 α，而 II 类错误（β）则与统计功效（1 – β）相关。
When interpreting academic articles, students should also note effect sizes, confidence intervals, and replication issues, moving beyond binary significant/non-significant thinking.
在解读学术文章时，学生还应关注效应量、置信区间和可重复性问题，超越二元显著/不显著的思维。
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