📚 Year 12 CIE Psychology: Formula & Theorem Quick Reference Guide | Year 12 CIE 心理学：公式定理速查手册
Mastering the essential statistical formulas and psychological laws is a cornerstone of the CIE AS Level Psychology (9990) syllabus. Whether you are analysing data from core studies or tackling research methods questions, being able to recall and apply these formulas accurately will give you confidence and raise your marks. This quick reference guide brings together all the key formulas, from measures of central tendency to inferential tests, alongside classic psychological theorems that are frequently examined. Use it to consolidate your revision, check your understanding, and ensure you are fully prepared for Paper 1 and Paper 2.
掌握核心统计公式和心理定律是 CIE AS 阶段心理学 (9990) 大纲的基础。无论你是在分析核心研究的数据，还是应对研究方法类的考题，能够准确地回忆并运用这些公式都将增强你的信心并提高分数。这本速查手册汇总了所有关键公式——从集中趋势测量到推断性检验，并搭配了考试中经常出现的经典心理学定理。用它来巩固复习、查漏补缺，确保你为卷一和卷二的考试做好充分准备。
1. Measures of Central Tendency | 集中趋势测量
The mean is the arithmetic average, calculated by summing all scores and dividing by the number of scores. It is the most sensitive measure because it uses every value in the data set, but it can be heavily distorted by outliers.
平均数是指算术平均值，即所有分数之和除以分数的个数。它是最灵敏的测量指标，因为它使用了数据集中的每一个数值，但极端值可能使其严重失真。
Mean (x̄) = Σx / n
The median is the middle value when scores are arranged in ascending order. If there is an even number of scores, the median is the average of the two middle values. The median is unaffected by extreme scores and is particularly useful for ordinal data or skewed distributions.
中位数是将所有分数按升序排列后位于中间位置的数值。如果有偶数个分数，则中位数是中间两个数的平均值。中位数不受极端值的影响，对于顺序数据或偏态分布尤其有用。
The mode is the most frequently occurring score in a data set. A set can have one mode, more than one mode (bimodal or multimodal), or no mode at all. 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。一组数据可能有一个众数、多个众数（双众数或多众数），也可能没有众数。众数是唯一适用于名义（分类）数据的集中趋势指标。
2. Measures of Dispersion | 离散程度测量
The range is the simplest measure of dispersion, found by subtracting the lowest score from the highest score. It gives a quick sense of spread but is highly sensitive to a single extreme value.
极差是最简单的离散程度测量指标，用最高分减去最低分得到。它能迅速反映数据的散布情况，但对单个极端数值高度敏感。
Range = highest score – lowest score
The variance and standard deviation describe how much scores deviate from the mean. A small standard deviation tells you the data points cluster closely around the mean; a large one indicates they are widely spread. For the AS examination you may be asked to calculate or interpret a standard deviation.
方差和标准差描述的是分数偏离平均数的程度。标准差小，说明数据点紧密聚集在平均数周围；标准差大，说明数据分布很宽。在 AS 考试中，你可能会被要求计算或解释标准差。
3. The Standard Deviation | 标准差
The standard deviation is the square root of the variance. When working with a sample (as in almost all psychology studies), the formula uses n–1 as the divisor to give an unbiased estimate of the population standard deviation.
标准差是方差的平方根。当以样本为对象时（几乎所有心理学研究都是如此），公式使用 n–1 作为分母，从而给出总体标准差的无偏估计。
s = √[Σ(x – x̄)² / (n – 1)]
Where s = sample standard deviation, x = each individual score, x̄ hat a given standard deviation reveals about the consistency of a participant’s performance.
其中 s = 样本标准差，x = 每个单独的分数，x̄ = 样本平均数，n = 分数个数。在 CIE 考试中，你可能需要将数值代入这个公式然后求平方根，或者评价一个给定的标准差究竟揭示了参与者表现的一致性如何。
4. The Chi-Square Test (χ²) | 卡方检验
Chi-square is a test of association for independent (unrelated) designs with nominal data. It compares observed frequencies with the frequencies we would expect if the null hypothesis were true, and determines whether any difference is large enough to be statistically significant.
