📚 Pre-U Edexcel Psychology Formula & Theorem Quick Reference Handbook | Pre-U Edexcel 心理学：公式定理速查手册
Psychology is not only about theories and case studies; it also relies on precise mathematical tools to analyse behaviour, test hypotheses, and quantify mental processes. This quick reference handbook gathers the key formulas, statistical tests, and psychophysical laws that every Pre-U Edexcel Psychology student needs to master. From descriptive statistics to decision theory, these equations form the backbone of psychological research methods and provide a common language for evaluating evidence.
心理学不只有理论和案例分析，它还依赖精确的数学工具来分析行为、检验假设并量化心理过程。这份速查手册汇集了每位 Pre-U Edexcel 心理学学生必须掌握的核心公式、统计检验和心理物理学定律。从描述统计到决策理论，这些方程构成了心理学研究方法的支柱，也为评估证据提供了通用语言。
1. Descriptive Statistics: Mean, Median, Mode | 集中趋势：均值、中位数、众数
The mean (M) is the arithmetic average of a set of scores, calculated by summing all values and dividing by the total number of observations N. The median is the middle score when data are arranged in order, and the mode is the most frequently occurring score. These three measures of central tendency summarise the typical or most representative value in a dataset.
均值（M）是一组分数的算术平均数，通过将所有数值相加再除以观测总数 N 得到。中位数是将数据按顺序排列后位于中间位置的分数，众数则是出现频率最高的分数。这三种集中趋势量度概括了数据集中最典型或最具代表性的数值。
M = Σx ÷ N
The formula for the mean uses the Greek letter sigma (Σ) to denote summation. When the distribution is symmetrical, the mean, median, and mode coincide; in skewed distributions, the median often provides a more robust indicator of central tendency. Understanding when to use each measure helps researchers avoid misleading conclusions caused by outliers.
均值公式使用希腊字母西格玛（Σ）表示求和。当分布对称时，均值、中位数和众数重合；在偏态分布中，中位数往往更能稳健地反映集中趋势。了解何时使用每种量度有助于研究者避免因异常值而得出错误结论。
2. Measures of Dispersion: Range, Variance, Standard Deviation | 离散度量：全距、方差、标准差
Dispersion tells us how spread out the scores are around the central tendency. The range is simply the difference between the highest and lowest values. Variance and standard deviation provide more sophisticated measures by quantifying the average squared deviation from the mean. For a population, variance is denoted σ², and its square root is the standard deviation σ, the most widely used index of variability in psychology.
离散度反映了分数在中心趋势周围的分散程度。全距即最大值与最小值之差。方差和标准差通过量化与均值的平均平方偏离，提供了更精细的度量。总体方差记作 σ²，其平方根为标准差 σ，是心理学中最常用的变异指标。
σ² = Σ(x – μ)² ÷ N
σ = √[ Σ(x – μ)² ÷ N ]
In these formulas, x represents each individual score, μ is the population mean, and N is the number of scores. The sample standard deviation uses N−1 in the denominator to correct for bias, and is denoted s. A larger standard deviation indicates greater individual differences within the group, which is crucial when comparing the effectiveness of interventions or the consistency of psychological traits.
在这些公式中，x 代表每个个体的分数，μ 是总体均值，N 是分数个数。样本标准差分母使用 N−1 以校正偏差，记作 s。标准差越大，说明组内个体差异越大，这在比较干预效果或心理特质的稳定性时至关重要。
3. Normal Distribution and Standard Scores | 正态分布与标准分数
Many psychological variables – such as IQ scores, reaction times, and personality traits – follow a normal distribution, characterised by the bell-shaped curve. Standard scores (z‑scores) transform raw data into units of standard deviation relative to the mean, allowing comparisons across different scales. A z‑score of 0 corresponds to the mean, positive values lie above the mean, and negative values below.
许多心理变量——如智商分数、反应时和人格特质——均服从正态分布，即钟形曲线。标准分数（z 分数）将原始数据转化为以标准差为单位、相对均值的数值，便于跨量表比较。z 分数为 0 对应均值，正值表示高于均值，负值表示低于均值。
z = (x – μ) ÷ σ
The probability density function of the normal curve is often written: f(x) = (1/σ√2π) e ^[–½((x–μ)/σ)²], but for Pre-U purposes, applying the z‑formula and using standard normal tables to find proportions is the key skill. Knowledge of the 68‑95‑99.7 rule – that approximately 68% of values fall within ±1σ, 95% within ±2σ – is also essential for interpreting confidence intervals and statistical significance.
