📚 Year 9 CAIE Psychology: Quick Reference to Formulas and Theorems | Year 9 CAIE 心理学：公式定理速查手册
This handbook brings together the essential quantitative relationships, laws, and models you will encounter in the Year 9 CAIE Psychology course. Although psychology is often seen as a descriptive subject, many of its most powerful ideas can be expressed as clear, testable principles — some even resembling mathematical formulas. Use this reference to consolidate your understanding of perception, memory, learning, and social behaviour.
本手册汇集了 Year 9 CAIE 心理学课程中你将遇到的核心数量关系、定律和模型。虽然心理学常被视为描述性学科，但它许多最有力的思想都可以被表达为清晰、可检验的原理，其中一些甚至接近于数学公式。使用这份速查资料来巩固你对感知、记忆、学习和社会行为的理解。
1. Weber's Law | 韦伯定律
Weber's Law describes the relationship between the intensity of a stimulus and the minimum change needed for a person to notice a difference. For a given sensory modality, the just noticeable difference (JND) is a constant proportion of the original stimulus intensity. The formula is ΔI / I = k, where ΔI is the JND, I is the initial stimulus intensity, and k is the Weber fraction (constant for that sense). This law helps explain why you can easily detect a whisper added to a quiet room but might miss the same whisper in a noisy concert.
韦伯定律描述了刺激强度与一个人察觉差异所需的最小变化之间的关系。对于特定的感觉通道，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JND，I 是初始刺激强度，k 是韦伯分数（对该感觉来说是常数）。这条定律有助于解释为什么在安静的房间中加入轻声细语你能轻易察觉，但在嘈杂的音乐会中却可能忽略同样的细微声音。
2. Fechner's Law | 费希纳定律
Fechner's Law extends Weber's findings by proposing that sensation magnitude increases as the logarithm of stimulus intensity. The formula is S = k log I, where S is the perceived sensation magnitude, I is the physical stimulus intensity, and k is a constant that depends on the sensory system. This logarithmic relationship means that to double the perceived brightness of a light, you must multiply its physical intensity by a factor much greater than two. Fechner's Law underpins modern psychophysics and explains why sensory perception compresses wide physical ranges into manageable subjective scales.
费希纳定律扩展了韦伯的发现，提出感觉量级随刺激强度的对数增加。公式为 S = k log I，其中 S 是感知的感觉量级，I 是物理刺激强度，k 是取决于感觉系统的常数。这种对数关系意味着，要让你感到灯光的亮度翻倍，其物理强度必须以远大于二的倍数增加。费希纳定律是现代心理物理学的基石，它解释了为何感觉系统将宽广的物理范围压缩为易于管理的主观尺度。
3. Ebbinghaus Forgetting Curve | 艾宾浩斯遗忘曲线
Hermann Ebbinghaus quantified memory decay by learning lists of nonsense syllables and measuring retention over time. The forgetting curve can be described by the formula R = e-t/s, where R is memory retention, t is time since learning, and s is the 'strength' of the memory trace. The curve drops sharply within the first hour after learning and then levels off, meaning the rate of forgetting is fastest immediately after encoding. This discovery highlights the importance of spaced repetition and timely review to strengthen long-term memory.
赫尔曼·艾宾浩斯通过学习无意义音节表并测量不同时间后的保持量，量化了记忆衰退。遗忘曲线可用公式 R = e-t/s 描述，其中 R 为记忆保持量，t 为学习后经过的时间，s 为记忆痕迹的“强度”。曲线在学习的第一个小时内急剧下降，然后趋于平缓，这意味着遗忘速率在编码后即刻最快。这一发现凸显了间隔重复和及时复习对巩固长时记忆的重要性。
4. Miller's Magic Number 7±2 | 米勒的神奇数字7±2
George Miller discovered that the capacity of short-term memory is roughly 7 ± 2 items or chunks. The formula can be written as STM capacity = 7 ± 2 chunks. By 'chunking' — grouping individual bits of information into meaningful units — we can increase the total amount of information held. For example, the letter sequence 'FBICIAUSA' is easier to remember as four chunks: FBI, CIA, USA. This principle is central to understanding working memory limitations and strategies for improving recall.
