📚 KS3 CCEA Psychology: Formula & Theorem Quick Reference Handbook | KS3 CCEA 心理学：公式定理速查手册
Welcome to your quick reference guide for key formulas, laws, and theorems in KS3 CCEA Psychology. Although psychology is not a subject of hard mathematics, it does contain several important numerical relationships, scientific laws, and theoretical principles that help explain how people think, feel, and behave. This handbook organises these essentials into bite‑sized, bilingual sections, so you can revise with confidence and clarity. Keep this article handy for homework, class tests, and end‑of‑topic assessments.
欢迎使用 KS3 CCEA 心理学关键公式、定律与定理的速查手册。虽然心理学不是一门以数学见长的学科，但它确实包含若干重要的数量关系、科学定律和理论原则，用以解释人的思维、情感和行为。本手册将这些要点整理成简洁的双语小节，方便你自信而清晰地复习。在完成作业、课堂测验和单元评估时，可随时查阅本文。
1. Weber’s Law | 韦伯定律
Weber’s Law states that the just noticeable difference (JND) between two stimuli is a constant proportion of the original stimulus intensity. The formula is ΔI / I = k, where ΔI is the smallest detectable change, I is the initial intensity, and k is the Weber fraction. This means that if you are holding a 100 g weight, you might need an extra 5 g to notice a difference; but for a 200 g weight, you would need around 10 g. The exact value of k depends on the sensory modality.
韦伯定律指出，两个刺激之间的最小可觉差（JND）与原始刺激强度之比是一个常数。公式为 ΔI / I = k，其中 ΔI 是最小可察觉的变化量，I 是初始强度，k 是韦伯分数。这意味着如果你手持 100 克的重物，可能需要再增加 5 克才能感觉到差异；但如果初始重量是 200 克，就大约需要增加 10 克。k 的具体值取决于感觉通道。
- Vision (brightness): k ≈ 0.08
- Hearing (pitch): k ≈ 0.003
- Touch (pressure): k ≈ 0.14
韦伯定律告诉我们，对变化的敏感度与背景强度成比例。这在日常感知和实验心理学中都有应用。
2. Fechner’s Law | 费希纳定律
Fechner’s Law builds on Weber’s findings and describes the relationship between the physical intensity of a stimulus and the perceived sensation magnitude. It is expressed as S = k log(I), where S is the subjective sensation, I is the physical intensity, and k is a constant. Psychologically, this means that equal steps in sensation correspond to equal ratios of physical intensity. For example, doubling the brightness of a light does not double the perceived brightness — instead, the perceived change follows a logarithmic scale.
费希纳定律在韦伯研究的基础上，描述了刺激的物理强度与感知到的感觉量之间的关系。其公式为 S = k log(I)，其中 S 是主观感觉，I 是物理强度，k 是常数。从心理学上讲，这意味着感觉的等量增加，对应物理强度按固定比例增加。例如，将灯光的物理亮度加倍，并不会让你感觉也亮了一倍——主观感觉的变化遵循对数尺度。
Fechner’s Law helps to explain why we are more sensitive to small changes in weak stimuli than in strong ones. It is one of the earliest psychophysical laws linking mind and body.
费希纳定律有助于解释为什么我们对微弱刺激的小幅变化比对强烈刺激的变化更敏感。这是最早将心理与身体联系起来的心理物理学定律之一。
3. Yerkes–Dodson Law | 耶克斯–多德森定律
The Yerkes–Dodson Law is a principle linking arousal level to performance. It states that performance increases with physiological or mental arousal up to an optimal point, after which performance declines. The curve is often drawn as an inverted U. For simple or well‑learned tasks, the optimal arousal level is relatively high; for complex or unfamiliar tasks, the optimal level is lower. There is no single formula, but the relationship can be summarised as: Performance = f(arousal), with f being an inverted‑U function.
