📚 IGCSE Cambridge Psychology: Formula & Theorem Quick Reference Handbook | IGCSE 剑桥心理学：公式与定理速查手册
This quick-reference handbook collates essential formulas, laws, and theorems you need to master for the Cambridge IGCSE Psychology examination. From statistical calculations to psychophysical laws and memory principles, each entry is paired with a concise explanation to aid rapid revision.
这本速查手册汇总了你在剑桥 IGCSE 心理学考试中需要掌握的关键公式、定律和定理。从统计计算到心理物理定律和记忆原理，每一条目都配有简洁的解释以帮助你快速复习。
1. Vital Statistical Formulas: Mean, Median, Mode, Range | 重要统计公式：均值、中位数、众数、极差
These measures of central tendency and dispersion allow psychologists to summarise data and spot patterns. The mean is the arithmetic average, calculated as the sum of all scores divided by the number of scores: Mean = Σx / N. The median is the middle value when scores are arranged in order, and the mode is the most frequently occurring score. To describe how spread out data are, we use the range: Range = maximum value – minimum value.
集中趋势和离散程度的量数帮助心理学家汇总数据并发现模式。均值是算术平均数，计算公式为所有分数的总和除以分数的个数：均值 = Σx / N。中位数是将分数按顺序排列后处于中间位置的那个值，众数则是出现频率最高的分数。为了描述数据的离散程度，我们使用极差：极差 = 最大值 – 最小值。
The mean is sensitive to extreme scores (outliers), while the median is resistant; the mode may reveal the most typical performance. Always check whether the distribution is symmetrical before choosing the best average to report.
均值对极端分数（异常值）敏感，而中位数则有抗干扰性；众数可以揭示最典型的表现。在选择报告哪种平均数之前，一定要检查数据分布是否对称。
2. Research Ratios and Percentage Calculations | 研究中的比率与百分比计算
IGCSE Psychology frequently asks you to interpret percentages, ratios, and proportions from studies. The key formula for converting a part into a percentage is: Percentage = (part / whole) × 100. A ratio expresses the relative size of two quantities, for example, the ratio of obedient participants to disobedient ones in Milgram’s study can be written as committed : defiant.
IGCSE 心理学经常要求你解释研究中的百分比、比率和比例。将部分转化为百分比的关键公式是：百分比 = (部分 / 整体) × 100。比率表示两个数量的相对大小，例如，在米尔格拉姆服从研究中，服从参与者与不服从参与者的人数比可以写作 服从人数 : 反抗人数。
When evaluating ethical cost–benefit in a study, you may informally compare the harm caused (cost) to the knowledge gained (benefit). A study is justifiable only if the benefit-to-cost ratio is judged to be greater than 1. Although this is not a numerical formula, it serves as a guiding theorem for ethical review.
当你评价一项研究的伦理成本-收益时，你可以非正式地比较所造成的伤害（成本）与所获得的知识（收益）。只有当收益与成本的比值被判定大于 1 时，研究才具有合理性。虽然这并非一个数值公式，但它可以作为伦理审查的指导性定理。
3. Miller’s Law: The Magical Number 7 ± 2 | 米勒定律：神奇的数字 7 ± 2
George Miller (1956) proposed that the capacity of short-term memory (STM) is limited to about 7 ± 2 items, or chunks of information. This means most people can hold between 5 and 9 meaningful units in immediate memory. The law is often written as: STM capacity = 7 ± 2 chunks.
乔治·米勒(1956)提出，短时记忆的容量大约局限于 7 ± 2 个项目或信息组块。这意味着大多数人可以在即时记忆中保持 5 到 9 个有意义的信息单元。这条定律常写作：短时记忆容量 = 7 ± 2 个组块。
Chunking is a strategy that transforms separate bits into larger, meaningful units, effectively expanding the amount of information retained. For example, a sequence of digits ‘1 9 1 4 1 9 1 8’ can be chunked into ‘1914 1918’, reducing the load from eight to two chunks. Miller’s law underpins techniques such as mnemonic devices and revision grouping.
组块化是一种将零散信息转化为更大、更有意义的单元的策略，从而有效扩大能够保留的信息量。例如，数字序列 “1 9 1 4 1 9 1 8” 可以被组块化为 “1914 1918”，将记忆负荷从八个组块降至两个组块。米勒定律为记忆术和分组复习等技巧提供了基础。
4. The Normal Distribution and the 68–95–99.7 Rule | 正态分布与 68–95–99.7 法则
Many psychological traits, such as IQ scores or reaction times (after transformation), approximate a normal distribution—a symmetrical bell-shaped curve. The empirical law governing the spread of data is: approximately 68% of observations fall within one standard deviation (σ) of the mean (μ), 95% within two standard deviations, and 99.7% within three standard deviations.