乔治·米勒发现，短时记忆的容量大约为 7 ± 2 个项目或组块。公式可写作 STM 容量 = 7 ± 2 组块。通过“组块化”——将单个信息片段组合成有意义的单元——我们能增加所保持的信息总量。例如，字母序列“FBICIAUSA”作为 FBI、CIA、USA 四个组块就更容易记住。这一原理对理解工作记忆的局限性和改善回忆的策略至关重要。
5. Yerkes-Dodson Law | 耶克斯-多德森定律
The Yerkes-Dodson Law describes the relationship between arousal and performance. It can be expressed as an inverted-U function: performance is optimal at a moderate level of arousal and deteriorates when arousal is too low or too high. For complex tasks, the peak shifts toward lower arousal; for simple tasks, toward higher arousal. The formula can be conceptualised as Performance = f(arousal), where f is optimal at medium arousal. This law is vital in explaining exam anxiety, sports performance, and everyday concentration.
耶克斯-多德森定律描述了唤醒水平与表现之间的关系。它可表达为一条倒U型函数：表现于中等唤醒水平时最佳，当唤醒过低或过高时都会下降。对于复杂任务，最佳点移向较低唤醒水平；对于简单任务，则移向较高唤醒水平。该公式可概念化为 表现 = f(唤醒)，其中 f 在中等唤醒时最优。这一定律在解释考试焦虑、运动表现和日常注意力时十分关键。
6. Signal Detection Theory (Sensitivity and Criterion) | 信号检测论（敏感性与判断标准）
Signal Detection Theory distinguishes between an observer's actual sensitivity (d') and their decision criterion (c). Sensitivity measures the ability to discriminate signal from noise: d' = z(Hit) - z(False Alarm), where z converts proportions to standard scores. The criterion c = -0.5 [ z(Hit) + z(False Alarm) ] reflects the observer's bias toward saying 'yes'. This framework explains why a radiologist may miss tumours (conservative criterion) or overcautiously flag everything (liberal criterion), even if their underlying sensitivity remains constant.
信号检测论区分观察者的实际敏感性（d'）与其判断标准（c）。敏感性衡量从噪声中辨别信号的能力：d' = z(击中) - z(虚报)，其中 z 将比例转化为标准分数。标准 c = -0.5 [ z(击中) + z(虚报) ] 反映观察者偏向说“是”的程度。该框架解释了为何放射科医生可能漏掉肿瘤（保守标准）或过度谨慎地标记一切（宽松标准），即便他们的潜在敏感性保持不变。
7. Thorndike's Law of Effect | 桑代克效果律
Edward Thorndike's Law of Effect states that responses followed by satisfying consequences become more likely, while responses followed by unpleasant consequences become less likely. This can be expressed behaviourally as Probability(Response) ∝ Past Satisfaction. The law laid the foundation for operant conditioning, where reinforcement and punishment shape voluntary behaviour. Skinner later formalised this with schedules of reinforcement, but the core empirical relationship remains the same: behaviour is controlled by its outcomes.
爱德华·桑代克的效果律指出，带来满意后果的反应变得更可能发生，而带来不愉快后果的反应则变得不太可能发生。这可在行为上表达为 反应概率 ∝ 过往满意程度。该定律为操作性条件作用奠定了基础，在此过程中强化和惩罚塑造着自发行为。斯金纳后来通过强化程式将其形式化，但核心的经验关系依然不变：行为由其后果所控制。
8. Bowlby's Maternal Deprivation Hypothesis (Time Window) | 鲍尔比的母爱剥夺假说（时间窗口）
John Bowlby proposed that there is a critical period — roughly the first 2.5 years of life — during which an infant must form a continuous, warm, and intimate relationship with a primary caregiver. The formula can be conceptualised as Secure Attachment = f(Continuous Care in First 30 Months). Failure to do so, according to the hypothesis, results in irreversible emotional and intellectual damage. While later research softened the idea of irreversibility, the principle of a sensitive period for attachment remains highly influential in developmental psychology.