耶克斯–多德森定律是一条将唤醒水平与绩效联系起来的原理。它指出，随着生理或心理唤醒度的升高，绩效会提升，直到达到一个最佳点，之后绩效就会下降。这一关系通常被绘制成倒 U 形曲线。对于简单或熟练掌握的任务，最佳唤醒水平相对较高；对于复杂或不熟悉的任务，最佳唤醒水平则较低。虽然不存在单一公式，但可将这一关系概括为：绩效 = f（唤醒度），其中 f 为倒 U 形函数。
Understanding this law can help you manage stress during exams: a moderate level of anxiety can motivate you, but too much fear may cause you to blank out.
理解该定律有助于你应对考试压力：适度的焦虑能激励你，但过度恐惧可能导致大脑一片空白。
4. Ebbinghaus Forgetting Curve | 艾宾浩斯遗忘曲线
Hermann Ebbinghaus discovered that memory loss follows a predictable pattern when there is no attempt to retain the information. The equation R = e^(-t/S) is sometimes used to model the proportion of retained material over time, where R is retention, t is time, and S is the strength of memory. In simple terms, forgetting is rapid at first and then levels off. After 20 minutes, we may forget nearly 40% of learned nonsense syllables; after a day, about 66%; and after a month, around 75% — though modern research shows varying rates.
赫尔曼·艾宾浩斯发现，在未曾刻意保持的情况下，记忆的遗忘遵循可预测的模式。有时会用 R = e^(-t/S) 来模拟随时间保留的材料比例，其中 R 为保持量，t 为时间，S 为记忆强度。简言之，遗忘先快后慢，逐渐趋于平缓。学习无意义音节 20 分钟后，可能遗忘近 40%；一天后约 66%；一个月后约 75%——不过现代研究显示不同材料的遗忘率有所不同。
The Ebbinghaus Curve highlights the importance of spaced repetition and regular review. Using revision timetables that revisit topics after increasing intervals can dramatically slow forgetting.
艾宾浩斯曲线强调了间隔重复和定期复习的重要性。利用复习时间表，按逐渐增加的间隔重温学习内容，可以显著减缓遗忘。
5. Multi‑Store Model of Memory (Atkinson & Shiffrin) | 记忆的多重存储模型
This model divides memory into three distinct stages: Sensory Memory, Short‑Term Memory (STM), and Long‑Term Memory (LTM). The flow of information is often described by the following sequence: environmental input → sensory memory (attention) → STM (rehearsal) → LTM. STM is limited in capacity (about 7±2 chunks) and duration (15–30 seconds without rehearsal). LTM is believed to have unlimited capacity and a much longer duration. Forgetting from STM occurs mainly through displacement; from LTM through interference or retrieval failure.
该模型将记忆分为三个不同的阶段：感觉记忆、短时记忆（STM）和长时记忆（LTM）。信息流动的顺序通常为：环境输入 → 感觉记忆（注意）→ 短时记忆（复述）→ 长时记忆。短时记忆的容量有限（约 7±2 个组块），持续时间短（若无复述，仅 15–30 秒）。长时记忆则被认为容量无限，且可持续极长时间。短时记忆的遗忘主要由于信息被替换，长时记忆的遗忘则源于干扰或提取失败。
Key terms to remember: encoding (converting information into a usable form), storage (keeping information over time), and retrieval (getting information back out).
需要记住的关键术语：编码（将信息转化为可用形式）、存储（随时间保存信息）和提取（取出信息）。
6. Miller’s Magic Number | 米勒的神奇数字
George Miller proposed that the capacity of short‑term memory is about 7 items, plus or minus 2. This is often called Miller’s Magic Number 7±2. The ‘items’ refer to meaningful units or chunks. Through chunking, we can combine several individual pieces of information into one meaningful whole, thereby increasing the total amount of detail we can hold in STM. For instance, the letter sequence ‘FBICIAUK’ is 8 letters but can be chunked as ‘FBI’, ‘CIA’, ‘UK’ — just 3 chunks.
乔治·米勒提出，短时记忆的容量大约为 7±2 个项目，这常被称为米勒的神奇数字 7±2。“项目”指有意义的信息单元或组块。通过组块化，我们可以把多个孤立的信息组合成一个有意义的整体，从而增加短时记忆中可容纳的细节总量。例如，字母序列 ‘FBICIAUK’ 有 8 个字母，但可以分组为 ‘FBI’、’CIA’、’UK’——仅 3 个组块。
Applying chunking in your revision can help you remember longer sequences of facts, numbers, or dates more efficiently.