许多心理特质，如智商分数或反应时（经转换后），都近似于正态分布——一条对称的钟形曲线。描述数据离散度的经验法则是：大约 68% 的观察值落在均值 (μ) 一倍标准差 (σ) 之内，95% 落在两倍标准差之内，99.7% 落在三倍标准差之内。
This theorem, often called the three-sigma rule, helps psychologists identify outliers (scores beyond ±3σ) and compare an individual’s performance to the population. In interpreting test scores, knowing that 2.5% of people will score higher than two standard deviations above the mean allows us to set clinical cut-offs.
这一定理常被称为三西格玛法则，有助于心理学家识别异常值（超出 ±3σ 的分数），并将个体的表现与总群体进行比较。在解释测验分数时，知道有 2.5% 的人成绩会高于均值两个标准差以上，就可以设定临床临界值。
5. Ebbinghaus’s Forgetting Curve and the Savings Formula | 艾宾浩斯遗忘曲线与节省量公式
Hermann Ebbinghaus (1885) quantified memory decay using the savings method. He measured how much shorter the relearning process was compared to original learning. The savings percentage is given by: Savings (%) = [(original learning time – relearning time) / original learning time] × 100.
赫尔曼·艾宾浩斯（1885）使用节省法量化了记忆的衰退。他测量了重新学习的过程比最初学习缩短了多少。节省量百分比的计算公式为：节省量 (%) = [(最初学习时间 – 重新学习时间) / 最初学习时间] × 100。
His forgetting curve shows a rapid loss of information shortly after learning, followed by a more gradual decline; roughly 50% of newly learned nonsense syllables are forgotten within an hour, and recall levels off over days. This exponential decay can be loosely expressed as: Retention ≈ 100 – k log(t), where t is time elapsed since learning. Although you do not need to compute this exactly, the theorem illustrates why distributed practice and regular revision are essential.
他的遗忘曲线表明，信息在学习后很快就会大量丢失，随后遗忘速度逐渐减缓；新学的无意义音节大约有 50% 会在一个小时内忘掉，几天后回忆水平趋于稳定。这种指数式衰退可以粗略地表达为：保持率 ≈ 100 – k log(t)，其中 t 是自学习后经过的时间。虽然你不需要进行精确计算，但这一定理说明了分散练习和定期复习为何如此重要。
6. Weber’s Law (ΔI / I = k) | 韦伯定律 (ΔI / I = k)
Ernst Weber discovered that the just noticeable difference (JND) between two stimuli is not an absolute amount but a constant proportion of the initial stimulus intensity. This is formalised as: ΔI / I = k, where ΔI is the change in intensity needed to detect a difference, I is the original intensity, and k is the Weber fraction, a constant for that sensory modality.
恩斯特·韦伯发现，两个刺激之间刚刚能被察觉的差异（最小可觉差）不是一个绝对数值，而是初始刺激强度的恒定比例。这一定律形式化为：ΔI / I = k，其中 ΔI 为察觉差异所需的强度变化量，I 为原始强度，k 为韦伯分数，即对该感觉通道恒定的常数。
For example, for lifted weights the Weber fraction is about 0.02–0.05, meaning a 100 g weight needs to change by roughly 2–5 g to feel noticeably different. Weber’s law holds across most sensory dimensions (brightness, loudness, etc.) except at extremely low intensities. It forms a cornerstone of psychophysics and explains why you are more likely to notice a price increase on a cheap item than on a luxury car.
例如，对于举重来说，韦伯分数约为 0.02–0.05，这意味着一个 100 克的重量大约需要变化 2–5 克才能让人感到明显的不同。除了在极低强度条件下，韦伯定律在大多数感觉维度（亮度、响度等）中都成立。它是心理物理学的基石，也解释了为什么人们更容易注意到廉价商品的价格上涨，而不是奢侈汽车的价格变化。
7. Fechner’s Law (S = k log I) | 费希纳定律 (S = k log I)
Gustav Fechner extended Weber’s findings into a law relating physical stimulus intensity to perceived sensation. Fechner’s law states: S = k log I, where S is the perceived sensation magnitude, I is the physical stimulus intensity, and k is a constant derived from the Weber fraction. The formula shows that sensation increases in proportion to the logarithm of the physical stimulus.
古斯塔夫·费希纳将韦伯的发现扩展为一条联系物理刺激强度与主观感觉的定律。费希纳定律表述为：S = k log I，其中 S 是感知到的感觉量，I 是物理刺激强度，k 是由韦伯分数导出的常数。该公式显示，感觉的增长与物理刺激的对数成正比。
Practically, doubling the physical intensity of a light will not make it seem twice as bright; a much larger physical increase is needed to double the perceived brightness. Fechner’s law reminds researchers that psychological scales do not map linearly onto physical scales, a principle central to designing perceptual experiments and surveys with equal-interval response formats.
