📚 Pre-U CAIE Psychology: Formula & Theorem Quick Reference Handbook | Pre-U CAIE 心理学：公式定理速查手册
This handbook provides a concise reference to the key mathematical formulas, statistical tests, and psychophysical laws you need for the CAIE Pre-U Psychology course. From Weber’s law to t-tests, each entry includes the standard notation, worked application, and connection to psychological research. Use it to reinforce your understanding of research methods and to check your calculations when analysing data.
本手册为你在CAIE Pre-U心理学课程中所需的关键数学公式、统计检验和心理物理学定律提供简明参考。从韦伯定律到t检验，每个条目都包含了标准符号、应用示例以及与心理学研究的关联。用它来巩固你对研究方法的理解，并在分析数据时检验你的计算。
1. Weber’s Law | 韦伯定律
Weber’s law describes the relationship between the physical intensity of a stimulus and the change required for an observer to notice a difference. The just noticeable difference (JND) is a constant proportion of the initial stimulus intensity. The formula is ΔI / I = k, where ΔI is the increment needed to produce a JND, I is the original intensity, and k is the Weber fraction. For example, the Weber fraction for lifted weights is approximately 0.02 – a 2% change is detectable.
韦伯定律描述了刺激的物理强度与观察者察觉差异所需变化之间的关系。最小可觉差（JND）是初始刺激强度的一个恒定比例。公式为ΔI / I = k，其中ΔI是产生JND所需的增量，I是原始强度，k是韦伯分数。例如，提举重量的韦伯分数约为0.02，即2%的变化可被察觉。
ΔI / I = k
2. Fechner’s Law | 费希纳定律
Fechner built on Weber’s work to propose that sensation magnitude increases as the logarithm of stimulus intensity. The equation S = k log I states that perceived sensation (S) is proportional to the log of physical intensity (I), with k being a scaling constant. This means equal ratios of intensity produce equal increments of sensation. The law holds well for moderate stimulus ranges, such as brightness or loudness estimates.
费希纳在韦伯研究的基础上提出，感觉强度随刺激强度的对数增加。方程S = k log I表明，感知到的感觉（S）与物理强度（I）的对数成比例，k为比例常数。这意味着相等的强度比率会产生相等的感觉增量。该定律在中等刺激范围内（例如亮度或响度估计）吻合良好。
S = k log I
3. Stevens’ Power Law | 斯蒂文斯幂定律
Stevens refined psychophysical scaling by proposing a power relationship: S = k Iⁿ, where n is an exponent that depends on the sensory modality. For electric shock, n is about 3.5 (sensation grows rapidly), while for brightness, n is around 0.33. This law accounts for both over- and under-estimation of intensity changes and is supported by magnitude estimation experiments.
斯蒂文斯通过提出幂关系对心理物理测量进行了完善：S = k Iⁿ，其中指数n取决于感觉通道。电击的n约3.5（感觉增长迅速），而亮度的n约0.33。该定律既能解释强度变化的高估也能解释低估，并得到数量估计实验的支持。
S = k Iⁿ
4. Yerkes-Dodson Law | 耶克斯–多德森定律
The Yerkes-Dodson law states that performance increases with arousal up to an optimal point, after which further arousal leads to a decline in performance. The relationship forms an inverted U-shaped curve. The optimal level of arousal depends on task difficulty: lower arousal is better for difficult tasks, while higher arousal suits simple or well-learned tasks. No single algebraic equation fully captures this law, but it is central to understanding stress and motivation in cognitive psychology.
耶克斯–多德森定律指出，表现随唤醒水平提高而提升，直到达到最佳点，之后进一步唤醒会导致表现下降。这种关系呈倒U型曲线。最佳唤醒水平取决于任务难度：困难任务在较低唤醒时表现最佳，而简单或熟练的任务需要较高唤醒。虽然没有单一代数方程能完全描述该定律，但它在认知心理学中对于理解压力与动机至关重要。
5. Descriptive Statistics: Measures of Central Tendency and Dispersion | 描述统计：集中趋势与离散程度
The three main measures of central tendency are the mean (x̄ = Σx / n), the median (middle value when ordered), and the mode (most frequent value). Dispersion is described by the range (max − min), variance (s² = Σ(x − x̄)² / (n − 1)), and standard deviation (s = √s²). The standard deviation is the most informative as it uses all data points and is expressed in the original units. In psychology, these statistics summarise data from experiments or surveys before inferential testing.