约翰·鲍尔比提出，存在一个关键期——大约生命最初的2.5年——在此期间婴儿必须与主要看护者建立起连续、温暖且亲密的关系。该公式可概念化为 安全依恋 = f(前30个月的连续照料)。根据该假说，若未能如此则会导致不可逆的情感和智力损伤。虽然后续研究软化了不可逆的观点，但依恋敏感期的原理在发展心理学中仍极具影响力。
9. Cognitive Dissonance: Magnitude as Function of Justification | 认知失调：失调量与理由间的函数关系
Festinger's theory of cognitive dissonance holds that when a person holds two conflicting cognitions, they experience psychological discomfort proportional to the importance of the elements and the degree of conflict. The magnitude of dissonance can be estimated as D = (Importance × Discrepancy) / (External Justification + 1). A small external reward for counter-attitudinal behaviour produces greater dissonance — and thus more attitude change — than a large reward, because the large reward provides ample justification. This inverse relationship explains findings in forced-compliance experiments.
费斯廷格的认知失调理论认为，当一个人持有两个冲突的认知时，会体验到与元素重要性和冲突程度成正比的心理不适。失调量可估计为 D = (重要性 × 不一致程度) / (外部理由 + 1)。微小的外部奖励引发与态度相悖的行为时，会产生更大的失调——从而导致更多的态度改变——而较大的奖励因提供了充分理由，失调反而更小。这种反比关系解释了被迫顺从实验中的发现。
10. Social Impact Theory: Latané's Equation | 社会影响理论：拉塔内方程
Bibb Latané's Social Impact Theory proposes that the amount of social influence a person experiences depends on strength (S), immediacy (I), and number (N) of sources. The formula is Impact = f(S × I × N), where S is the source's status or power, I is closeness in space or time, and N is the number of sources. However, as N increases, each additional source adds less influence, following a power function with an exponent less than 1. This explains why a single authority figure can exert immense pressure, and why the presence of a few allies dramatically reduces conformity.
比布·拉塔内的社会影响理论提出，一个人所承受的社会影响量取决于影响源的力量（S）、接近性（I）和数量（N）。公式为 影响 = f(S × I × N)，其中 S 是源的地位或权力，I 是时空上的接近性，N 是源的数量。然而，随着 N 增加，每增加一个源所带来的额外影响递减，遵循指数小于1的幂函数。这解释了为何单一的权威人物能施加巨大压力，以及为何少数同盟者的存在会大幅降低从众行为。
11. Zeigarnik Effect: Unfinished Tasks in Memory | 蔡格尼克效应：记忆中的未完成工作
Bluma Zeigarnik found that people remember interrupted or unfinished tasks about twice as well as completed ones. The formula can be stated as Recall(Unfinished) / Recall(Finished) ≈ 2.0. This cognitive bias arises from a mental tension that persists until the task is completed, keeping it active in memory. The effect has practical applications in study strategies — leaving a complex problem partially solved before a break can enhance later recall — and it underscores the role of intrinsic motivation in memory consolidation.
布卢玛·蔡格尼克发现，人们对被打断或未完成的任务的记忆大约是已完成任务的两倍。公式可表述为 回忆(未完成) / 回忆(已完成) ≈ 2.0。这一认知偏差源于任务完成前持续存在的心理张力，使其在记忆中保持活跃。该效应在学习策略中有实际应用——休息前留下部分未解决的复杂问题可增强稍后的回忆——它凸显了内在动机在记忆巩固中的作用。
12. The Peak-End Rule (Kahneman) | 峰终定律（卡尼曼）
Daniel Kahneman's Peaked-End Rule explains how we evaluate past experiences: we do not recall the sum or average of all moments, but rather focus on the most intense moment (the peak) and the ending. The formula is Remembered Utility = Average(Peak Intensity, End Intensity). This has been demonstrated in experiments where patients undergoing painful medical procedures reported a better overall experience when a slightly less painful ending was added, even though it extended the total duration of discomfort. The rule profoundly impacts how experiences should be designed and recalled.
丹尼尔·卡尼曼的峰终定律解释了我们如何评估过往经历：我们并非回忆所有时刻的总和或平均，而是关注最强烈的时刻（峰值）和结尾。公式为 记忆中的效用 = 平均值(峰值强度, 结尾强度)。实验已证明，接受疼痛医疗操作的患者，若在最后添加一个稍缓和的疼痛结尾，即使延长了总不适时间，他们对整个体验的评价却更好。该定律深刻影响着如何设计体验和塑造回忆。
Published by TutorHao | CAIE Psychology Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图