卡方检验是一种针对名义数据和独立（无关）设计的关联性检验。它把观察到的频数与在虚无假设为真的前提下所期望的频数进行比较，从而判断差异是否大到具有统计显著性。
χ² = Σ (O – E)² / E
y the column total and dividing by the grand total. The larger the χ² value, the more likely the result is to be significant. The critical value depends on the degrees of freedom (df), where df = (number of rows – 1) × (number of columns – 1).
O 代表每个单元格中的观察频数，E 代表期望频数。期望频数的计算方法是行总计乘以列总计再除以总合计。χ² 值越大，结果显著的可能性越高。临界值取决于自由度 (df)，df = (行数 – 1) × (列数 – 1)。
5. The Sign Test | 符号检验
The Sign Test is a non-parametric inferential test used for repeated measures or matched pairs designs when the data are nominal or when the investigator simply records the direction of a difference (plus or minus). It requires no assumptions about the population distribution.
符号检验是一种非参数推断性检验，适用于重复测量或配对设计，且数据为名义数据，或者研究者仅记录差异的方向（正或负）。它对总体的分布没有任何假定。
S = number of positive signs (or negative signs, whichever is fewer)
The calculated value S is the smaller of the two totals of plus and minus signs. Zero differences are omitted from the analysis. The obtained S is then compared against a critical value from a Sign Test table for the relevant N (number of pairs). For statistical significance the obtained value must be equal to or less than the critical value.
计算值 S 取正号总数和负号总数中较少的那一个。零差异从分析中剔除。然后将得到的 S 值与符号检验表中对应 N（配对数目）的临界值进行比较。要达到统计显著性，所得的 S 值必须小于或等于临界值。
6. Wilcoxon Signed-Rank Test | 威尔科克森符号秩检验
The Wilcoxon test is used for repeated measures or matched pairs designs with ordinal (or interval) data. It takes into account both the direction and the magnitude of the differences, making it more powerful than the Sign Test.
威尔科克森检验用于顺序（或等距）数据的重复测量或配对设计。它同时考虑了差异的方向和大小，因此比符号检验的统计检验力更强。
T = sum of ranks of the less frequent sign
After calculating the difference for each pair, ignoring zeros, rank the absolute differences from 1 (smallest) to N (largest). Then assign each rank the sign of the original difference. T is the smaller sum of the signed ranks. The obtained T must be equal to or less than the critical T in the table to reject the null hypothesis.
在计算出每一对的差值并忽略零之后，按绝对值大小从 1（最小）到 N（最大）排序。然后给每个秩次赋予原始差值的符号。T 是带符号秩次中总和较小的那个。要拒绝虚无假设，所得的 T 值必须小于或等于表格中的临界 T 值。
7. Mann–Whitney U Test | 曼-惠特尼 U 检验
The Mann–Whitney U test is the independent-groups equivalent of the Wilcoxon test. It is used when an experiment has an independent measures design and the dependent variable yields at least ordinal data.
曼-惠特尼 U 检验是独立组设计的威尔科克森检验对等项。当实验采用独立测量设计，且因变量至少给出顺序数据时，便使用该检验。
U₁ = R₁ – n₁(n₁ + 1)/2
U₂ = R₂ – n₂(n₂ + 1)/2
All scores from both groups are pooled and ranked together. R₁ is the sum of ranks in Group 1, and R₂ is the sum of ranks in Group 2. n₁ and n₂ are the sizes of the two groups. The smaller U value is taken as the test statistic and compared against critical values. If the smaller U is less than or equal to the critical U, the result is significant.
将两组的全部分数合并并统一排序。R₁ 是第一组的秩和，R₂ 是第二组的秩和。n₁ 与 n₂ 是两组的样本量。取较小的 U 值作为检验统计量并与临界值比较。若较小的 U 值小于或等于临界 U 值，结果具有显著性。
8. Spearman's Rank Correlation Coefficient (rₛ) | 斯皮尔曼等级相关系数
Spearman’s rho is a non-parametric test for correlation that works with ordinal data or data that has been ranked. It measures the strength and direction of a monotonic relationship between two co-variables.
斯皮尔曼等级相关系数是一种非参数相关检验，适用于顺序数据或已转换成秩次的数据。它测量的是两个协变量之间单调关系的强度和方向。
rₛ = 1 – (6 Σd²) / (n(n² – 1))
ions, and n = number of pairs. An rₛ value of +1 indicates a perfect positive correlation, – is is rejected if the absolute value of rₛ is greater than or equal to the critical value.
