📚 KS3 CIE Psychology Formula & Theorem Quick Reference Handbook | KS3 CIE 心理学：公式定理速查手册
Psychology is often thought of as a subject of theories and words, but it also contains many important formulas and mathematical laws that help us measure mental processes. This quick reference handbook gathers the essential formulas, laws, and theorems covered in the KS3 CIE Psychology curriculum. From the classic Weber–Fechner laws in psychophysics to Ebbinghaus’ forgetting curve and the IQ formula, each entry is explained with a clear definition and a practical example. Whether you are revising for an end‑of‑topic test or building a solid foundation for IGCSE Psychology, use this guide to memorise and apply the key quantitative ideas that shape our understanding of the mind and behaviour.
心理学常被认为是一门充满理论和文字的学科，但它同样包含许多帮助我们测量心理过程的重要公式和数学定律。这本速查手册汇集了 KS3 CIE 心理学课程中涉及的核心公式、定律和定理。从心理物理学中经典的韦伯–费希纳定律，到艾宾浩斯遗忘曲线和智商公式，每个条目都配有清晰的定义和实例。无论你是在准备单元测验，还是想为 IGCSE 心理学打下扎实基础，都可以借助这本手册记住并运用那些塑造我们对心智与行为理解的关键量化思想。
1. Weber’s Law of Just Noticeable Difference | 韦伯差别阈限定律
Weber’s Law states that the size of the just noticeable difference (JND) – the smallest change in a stimulus that can be detected – is a constant proportion of the original stimulus intensity. The formula is written as ΔI / I = k, where ΔI is the increase in intensity needed to notice a change, I is the initial intensity, and k is the Weber fraction (a constant that varies between senses). For example, if you are holding a 100 g weight, you might need to add about 2 g (k ≈ 0.02) before you can feel that it is heavier. If the weight were 200 g, you would need around 4 g extra. This shows our sensory systems are relative, not absolute, detectors of change.
韦伯定律指出，最小可觉差（刚刚能够察觉到刺激发生变化的最小量）的大小与初始刺激强度成恒定比例。公式写作 ΔI / I = k，其中 ΔI 是察觉变化所需增加的强度，I 是初始强度，k 是韦伯分数（一个随感觉通道不同而变化的常数）。例如，你手里拿着一个 100 g 的砝码，可能需要增加大约 2 g（k ≈ 0.02）才能感觉到它变重了；如果初始重量是 200 g，就需要增加约 4 g。这表明我们的感觉系统是相对而非绝对的探测器。
| Sense / 感觉 | Approximate Weber fraction (k) / 近似韦伯分数 |
|---|---|
| Vision (brightness) / 视觉（亮度） | 0.08 |
| Hearing (pitch) / 听觉（音高） | 0.003 |
| Touch (pressure) / 触觉（压力） | 0.14 |
| Taste (saltiness) / 味觉（咸度） | 0.20 |
2. Fechner’s Law of Sensation Magnitude | 费希纳感觉强度定律
Fechner extended Weber’s work by proposing a logarithmic relationship between physical stimulus intensity and perceived sensation. Fechner’s Law is expressed as S = k log I, where S is the perceived magnitude of sensation, I is the physical intensity of the stimulus, and k is a constant that depends on the sensory dimension. A small increase in a weak stimulus produces a much larger jump in sensation than the same increase added to a strong stimulus. Imagine a room with one lit candle: adding a second candle makes the room seem much brighter. But if the room already has 100 candles, adding one more barely changes the perceived brightness.
费希纳在韦伯研究的基础上提出，物理刺激强度与主观感觉量之间存在对数关系。费希纳定律的表达式为 S = k log I，其中 S 是主观感觉量，I 是刺激的物理强度，k 是取决于感觉通道的常数。对微弱刺激施加一个小的增量，会引起比对强刺激施加同样增量时大得多的感觉变化。想象一间只点了一支蜡烛的房间：再添一支蜡烛会让房间看起来亮很多；但如果房间里已经点了 100 支蜡烛，再点多一支几乎不会改变感知到的亮度。
In psychological experiments, this logarithmic relationship explains why people are much better at telling the difference between two very quiet sounds than between two very loud sounds. Fechner’s Law is the foundation of psychophysics and remains useful in understanding sensory scales such as decibels for loudness or the apparent magnitude of stars.