在复习中运用组块法，能帮你更高效地记住较长的事实、数字或日期序列。
7. Classical Conditioning Formula (Pavlov) | 经典条件作用公式
Classical conditioning is a learning process in which a neutral stimulus becomes associated with an unconditioned stimulus to produce a conditioned response. A simplified formula is: NS (neutral stimulus) + UCS (unconditioned stimulus) → UCR (unconditioned response). After repeated pairings: CS (conditioned stimulus) → CR (conditioned response). For example, in Pavlov’s experiment, food (UCS) naturally causes salivation (UCR). The bell (NS) is paired with food, and eventually the bell alone (CS) elicits salivation (CR).
经典条件作用是一种学习过程，其中中性刺激与无条件刺激建立联结，从而引发条件反应。简化的公式为：中性刺激（NS）+ 无条件刺激（UCS）→ 无条件反应（UCR）。经过多次配对后：条件刺激（CS）→ 条件反应（CR）。例如，在巴甫洛夫的实验中，食物（UCS）天然地引起唾液分泌（UCR）。将铃声（NS）与食物配对，最终铃声单独（CS）就能引起唾液分泌（CR）。
Other key concepts: extinction (CR disappears when CS is presented without UCS), spontaneous recovery (reappearance of CR after a rest period), generalisation (responding to similar stimuli), and discrimination (not responding to different stimuli).
其他关键概念：消退（单独呈现 CS 而无 UCS 时，CR 消失）、自发恢复（经休息后 CR 再次出现）、泛化（对相似刺激作出反应）和辨别（不对不同刺激作出反应）。
8. Operant Conditioning Reinforcement Schedules | 操作性条件作用的强化程序
Operant conditioning, proposed by B.F. Skinner, explains how behavior is shaped by consequences. The four main types of reinforcement schedules are described by their mathematical patterns of delivery — though no single formula exists, we can think of them as rules: 1) Fixed Ratio (FR): reward after a set number of responses; 2) Variable Ratio (VR): reward after an unpredictable number of responses (average given); 3) Fixed Interval (FI): reward for the first response after a fixed time has passed; 4) Variable Interval (VI): reward after varying time intervals, making timing unpredictable.
操作性条件作用由 B.F. 斯金纳提出，解释行为如何被后果所塑造。四种主要的强化程序可以通过其传递的数学模式来描述——尽管没有单一公式，但我们可以视其为规则：1）固定比率（FR）：完成设定次数的反应后给予奖励；2）可变比率（VR）：经过不可预测的反应次数后给予奖励（给定平均值）；3）固定时距（FI）：固定时间过后，对第一次反应给予奖励；4）可变时距（VI）：不同时间间隔后给予奖励，使得时间难以预测。
VR and FR schedules typically produce higher response rates than interval schedules. Variable schedules are more resistant to extinction.
VR 和 FR 程序产生的反应率通常高于时距程序。可变程序更难以消退。
| Schedule | Pattern | Example |
| FR | Reward after every nth response | Getting paid for every 10 shirts sewn |
| VR | Reward after an average of n responses | Slot machines — win after an unpredictable number of pulls |
| FI | Reward after first response after t minutes | Checking the post arrives at 9 am |
| VI | Reward after varying times, averaging t minutes | Checking email — messages arrive at unpredictable intervals |
操作性条件作用强化程序表，包含固定比率、可变比率、固定时距和可变时距。
9. IQ Formula | 智商公式
Although modern IQ tests use deviation scores, the historical concept of intelligence quotient is calculated using the simple formula: IQ = (Mental Age / Chronological Age) × 100. A child whose mental age equals their chronological age would have an IQ of 100, which is exactly average. If a 10‑year‑old performs at the level of an average 12‑year‑old, their IQ would be (12/10)×100 = 120. This formula is less reliable for adults, which is why modern tests use standard deviations.