正态曲线的概率密度函数常写作：f(x) = (1/σ√2π) e ^[–½((x–μ)/σ)²]，但对 Pre-U 而言，熟练应用 z 公式并查标准正态分布表求比例才是关键。掌握 68‑95‑99.7 规则——约 68% 的数值落在 ±1σ 内，95% 落在 ±2σ 内——对于解读置信区间和统计显著性同样必要。
4. Correlation Analysis: Pearson’s r | 相关分析：皮尔逊相关系数
Pearson’s product‑moment correlation coefficient (r) measures the strength and direction of a linear relationship between two continuous variables. Its value ranges from –1 (perfect negative correlation) through 0 (no linear relationship) to +1 (perfect positive correlation). Correlation does not imply causation, but it is a foundational tool for exploring associations in observational studies.
皮尔逊积矩相关系数（r）衡量两个连续变量之间线性关系的强度和方向。其取值范围从 –1（完全负相关）到 0（无线性关系）再到 +1（完全正相关）。相关系数本身不意味因果关系，但它是探索观察性研究中关联关系的基础工具。
r = Σ[(x – Mx)(y – My)] ÷ √[ Σ(x – Mx)² Σ(y – My)² ]
Each pair of scores (x,y) contributes to the numerator (cross‑product deviations), while the denominator standardises the value. Squaring r gives the coefficient of determination (R²), which indicates the proportion of variance in one variable that can be accounted for by the other. Researchers must also examine scatterplots to check for outliers and non‑linear patterns before relying on r.
每对分数 (x,y) 都贡献于分子（离均差乘积和），分母则将其标准化。将 r 平方得到决定系数（R²），它表示一个变量的变异中有多大比例可以由另一个变量解释。研究者还必须检查散点图，以排查异常值和非线性模式，再依赖 r 进行推断。
5. Hypothesis Testing and p‑value | 假设检验与 p 值
Hypothesis testing is a decision‑making procedure that determines whether observed data provide enough evidence to reject a null hypothesis (H₀). The p‑value represents the probability of obtaining results at least as extreme as those observed, assuming that H₀ is true. A small p‑value (typically p < 0.05) suggests that such data would be unlikely if H₀ were correct, leading researchers to favour the alternative hypothesis (H₁).
假设检验是一套决策程序，用以判断观测数据是否提供足够证据来拒绝虚无假设（H₀）。p 值表示在 H₀ 为真的情况下，获得至少与观测结果一样极端的数据的概率。较小的 p 值（通常 p < 0.05）意味着如果 H₀ 成立，这类数据不太可能出现，研究者因此倾向于备择假设（H₁）。
Decision: if p ≤ α, reject H₀; if p > α, retain H₀
The significance level α (often 0.05) is the threshold for rejecting H₀. Type I error (false positive) occurs when H₀ is rejected though true; Type II error (false negative) occurs when H₀ is retained though false. Power, the complement of Type II error, depends on sample size, effect size, and α. Understanding this framework underpins all inferential statistics in psychology.
显著性水平 α（通常 0.05）是拒绝 H₀ 的门槛。当 H₀ 为真却被拒绝时，发生第一类错误（假阳性）；当 H₀ 为假却被保留时，发生第二类错误（假阴性）。统计检验力是第二类错误的补集，取决于样本量、效应量和 α。理解这一框架是心理学所有推论统计的基础。
6. Independent‑Samples t‑test | 独立样本 t 检验
The independent‑samples t‑test compares the means of two unrelated groups to determine whether they are significantly different. It uses the difference between group means divided by the standard error of the difference. The assumption is that both populations are normally distributed with equal variances (homogeneity of variance). When sample sizes are equal or Levene’s test is not significant, the equal‑variance version is applied.
独立样本 t 检验比较两个不相关组的均值，以判断它们是否存在显著差异。它使用组均值之差除以差值的标准误。假设两组总体均呈正态分布且方差齐性。当样本量相等或莱文检验不显著时，使用方差齐性版本。
t = (M₁ – M₂) ÷ SEdiff
SEdiff = √[ (sp² ÷ n₁) + (sp² ÷ n₂) ]
sp² = [ (n₁–1)s₁² + (n₂–1)s₂² ] ÷ (n₁ + n₂ – 2)
Here, M₁ and M₂ are sample means, sp² is the pooled variance, n₁ and n₂ are sample sizes, and degrees of freedom (df) = n₁ + n₂ – 2. The obtained t is compared against critical values from the t‑distribution table. If |t| exceeds the critical value, the difference is statistically significant, indicating a real effect beyond chance.