在心理学实验中，这种对数关系解释了为什么人们对两个极其微弱声音的辨别能力远强于对两个极强声音的辨别。费希纳定律是心理物理学的基础，至今仍被用于理解如响度分贝、星等视亮度等感官量表的构建。
3. Ebbinghaus Forgetting Curve | 艾宾浩斯遗忘曲线
Hermann Ebbinghaus studied his own memory by learning lists of nonsense syllables and testing himself after various delays. He discovered that forgetting follows an exponential pattern, which can be approximated by the formula R = e^(−t / τ), where R is the proportion of material retained, t is the time elapsed since learning, and τ is a time constant that reflects the rate of forgetting. In simpler classroom terms, memory retention drops rapidly in the first few hours and days, then levels off. After one hour, people may forget about 50% of newly learned information unless they review it.
赫尔曼·艾宾浩斯通过记忆无意义音节表、并在不同时间间隔后测试自己的方式来研究记忆。他发现遗忘遵循指数规律，可用公式 R = e^(−t / τ) 近似表示，其中 R 是保留材料所占的比例，t 是学习之后经过的时间，τ 是一个反映遗忘速率的时间常数。在学校常用的通俗说法中，记忆在最初几小时和几天内急剧下降，随后趋于平缓。如果不复习，人们在学习后一小时可能已经忘掉大约一半的新学信息。
This curve teaches us the importance of spaced repetition. Each time you review material, the rate of forgetting slows down and the retention after the same time interval becomes higher. Teachers often recommend reviewing notes after one day, one week and one month to build long‑term memory. Ebbinghaus’ work is one of the first quantitative theories in cognitive psychology and remains a cornerstone of revision strategies.
这条曲线告诉我们间隔复习的重要性。每当你复习一次材料，遗忘的速率就会变慢，相同时隔后的保持量也会更高。老师们常建议在学习后一天、一周和一个月分别进行复习，以建立长期记忆。艾宾浩斯的研究是认知心理学中最早的定量理论之一，至今仍是复习策略的基石。
4. IQ Formula (Stanford‑Binet) | 智商公式（斯坦福‑比奈）
The intelligence quotient (IQ) was originally computed using a simple ratio method. The formula is IQ = (Mental Age / Chronological Age) × 100. Mental age (MA) is a measure of a child’s intellectual performance relative to the average performance of children of a particular chronological age (CA). For instance, if an 8‑year‑old child can solve problems that the average 10‑year‑old can solve, the child’s mental age is 10. Then IQ = (10 / 8) × 100 = 125. An IQ of 100 indicates performance exactly at age level.
智商最初通过简单的比率法计算。公式为智商 = （智力年龄 / 实际年龄） × 100。智力年龄（MA）是衡量儿童智力表现相对于特定实际年龄（CA）儿童平均水平的指标。例如，如果一个 8 岁的孩子能够解决平均 10 岁孩子才能解决的问题，其智力年龄就是 10 岁。那么 IQ = (10 / 8) × 100 = 125。IQ 为 100 表示其表现恰好与同龄水平相当。
Today most IQ tests use deviation scores rather than the ratio IQ, but the concept of IQ as a standardised score remains central to psychometrics. Studying the IQ formula helps KS3 students understand that intelligence tests are designed to compare an individual’s performance with that of a normative sample. However, it is also important to remember that intelligence is multi‑faceted and not fully described by a single number.
如今多数智商测验采用离差分数而非比率智商，但将 IQ 视为标准化分数的概念仍是心理测量学的核心。学习智商公式有助于 KS3 学生理解智力测验旨在将个体表现与常模样本进行比较。然而，同样重要的是要记住智力是多维的，不能单用一个数字来完整描述。
5. Yerkes–Dodson Law of Arousal and Performance | 耶基斯‑多德森唤醒与绩效定律
The Yerkes–Dodson Law states that performance increases with physiological or mental arousal up to an optimal point, after which further arousal leads to a decline in performance. This relationship is often drawn as an inverted‑U‑shaped curve. There is no single mathematical formula, but the principle can be summarised as: there exists an optimal arousal level that produces peak performance, and that optimal level is lower for difficult tasks and higher for simple tasks.