虽然现代智力测验采用离差分数，但智力商数的历史概念可以通过简单公式计算：IQ =（智力年龄 / 实际年龄）× 100。智力年龄与实际年龄相等的儿童，其 IQ 为 100，恰好是平均水平。如果一个 10 岁的儿童表现达到 12 岁平均水平，其 IQ 为 (12/10)×100 = 120。这一公式对成年人的可靠性较低，因此现代测验改用标准差计算。
- Average IQ range: 85–115
- Above 130: often considered gifted
- Below 70: may indicate intellectual disability
平均 IQ 范围：85–115；130 以上通常被视为天才；70 以下可能表示智力障碍。
10. Signal Detection Theory (d’ and Criterion) | 信号检测论
Signal Detection Theory (SDT) provides a mathematical framework for understanding how we make decisions under uncertainty. It separates two factors: sensitivity (d’) — how well you can distinguish a signal from noise, and criterion (β or c) — the threshold or willingness to say ‘yes’. The formula d’ = z(Hit) − z(False Alarm) captures discrimination ability. A liberal criterion leads to many hits but also many false alarms; a conservative criterion reduces false alarms but may miss signals.
信号检测论（SDT）提供了一个数学框架，用以理解我们如何在不确定条件下做出决策。它将两个因素分开：敏感性（d’）——你分辨信号与噪音的能力，以及判断标准（β 或 c）——即回答“是”的阈值或意愿。公式 d’ = z(击中率) − z(虚报率) 反映了辨别能力。宽松的判断标准会导致大量击中，但也带来许多虚报；保守的标准则减少虚报，但可能漏掉信号。
This theory is widely applied in medical diagnosis, eyewitness testimony, and even everyday decisions like deciding whether you heard your phone ring in a noisy room.
这一理论广泛应用于医学诊断、目击证词，甚至日常决策，比如在嘈杂的房间中判断手机是否响起。
11. S–O–R Model (Stimulus–Organism–Response) | S–O–R 模型（刺激–机体–反应）
Moving beyond simple S–R (stimulus–response) psychology, the S–O–R model adds an intervening Organism variable to account for internal mental processes. The ‘formula’ can be written as: S → O → R, where the organism includes motives, emotions, perceptions, and cognitive interpretations. For example, a grade on a test (S) may lead to different responses depending on whether the student interprets it as a challenge (O produces excitement and hard work) or as a threat (O produces anxiety and avoidance).
S–O–R 模型超越了简单的 S–R（刺激–反应）心理学，加入了一个中介变量“机体”，以解释内部心理过程。该“公式”可写作：S → O → R，其中机体包括动机、情绪、知觉和认知解释。例如，一次考试成绩（S）可能导致不同的反应，取决于学生是将它视为挑战（O 产生兴奋和努力）还是威胁（O 产生焦虑和回避）。
This model is central to cognitive psychology and helps explain why the same event can cause different reactions in different people.
该模型是认知心理学的核心，有助于解释为何同一事件会在不同的人身上引起不同的反应。
12. Social Impact Theory Formula (Latané) | 社会影响理论公式
Bibb Latané’s Social Impact Theory proposes that the amount of influence a source has on a target depends on three factors: Strength (S), Immediacy (I), and Number (N) of sources. The impact can be represented as: Impact = f(SIN), meaning the multiplicative combination of strength, immediacy, and number. Strength refers to the source’s power/status; immediacy is closeness in space/time; number is the quantity of people. However, the effect of number follows a power law: the nth person adds less impact than the previous one (marginal impact diminishes).
比布·拉坦内的社会影响理论提出，影响源对目标个体的影响量取决于三个因素：强度（S）、接近性（I）和数量（N）。影响可表示为：Impact = f(SIN)，即强度、接近性和数量的乘积组合。强度指影响源的权力/地位，接近性指空间/时间上的亲近程度，数量指人数。然而，数量的效果遵循幂律：第 n 个人带来的影响增量要小于前一个人（边际影响递减）。
This theory helps explain why a single close friend (high immediacy, moderate strength) can sometimes persuade you more than a large online audience (high number, low immediacy).
该理论有助于解释为什么一个亲密的朋友（高接近性、中等强度）有时比一个庞大的线上观众（高数量、低接近性）更能说服你。
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