这里 M₁ 和 M₂ 是样本均值，sp² 是合并方差，n₁ 和 n₂ 是样本量，自由度 df = n₁ + n₂ – 2。计算得到的 t 值与 t 分布表中的临界值比较；若 |t| 超过临界值，则差异具有统计显著性，表明存在超越偶然的真实效应。
7. Chi‑Square Test of Independence | 卡方独立性检验
The chi‑square (χ²) test assesses whether two categorical variables are associated in a population. It compares observed frequencies (O) with the frequencies expected (E) under the null hypothesis of no association. A large χ² indicates a discrepancy between observed and expected counts, suggesting a significant relationship. This test is widely used in surveys and experiments involving nominal data, such as gender and answer choice.
卡方（χ²）检验评估两个类别变量在总体中是否相关。它比较实际观测频次（O）与在无关联虚无假设下的期望频次（E）。较大的 χ² 值表示观测与期望计数之间存在较大差异，提示显著关系。该检验广泛用于涉及名义数据（如性别与答题选择）的问卷和实验。
χ² = Σ [ (O – E)² ÷ E ]
The expected frequency for each cell is calculated by (row total × column total) ÷ grand total. Degrees of freedom are (number of rows – 1) × (number of columns – 1). The computed χ² is compared to critical values in the chi‑square distribution table. A significant result implies that the variables are not independent, but it does not reveal the direction or strength of association – for that, measures like Cramér’s V are used.
每个格子的期望频次计算公式为（行合计 × 列合计）÷ 总计。自由度 = (行数 – 1) × (列数 – 1)。计算出的 χ² 需与卡方分布表中的临界值比较。显著结果意味着变量不是独立的，但无法揭示关联方向或强度——为此需使用 Cramér’s V 等量度。
8. Weber’s Law in Psychophysics | 心理物理学中的韦伯定律
Weber’s Law describes the relationship between the physical intensity of a stimulus and the just noticeable difference (JND) – the smallest change in intensity that can be detected. The law states that the JND is a constant proportion (k) of the initial stimulus intensity. This principle applies to many sensory modalities, including brightness, loudness, and weight discrimination, and is foundational to understanding sensory thresholds.
韦伯定律描述了刺激物理强度与最小可觉差（JND）——即能够被觉察的最小强度变化——之间的关系。该定律指出，JND 是初始刺激强度的一个恒定比例（k）。这一原理适用于多种感觉通道，包括亮度、响度和重量辨别，是理解感觉阈限的基础。
ΔI ÷ I = k
Where ΔI is the increment needed to produce a JND, I is the initial stimulus intensity, and k is the Weber fraction (a constant specific to the sensory modality). For example, for lifted weights k ≈ 1/30. While Weber’s Law holds reasonably well for moderate intensities, it tends to break down at very low and very high stimulus levels. Fechner later built on this law to create the logarithmic psychophysical scale.
式中 ΔI 为产生 JND 所需的增量，I 为初始刺激强度，k 为韦伯分数（特定于感觉通道的常数）。例如，对于举重，k 约 1/30。尽管韦伯定律在中等强度下成立得相当好，但在极低和极高刺激水平下往往会失效。费希纳后在此基础上构建了对数心理物理量表。
9. Fechner’s Law and Perceived Magnitude | 费希纳定律与感知强度
Fechner’s Law extends Weber’s findings by proposing a logarithmic relationship between the physical magnitude of a stimulus and its perceived sensation intensity. As stimulus intensity increases geometrically, perceived sensation grows arithmetically. This explains why adding one more light bulb to 100 bulbs produces a smaller perceived change than adding one to 10 bulbs, reflecting the compressive nature of many sensory systems.
费希纳定律基于韦伯的研究，提出刺激物理强度与其主观感觉强度之间存在对数关系。当刺激强度按几何级数增加时，感知感觉仅按算术级数增长。这解释了为什么向 100 盏灯中再添加一盏灯，其主观变化远小于向 10 盏灯中添加一盏——反映了许多感觉系统的压缩特性。
S = k log I
Here S is the subjective sensation magnitude, I is the physical intensity, and k is a scaling constant. Although modern psychophysics often uses Stevens’ power law (S = a Iᵇ) instead, Fechner’s logarithmic formulation remains important for understanding classic psychophysical experiments and the historical development of experimental psychology. Pre-U students should be able to contrast the two perspectives.
这里 S 是主观感觉强度，I 是物理强度，k 是尺度常数。尽管现代心理物理学常使用史蒂文斯幂定律（S = a Iᵇ），但费希纳的对数公式对于理解经典心理物理实验以及实验心理学的历史发展仍至关重要。Pre-U 学生应能对比这两种视角。
10. Signal Detection Theory | 信号检测论
Signal Detection Theory (SDT) provides a framework for quantifying decision‑making under uncertainty, especially when distinguishing a signal from background noise. It separates sensitivity (d’) – the ability to discriminate signals from noise – from response bias (β or criterion c), which reflects the observer’s willingness to respond. SDT is applied to memory recognition, medical diagnosis, and eyewitness testimony, making it a versatile tool in cognitive psychology.