耶基斯‑多德森定律指出，生理或心理唤醒水平提升时，绩效会随之提高，直至达到最佳点；超过这一最佳点后，继续上升的唤醒反会导致绩效下降。这种关系通常画成倒 U 形曲线。虽然没有单一的数学公式，但该原理可归纳为：存在一个能产生最佳绩效的最优唤醒水平，且复杂任务的最优唤醒水平较低，简单任务则较高。
For example, a little anxiety before an exam can help you focus and perform better. However, extreme nerves may cause mind‑blanking and poor recall. When you are taking a tricky maths paper, staying calm (moderate‑low arousal) is better; for a straightforward sprint race, higher arousal gives an edge. This law is widely used in sports psychology and stress management.
例如，考前适度的紧张能帮助你集中精神，发挥出色；但过度焦虑则可能导致大脑一片空白、回忆困难。做难度较大的数学卷子时，保持平静（中低唤醒）更为有利；而对于简单的短跑比赛，较高的唤醒水平则能带来优势。该定律广泛用于运动心理学与压力管理。
6. Miller’s Magic Number 7 ± 2 | 米勒的神奇数字 7 ± 2
George Miller’s classic 1956 paper proposed that the capacity of short‑term memory (STM) is limited to about seven chunks of information, plus or minus two. This is not a formula in the algebraic sense but a robust quantitative finding. The chunk is the unit of information that is meaningful to the individual – it can be a digit, a letter, a word, or even an entire phrase. Miller’s law explains why phone numbers without area codes are typically seven digits long and why we can repeat back a short list of words more accurately than a long sentence.
乔治·米勒在 1956 年的经典论文中提出，短时记忆的容量大约为七个信息组块，上下浮动两个。这并非代数意义上的公式，却是一个稳健的定量发现。组块是对个体有意义的信息单元——可以是一个数字、一个字母、一个单词，甚至是一个完整短语。米勒定律解释了为什么不带区号的电话号码通常为七位数，以及为什么我们能准确复述一列短单词，却很难逐字记住一个长句子。
Chunking is a powerful memory technique. For instance, the letter string FBICIAUK is hard to hold as eight separate items, but it becomes three easy chunks if you spot the abbreviations FBI, CIA and UK. By grouping smaller pieces of information into larger, meaningful units, we can expand the effective capacity of short‑term memory and improve learning.
组块是一种强大的记忆技巧。例如，字母串 FBICIAUK 作为八个独立项目很难记住，但如果你识别出 FBI、CIA 和 UK 这三个缩写，就变成了三个容易记忆的组块。通过将零散信息组合成更大的、有意义的单元，我们可以扩展短时记忆的有效容量并改善学习效果。
7. Signal Detection Theory (Basic d′ Concept) | 信号检测论（基本 d′ 概念）
Signal Detection Theory (SDT) provides a mathematical framework for understanding how decisions are made under uncertainty. The key index is d′ (d‑prime), which measures how easily a signal can be distinguished from background noise. In its simplest form, d′ = z(Hit) – z(False Alarm), where z(Hit) and z(False Alarm) are the z‑scores for the proportion of correctly detected signals and the proportion of false alarms. A larger d′ indicates greater sensitivity – the observer can tell signal from noise more reliably. For KS3 learners, think of it as a way to separate a person’s true perceptual ability from their response bias (how willing they are to say ‘yes’).
信号检测论提供了一个理解个体如何在不确定条件下做出决策的数学框架。核心指标是 d′（d‑prime），它衡量信号与背景噪音的易区分程度。最简单的形式为 d′ = z(击中率) – z(虚报率)，其中 z(击中率) 和 z(虚报率) 分别代表正确检测到信号的比例与误报比例对应的 z 分数。d′ 越大表示敏感性越高——观察者能更可靠地区分信号与噪音。对 KS3 学习者来说，可以将其视为一种分离真正感知能力与反应偏向（个体说“是”的意愿）的方法。
For example, a radiologist looking for a tumour on a scan must decide whether a shadow is a real signal or just noise. A very cautious radiologist might have few false alarms but could miss real tumours, while a less cautious one might find more tumours but also flag healthy tissue unnecessarily. SDT helps researchers account for both sensitivity and response bias.