三种主要的集中趋势量度是均值（x̄ = Σx / n）、中位数（排序后的中间值）和众数（出现频率最高的值）。离散程度由全距（最大值 − 最小值）、方差（s² = Σ(x − x̄)² / (n − 1)）和标准差（s = √s²）来描述。标准差因使用了所有数据点且以原始单位表示而最有信息量。在心理学中，这些统计量在推断检验之前用于概括实验或调查数据。
s = √[Σ(x − x̄)² / (n − 1)]
6. Z-Scores and the Normal Distribution | Z分数与正态分布
A z-score expresses how many standard deviations a raw score is from the population mean: z = (x − μ) / σ. For a sample, z = (x − x̄) / s. Z-scores allow comparisons across different distributions and are fundamental to determining probabilities under the normal curve. In psychology, z-scores are used to standardise test scores, identify outliers, and calculate the proportion of participants scoring above or below a given value.
Z分数表示原始分数距离总体均值多少个标准差：z = (x − μ) / σ。对于样本，z = (x − x̄) / s。Z分数使得跨不同分布的比较成为可能，并且是确定正态曲线下概率的基础。在心理学中，z分数用于标准化测验分数、识别离群值以及计算得分高于或低于某给定值的参与者比例。
z = (x − μ) / σ
7. Independent-Samples t-Test | 独立样本t检验
The independent-samples t-test compares the means of two separate groups. The formula is t = (x̄₁ − x̄₂) / √[sₚ² (1/n₁ + 1/n₂)], where sₚ² is the pooled variance: sₚ² = [(n₁−1)s₁² + (n₂−1)s₂²] / (n₁ + n₂ − 2). The degrees of freedom are df = n₁ + n₂ − 2. If the calculated t exceeds the critical value at a chosen alpha level (e.g., 0.05), the null hypothesis of no difference is rejected. Pre-U psychology exam questions often require calculating t and interpreting the result in the context of an experiment.
独立样本t检验比较两个独立组的均值。公式为t = (x̄₁ − x̄₂) / √[sₚ² (1/n₁ + 1/n₂)]，其中sₚ²是合并方差：sₚ² = [(n₁−1)s₁² + (n₂−1)s₂²] / (n₁ + n₂ − 2)。自由度为df = n₁ + n₂ − 2。若计算出的t值大于选定α水平（如0.05）的临界值，则拒绝无差异的零假设。Pre-U心理学考试常要求计算t值并在实验情境中解释结果。
t = (x̄₁ − x̄₂) / √[sₚ² (1/n₁ + 1/n₂)]
sₚ² = [(n₁−1)s₁² + (n₂−1)s₂²] / (n₁ + n₂ − 2)
8. Related-Samples t-Test | 相关样本t检验
When the same participants are tested twice (repeated measures) or matched pairs are used, the related-samples t-test applies. Compute the difference score d for each pair, then t = d̄ / (s_d / √n), where d̄ is the mean difference and s_d is the standard deviation of the differences. The degrees of freedom are n − 1 (n = number of pairs). This test is more powerful than the independent test because it controls for individual differences.
当相同参与者被测试两次（重复测量）或使用配对设计时，适用相关样本t检验。计算每对的差值d，然后t = d̄ / (s_d / √n)，其中d̄是差值的均值，s_d是差值的标准差。自由度为n − 1（n为对数）。该检验比独立样本检验更有效，因为它控制了个体差异。
t = d̄ / (s_d / √n)
9. Chi-Square Tests | 卡方检验
The chi-square formula is χ² = Σ[(O − E)² / E], where O is the observed frequency and E is the expected frequency. For goodness-of-fit, E is based on a theoretical distribution (e.g., equal preference). For independence, E is calculated as (row total × column total) / grand total. Degrees of freedom: for goodness-of-fit, df = number of categories − 1; for independence, df = (rows − 1) × (columns − 1). A significant χ² indicates a departure from the expected pattern. In psychology, it is frequently used with categorical data such as choices or clinical diagnoses.