信号检测论（SDT）提供了一套在不确定情况下量化决策过程的框架，尤其用于区分信号与背景噪音。它将感受性（d’）——辨别信号与噪音的能力——与反应偏差（β 或判断标准 c）分离开来，后者反映观察者作出反应的主观倾向。SDT 被应用于记忆再认、医学诊断及目击证词，是认知心理学中的万能工具。
d’ = zHit – zFA
c = –0.5 (zHit + zFA)
Hit rate is the proportion of correctly identified signals, and false‑alarm rate is the proportion of noise trials incorrectly called signals. These proportions are converted to z‑scores using the inverse normal distribution. A higher d’ indicates greater sensitivity, while c reflects the criterion: negative values mean a liberal bias (tendency to say ‘yes’). The Receiver Operating Characteristic (ROC) curve plots hit rate against false‑alarm rate at different criteria.
击中率是正确识别信号的比例，虚报率是将噪音试次误报为信号的比例。这些比例经逆正态分布转换为 z 分数。d’ 越高表明感受性越强，c 反映判断标准：负值表示宽松偏向（倾向于说“是”）。操作者特征曲线（ROC）绘制了不同判断标准下击中率与虚报率的关系。
11. Expected Utility and Prospect Theory | 期望效用与前景理论
Rational choice theories assume that individuals make decisions by weighing potential outcomes by their probabilities, maximizing expected utility. The classic expected utility formula multiplies each outcome’s utility (U) by its probability (p). However, psychological research by Kahneman and Tversky demonstrated that people systematically violate these axioms, valuing losses more heavily than equivalent gains (loss aversion) and distorting probabilities. Prospect theory replaces final wealth with gains and losses relative to a reference point.
理性选择理论假设个体通过将各结果的概率乘以其效用，从而最大化期望效用来做决策。经典的期望效用公式将各结果的效用（U）与其概率（p）相乘。但卡尼曼和特沃斯基的心理学研究表明，人们会系统性地违背这些公理：与同等收益相比，损失被赋予更大的权重（损失厌恶），并对概率进行扭曲。前景理论以相对于参照点的收益和损失取代了终末财富。
EU = Σ [ p(O) × U(O) ]
Prospect value V = Σ [ w(p) × v(Δ) ]
In prospect theory, the value function v(Δ) is concave for gains and convex for losses, and it is steeper in the loss domain; w(p) is a probability‑weighting function that overweights small probabilities and underweights moderate probabilities. These equations formalise phenomena such as the framing effect and the certainty effect, which are core topics in the cognitive and social psychology syllabus. Understanding these models helps students explain why human decision‑making often deviates from economic rationality.
在前景理论中，价值函数 v(Δ) 在收益区域凹、损失区域凸，且在损失区域更陡峭；w(p) 是概率权重函数，对小概率赋予过高权重，对中高概率赋予过轻权重。这些方程将框架效应和确定性效应等现象形式化，是认知与社会心理学课程中的核心主题。理解这些模型有助于学生解释为何人类决策常偏离经济理性。
12. Bayes’ Theorem in Psychological Reasoning | 心理学推理中的贝叶斯定理
Bayes’ Theorem provides a mathematical method for updating the probability of a hypothesis in light of new evidence. In psychology, it is used to model human judgment under uncertainty, such as diagnostic reasoning and belief revision. Research on base‑rate neglect shows that people often fail to use prior probabilities correctly, leading to errors that Bayes’ theorem can help quantify and correct.
贝叶斯定理为根据新证据更新假设概率提供了数学方法。在心理学中，它用于模拟不确定情况下的人类判断，如诊断推理和信念修正。关于基础比率忽视的研究表明，人们常常未能正确使用先验概率，从而导致错误，而贝叶斯定理可以帮助量化和纠正这些错误。
P(H|D) = [ P(D|H) × P(H) ] ÷ P(D)
Here P(H|D) is the posterior probability: the probability of the hypothesis given the data. P(H) is the prior probability of the hypothesis before seeing the data, P(D|H) is the likelihood of observing the data under that hypothesis, and P(D) is the total probability of the data across all hypotheses. This theorem underpins Bayesian cognitive modelling, which proposes that the brain performs probabilistic inference to perceive and learn.
这里 P(H|D) 是后验概率：给定数据下假设成立的概率。P(H) 是看到数据前假设的先验概率，P(D|H) 是在该假设下观察到数据的似然，P(D) 是所有假设下数据的总概率。贝叶斯定理是贝叶斯认知建模的基石，该框架提出大脑通过概率推理来感知和学习。
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