举例来说，一名放射科医生在扫描影像中寻找肿瘤时，必须判断一个阴影是真实的信号还是噪音。非常谨慎的医生可能很少误报，但也可能遗漏真正的肿瘤；而不太谨慎的医生可能发现更多肿瘤，同时也会对健康组织发出不必要的警报。信号检测论帮助研究者同时将敏感性和反应偏向纳入考量。
8. Piaget’s Stages of Cognitive Development | 皮亚杰认知发展阶段理论
Jean Piaget’s theory describes four universal stages through which children’s thinking changes in qualitative ways. There is no numerical formula, but the stage theorem is fundamental to developmental psychology. The four stages are: Sensorimotor (0–2 years) – infants learn through senses and actions, developing object permanence; Preoperational (2–7 years) – children use symbols and language but are egocentric and struggle with conservation; Concrete Operational (7–11 years) – logical thinking about concrete events emerges, and children master conservation and classification; Formal Operational (11+ years) – individuals can reason abstractly and systematically, testing hypotheses like a scientist.
让·皮亚杰的理论描述了儿童思维发生质性变化的四个普遍阶段。虽然其中没有数字公式，但这一发展阶段定理是发展心理学的基石。四个阶段分别是：感知运动阶段（0–2 岁）—— 婴儿通过感觉和动作学习，发展出客体永久性；前运算阶段（2–7 岁）—— 儿童开始使用符号和语言，但具有自我中心且难以掌握守恒概念；具体运算阶段（7–11 岁）—— 出现对具体事件的逻辑思维，儿童掌握守恒与分类；形式运算阶段（11 岁以上）—— 个体能够进行抽象、系统的推理，像科学家一样检验假设。
Understanding these stages helps teachers plan age‑appropriate activities. For example, a Year 7 student (around 11–12 years) is typically entering Formal Operations, so they can handle abstract “what if” questions in science. However, each child’s progression can vary, and later research shows that culture and education play significant roles in shaping cognitive development.
理解这些阶段有助于教师设计适龄活动。例如，7 年级学生（约 11–12 岁）通常正步入形式运算阶段，因此能够应对科学课中抽象的“如果……会怎样”问题。不过，每个孩子的进展速度各不相同，后续研究也显示文化与环境教育在塑造认知发展中起着重要作用。
9. Maslow’s Hierarchy of Needs | 马斯洛需求层次理论
Abraham Maslow proposed that human motivation is driven by a hierarchy of needs, usually represented as a pyramid. The five levels from base to peak are: Physiological needs (food, water, shelter); Safety needs (security, stability); Love and belonging (friendship, intimacy); Esteem needs (achievement, respect); and Self‑actualisation (fulfilling one’s potential). The theorem states that lower‑level needs must be reasonably satisfied before a person can consistently focus on higher‑level ones. It is not a strict formula, but it predicts behaviour patterns – a hungry student will struggle to concentrate on algebra.
亚伯拉罕·马斯洛提出，人类行为由一套需求层次所驱动，通常呈现为金字塔形。从底层到顶层依次为：生理需求（食物、水、住所）；安全需求（保障、稳定）；爱与归属（友谊、亲密关系）；尊重需求（成就、受人尊重）；以及自我实现（发挥个人潜能）。该理论认为，只有当较低层次的需求得到基本满足后，个体才能持久地关注更高层次的需求。这虽非严格的公式，却能预测行为模式——一个饥饿的学生很难专心于代数题目。
Maslow’s thinking reminds educational psychologists that a child who feels unsafe or unloved at home may find it extremely difficult to learn. Schools often provide free breakfast programmes and pastoral support precisely to help meet those foundational physiological and safety needs so that students can engage in learning and eventually aim for self‑actualisation.