实际上，将灯光的物理强度加倍并不会让它看起来亮两倍；要使感知亮度翻倍，需要大得多的物理增量。费希纳定律提醒研究者，心理量表的刻度并不与物理刻度线性对应，这条原则对于设计感知实验和等距作答格式的问卷至关重要。
8. Yerkes–Dodson Law: The Inverted-U Hypothesis | 耶克斯–多德森定律：倒U假设
The Yerkes–Dodson law (1908) describes the relationship between arousal or motivation and performance. It is typically represented as an inverted-U-shaped curve, which can be captured by the proposition: Performance = f(arousal) with an optimal level of arousal for any given task. Performance improves as arousal rises to a moderate level, then declines if arousal continues to increase.
耶克斯–多德森定律（1908）描述了唤醒或动机水平与行为表现之间的关系。它通常被表示为一条倒U型曲线，可以用以下命题概括：绩效 = f(唤醒)，且对于任何给定的任务都存在一个最佳的唤醒水平。随着唤醒水平上升到中等程度，绩效会提高；之后若唤醒继续增强，绩效则会下降。
The precise shape of the inverted-U depends on task difficulty: for simple or well-learned tasks, optimal arousal is relatively high; for complex or novel tasks, optimal arousal is lower. While no single mathematical equation is universally exact, the law is often modelled as: Performance = a – b(c – arousal)², where c represents the optimum arousal point. Remember this when explaining why moderate exam anxiety can sharpen focus, but excessive anxiety impairs recall.
倒U的具体形状取决于任务难度：对于简单或高度熟练的任务，最佳唤醒水平相对较高；对于复杂或新异的任务，最佳唤醒水平较低。虽然没有一个完全精确的通用数学方程，但这一定律常被模型化为：绩效 = a – b(c – 唤醒)²，其中 c 代表最佳唤醒点。在解释为何适度的考试焦虑能提升注意力，但过度焦虑会损害回忆时，请牢记这一定律。
9. The Serial Position Effect: Primacy and Recency Laws | 序列位置效应：首因与近因定律
When people attempt to recall a list of items in any order (free recall), the probability of recalling an item is a function of its serial position. Murdock (1962) demonstrated this effect as a U-shaped curve: Recall probability ≈ position, with items presented first (primacy effect) and last (recency effect) recalled better than those in the middle.
当人们试图按任意顺序回忆一系列项目时（自由回忆），回忆起某个项目的概率是其序列位置的函数。默多克（1962）用一个U形曲线展示了这一效应：回忆概率 ≈ 序列位置，最先呈现的项目（首因效应）和最后呈现的项目（近因效应）的回忆效果优于中间的项目。
The primacy effect is attributed to stronger long-term memory encoding through rehearsal, while the recency effect relies on items still being available in short-term memory. These twin laws underpin the advice to structure revision so that the most critical points appear at the beginning and end of study sessions. Distractor tasks between learning and test can abolish the recency effect but leave the primacy effect intact.
首因效应被归因于通过复述而形成的更强的长时记忆编码，近因效应则依赖于项目仍停留在短时记忆中。这两条定律构成了如下建议的基础：在复习时，应将最关键的知识点安排在学习时段的开头和结尾。学习与测验之间插入干扰任务可以消除近因效应，但首因效应会保持完整。
10. Piaget’s Stage Theorem: Cognitive Development Laws | 皮亚杰的阶段定理：认知发展定律
Jean Piaget’s theory of cognitive development proposes that children pass through a fixed sequence of four stages, each characterised by qualitatively different logical structures. The theorem states that these stages unfold in an invariant order, with approximate age ranges: sensorimotor (0–2 years), preoperational (2–7 years), concrete operational (7–11 years), and formal operational (11 years onwards).
让·皮亚杰的认知发展理论提出，儿童会依次经历四个固定的阶段，每个阶段都以质上不同的逻辑结构为特征。该定理指出这些阶段以不变的顺序展开，具有近似的年龄范围：感知运动阶段（0–2 岁）、前运算阶段（2–7 岁）、具体运算阶段（7–11 岁）和形式运算阶段（11 岁及以上）。
Each stage brings new capabilities: object permanence emerges in the sensorimotor stage; symbolic thought but also egocentrism typify preoperational thinking; conservation and reversibility mark concrete operations; and abstract, hypothetical reasoning defines formal operations. The theory is often treated as a developmental law, informing education and the design of age-appropriate tasks. Although the exact age boundaries vary, the sequence itself is considered universal.
每个阶段都带来新的能力：客体永久性在感知运动阶段出现；符号思维及自我中心是前运算思维的典型特征；守恒和可逆性是具体运算阶段的标志；而抽象、假设性推理则定义了形式运算阶段。该理论常被当作一条发展定律，为教育以及设计符合年龄的任务提供指导。尽管具体的年龄边界有所变化，但这一顺序本身被视为普遍适用的。
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