卡方公式为χ² = Σ[(O − E)² / E]，其中O为观察频数，E为期望频数。拟合优度检验中，E基于理论分布（如等偏好）。独立性检验中，E按（行合计 × 列合计）/ 总计计算。自由度：拟合优度中，df = 类别数 − 1；独立性检验中，df = (行数 − 1) × (列数 − 1)。显著的χ²值表明观察结果偏离了期望模式。心理学中，卡方检验常用于选择或临床诊断等类别数据。
χ² = Σ[(O − E)² / E]
10. Correlation Coefficients | 相关系数
Pearson’s r measures the linear relationship between two interval/ratio variables: r = Σ[(x − x̄)(y − ȳ)] / √[Σ(x − x̄)² Σ(y − ȳ)²]. Spearman’s rho (ρ) is used for ordinal data or when assumptions are violated; it ranks the data and then applies: ρ = 1 − [6 Σd² / (n(n² − 1))], where d is the difference between ranks. Both range from −1 to +1. In Pre-U, you must select the appropriate coefficient based on the level of measurement and interpret the strength and direction of the relationship.
皮尔逊积差相关系数r度量两个等距/等比变量间的线性关系：r = Σ[(x − x̄)(y − ȳ)] / √[Σ(x − x̄)² Σ(y − ȳ)²]。斯皮尔曼等级相关系数ρ用于顺序数据或违反假设时；先对数据排序，然后应用：ρ = 1 − [6 Σd² / (n(n² − 1))]，其中d为等级差。两者取值范围均为−1到+1。Pre-U考试中，你必须根据测量水平选择合适的系数，并解释关系的强度与方向。
r = Σ[(x − x̄)(y − ȳ)] / √[Σ(x − x̄)² Σ(y − ȳ)²]
ρ = 1 − [6 Σd² / (n(n² − 1))]
11. Effect Size: Cohen’s d | 效应量：科恩d值
Cohen’s d quantifies the size of the difference between two means independent of sample size: d = (M₁ − M₂) / s_pooled, where s_pooled is typically the pooled standard deviation. Conventionally, d = 0.2 is small, 0.5 medium, and 0.8 large. Reporting d alongside a t-test allows readers to assess the practical significance of a finding. The pooled standard deviation is calculated as √[((n₁−1)s₁² + (n₂−1)s₂²) / (n₁ + n₂ − 2)], mirroring the denominator of the independent t-test.
科恩d值量化了两个均值之间不受样本量影响的差异大小：d = (M₁ − M₂) / s_pooled，其中s_pooled通常是合并标准差。通常认为d = 0.2为小效应，0.5为中等，0.8为大效应。在t检验同时报告d值能让读者评估研究结果的实际显著性。合并标准差计算为√[((n₁−1)s₁² + (n₂−1)s₂²) / (n₁ + n₂ − 2)]，与独立样本t检验的分母相同。
d = (M₁ − M₂) / s_pooled
12. Behavioural Laws: Matching Law and Social Impact Theory | 行为定律：匹配律与社会冲击理论
The Matching Law, introduced by Herrnstein, describes how an organism allocates its behaviour relative to the rate of reinforcement. For two response options, B₁/(B₁ + B₂) = R₁/(R₁ + R₂), where B is the response rate and R is the obtained reinforcement. Social Impact Theory (Latane) proposes that influence in a social setting is a multiplicative function of strength, immediacy, and number of sources: Impact ∝ S × I × N. Both laws provide parsimonious mathematical descriptions of complex behaviour and are examined in Pre-U options like psychology and law or behaviourist applications.
赫恩斯坦提出的匹配律描述了有机体如何根据强化率分配行为。对于两个反应选项，B₁/(B₁ + B₂) = R₁/(R₁ + R₂)，其中B为反应速率，R为获得的强化。社会冲击理论（拉塔内）提出，社会环境中的影响力是来源的强度、接近性和数量的乘积函数：Impact ∝ S × I × N。这两个定律都为复杂行为提供了简洁的数学描述，并在Pre-U的心理学与法律或行为主义应用等选项中考查。
B₁/(B₁ + B₂) = R₁/(R₁ + R₂)
Impact ∝ S × I × N
Published by TutorHao | Psychology Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导