马斯洛的观点提醒教育心理学家，一个在家中感到不安全或不被关爱的孩子可能极难投入学习。学校往往提供免费早餐和关怀辅导，正是为了帮助满足那些基础的生理与安全需求，使学生能够投身学习，并最终追求自我实现。
10. Thorndike’s Law of Effect | 桑代克效果律
Edward Thorndike’s Law of Effect states that responses that produce a satisfying outcome in a particular situation become more likely to occur again in that situation, whereas responses that produce an uncomfortable outcome become less likely. This can be expressed in a principle‑driven way: probability(repetition) ∝ satisfaction gained. Although the law is qualitative, it laid the groundwork for operant conditioning and reinforcement schedules later formalised by B. F. Skinner. Essentially, behaviours are stamped in by rewards and stamped out by punishments.
爱德华·桑代克的效果律指出，在某一情境中带来满意结果的反应，今后在该情境中重复出现的可能性更高；带来不适结果的反应，则更不易重复。该定律可用原则性方式表达：重复概率 ∝ 所获满意度。尽管该律是定性陈述，却为后来 B. F. 斯金纳的操作性条件反射和强化计划奠定了基础。本质上，行为会因奖赏而“印入”，因惩罚而“印出”。
For example, if a cat accidentally pulls a string and gets out of a puzzle box to obtain food, the pulling behaviour is reinforced and the cat will pull the string faster on subsequent trials. In the classroom, a student who answers a question correctly and receives praise is likely to volunteer again. The Law of Effect remains a central idea in behaviourist psychology.
例如，如果一只猫偶然拉动了绳索，逃出谜箱并获得食物，拉绳行为就受到强化，这只猫在后续试验中会更快地拉绳。在课堂上，正确回答问题并受到表扬的学生更可能再次主动举手。效果律至今仍是行为主义心理学的核心思想。
11. Bandura’s Social Learning Theory (Observational Learning) | 班杜拉社会学习理论（观察学习）
Albert Bandura’s Social Learning Theory proposes that people learn not only through direct reinforcement but also by observing others (models). The key processes are attention, retention, reproduction, and motivation. There is no mathematical formula, but the theory can be summarised by the principle: Learning = Observation × Motivation. This means that for observational learning to occur, the learner must pay attention to the model’s behaviour, remember it, be physically able to copy it, and have a reason (motivation) to do so – often because they expect a reward or see the model being reinforced.
阿尔伯特·班杜拉的社会学习理论认为，人不仅通过直接强化学习，还通过观察他人（榜样）进行学习。关键过程包括注意、保持、动作再现和动机。虽然没有数学公式，但该理论可用原理概括为：学习 = 观察 × 动机。这意味着，要发生观察学习，学习者必须注意榜样的行为、将其记住、有能力再现该行为，并有理由（动机）去做——通常是因为他们预计会获得奖赏，或看到榜样受到了强化。
The famous Bobo doll experiment showed that children who watched an adult act aggressively towards an inflatable doll later imitated the aggressive actions, especially if the adult was praised. This highlights that media and role models can significantly influence behaviour, a lesson that is very relevant in the age of social media.
著名的波波玩偶实验显示，观察到成人对充气娃娃做出攻击行为的儿童，随后会模仿攻击动作，尤其是当成人的行为受到表扬时。这凸显了媒体与榜样能够显著影响行为，在社交媒体时代这一教训格外具有现实意义。
12. Hick’s Law of Decision Time | 希克决策时间定律
Hick’s Law describes the time it takes for a person to make a decision as the number of possible choices increases. The equation is RT = a + b log₂(n), where RT is reaction time, n is the number of equally probable alternatives, and a and b are constants. The logarithm to base 2 means that doubling the number of choices adds a constant amount of time to the decision. This law is valuable in human‑computer interaction: menus with too many items slow down user response. In an experiment, if you have 2 possible response buttons, your reaction time might be 400 ms; with 8 buttons it rises, but not fourfold – because the relationship is logarithmic.
希克定律描述了当可能选项数量增加时，个人做出决定所需的时间。公式为 RT = a + b log₂(n)，其中 RT 为反应时间，n 是等概率备选项的数量，a 和 b 是常数。以 2 为底的对数意味着选项数量翻倍只会使决策时间增加一个常量。这一定律在人机交互领域很有用：菜单选项过多会拖慢用户反应。在实验中，若有两个可能的反应按钮，你的反应时可能是 400 ms；当按钮变成 8 个时，反应时虽会增加，但并非变为四倍——因为关系是对数的。
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