其中 d = 每一对观测值秩次之差，n = 数据对的数量。rₛ 值为 +1 表示完全正相关，–1 表示完全负相关，0 表示没有相关。将得到的 rₛ 与给定 N 下的临界值进行比较。若 rₛ 的绝对值大于或等于临界值，则拒绝虚无假设。
9. Effect Size: Cohen's d | 效应量：科恩 d
Effect size tells us not just whether a result is statistically significant but how large the difference or relationship actually is. Cohen’s d is a widely used measure of effect size for two-group comparisons.
效应量不仅告诉我们一个结果是否具有统计显著性，还告诉我们差异或关系实际上有多大。科恩 d 是一种广泛用于两组比较的效应量指标。
Cohen's d = (x̄₁ – x̄₂) / s pooled
rd deviation of the two groups. By convention, d = 0.2 is a small effect, d = 0.5 a medium effect, and d = 0.8 a large effect.
分子是两个组平均数的差值。分母是合并标准差，大致相当于两组标准差的平均值。按照惯例，d = 0.2 为小效应，d = 0.5 为中等效应，d = 0.8 为大效应。
10. 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 relationship is crucial for understanding sensory thresholds.
韦伯定律指出，两个刺激之间的最小可觉差 (JND) 是原刺激强度的一个恒定比例。这一关系对于理解感觉阈限至关重要。
k = ΔI / I
y modalities (e.g., around 0.02 for light intensity, 0.15 for loudness). Fechner later extended this into a logarithmic law: S = k log I, where S is the sensation magnitude and I is the physical intensity. Together these laws explain why the perceived brightness of a light does not double when its physical intensity is doubled.
其中 I 为初始刺激强度，ΔI 为可察觉的最小强度变化，k 为韦伯常数，该常数因感觉通道而异（例如，对光强度约为 0.02，对响度约为 0.15）。费希纳后来将其拓展为对数定律：S = k log I，其中 S 是感觉量，I 是物理强度。这两条定律共同解释了为什么一盏灯感觉上并不会在物理强度翻倍时亮度也翻倍。
11. Miller's Magic Number | 米勒的魔法数字
bes the number of chunks of information we can hold in immediate memory without rehearsal.
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。这个神奇的数字 7 ± 2 描述了我们在无复述时能够在即时记忆中保持的信息组块数量。
STM capacity ≈ 7 ± 2 chunks
A ‘chunk’ is a meaningful unit of information, such as a digit, a letter, or a familiar pattern. Chunking allows us to increase the amount of information stored by grouping individual elements into larger, meaningful units. This principle is frequently tested in cognitive psychology questions on the Multi-Store Model of Memory.
一个“组块”是一个有意义的信息单元，例如一个数字、一个字母或一个熟悉的模式。组块化通过将单个元素组合成更大的、有意义的单元，使我们能够增加存储的信息量。这一原理在认知心理学关于多存储记忆模型的题目中经常被考查。
12. Yerkes–Dodson Law | 耶克斯-多德森定律
The Yerkes–Dodson Law describes the relationship between arousal and performance. It posits that performance increases with physiological or mental arousal up to an optimal point, after which further arousal leads to a decline in performance. The curve is often described as an inverted-U.
耶克斯-多德森定律描述了唤醒水平与表现之间的关系。该定律认为，随着生理或心理唤醒水平的提高，表现会提升到一个最佳点，之后唤醒继续升高则导致表现下降。这一曲线常被描绘为一个倒 U 型。
Performance increases moderately with arousal → optimal performance → then decreases
The optimal level of arousal depends on the complexity of the task: simple or well-practised tasks require higher arousal for peak performance, whereas complex or novel tasks benefit from lower arousal. This law is relevant to topics in biological psychology, especially when discussing stress, anxiety, and exam performance.
最佳唤醒水平取决于任务的复杂程度：简单或熟练的任务需要较高的唤醒才能达到最佳表现，而复杂或新颖的任务则从较低的唤醒中受益。这条定律与生物心理学的若干主题相关，尤其在探讨压力、焦虑和考试表现时。
Published by TutorHao | Psychology Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图