📚 Year 8 CAIE Psychology: Formula & Theorem Quick-Reference Handbook | Year 8 CAIE 心理学：公式定理速查手册
Welcome to your essential quick-reference guide for Year 8 CAIE Psychology. Just like in mathematics or physics, psychology has its own set of 'formulas' and 'theorems' — powerful statements that summarise how our mind works, how we remember, learn, and interact. This handbook gathers the most important relationships, models, and principles you will encounter in your course, expressed in a concise, ready-to-review format. Whether you need to recall Miller's magic number, the flow of the multi-store memory model, or the rule for calculating a mean, this document puts the key formulas at your fingertips.
欢迎使用 Year 8 CAIE 心理学必备速查手册。与数学或物理一样，心理学也有自己的一套“公式”和“定理”——这些强有力的陈述概括了我们的思维如何运作、如何记忆、学习和互动。本手册汇集了课程中最重要的关系、模型和原则，以简洁的、便于复习的形式呈现。无论你需要回忆米勒的神奇数字、多储存记忆模型的流程，还是计算平均数的规则，这份文件都让你能快速查阅关键公式。
1. Miller's Magic Number 7 ± 2 | 米勒的神奇数字 7 ± 2
George Miller (1956) proposed that the capacity of short-term memory (STM) is about 7 items, plus or minus 2. This means most people can hold between 5 and 9 chunks of information in their immediate memory at any one time. A 'chunk' can be a single digit, a letter, a word, or even a familiar pattern. The formula for STM capacity can be written as: Capacity = 7 ± 2 chunks. By grouping information into larger, meaningful chunks, we can effectively expand how much we store.
乔治·米勒（1956年）提出，短时记忆的容量约为7个项目，上下浮动2个。这意味着大多数人在任何时刻都能在即时记忆中保持5到9个信息组块。一个“组块”可以是一个数字、一个字母、一个单词，甚至是一个熟悉的模式。短时记忆容量的公式可以写作：容量 = 7 ± 2 个组块。通过将信息组合成更大、有意义的组块，我们可以有效地扩展存储量。
2. Ebbinghaus Forgetting Curve | 艾宾浩斯遗忘曲线
Hermann Ebbinghaus discovered that memory loss follows a predictable pattern. His forgetting curve shows that after learning new material, retention drops sharply within the first hour, then the rate of forgetting slows down. The relationship can be approximated by the retention formula: R ≈ 100k / ((log₁₀ t)ᶜ + k), where R is percentage retained, t is time since learning, and k and c are constants. In simpler terms, you might forget about 40% within the first 30 minutes, but after a day or two, the loss levels off. Spaced repetition is the direct application of this theorem to boost long-term memory.
赫尔曼·艾宾浩斯发现，记忆的遗忘遵循一种可预测的模式。他的遗忘曲线表明，在学习新材料后，记忆保持量在第一个小时内急剧下降，然后遗忘速度放缓。这种关系可以用保持量公式近似表达：R ≈ 100k / ((log₁₀ t)ᶜ + k)，其中R是保持百分比，t是学习后的时间，k和c是常数。简单来说，你可能在前30分钟内忘记约40%的内容，但一两天后遗忘趋于平缓。间隔重复就是应用这一定理来增强长期记忆的直接方法。
3. Classical Conditioning Formula | 经典条件作用公式
Ivan Pavlov's classical conditioning can be expressed as a sequence of relationships. Before conditioning, the unconditioned stimulus (UCS) naturally triggers the unconditioned response (UCR): UCS → UCR. A neutral stimulus (NS) produces no relevant response: NS → no response. During acquisition, the NS is paired with the UCS repeatedly. The formula for the learned association is: NS + UCS → UCR. After conditioning, the NS becomes the conditioned stimulus (CS), which now elicits the conditioned response (CR): CS → CR. Extinction, spontaneous recovery, and discrimination are all explained by variations in this pairing regularity.
伊万·巴甫洛夫的经典条件作用可以用一系列关系来表达。在条件作用之前，无条件刺激（UCS）自然引发无条件反应（UCR）：UCS → UCR。中性刺激（NS）不产生相关反应：NS → 无反应。在习得阶段，NS与UCS反复匹配。习得的关联公式是：NS + UCS → UCR。条件作用之后，NS变成条件刺激（CS），此时它能引发条件反应（CR）：CS → CR。消退、自然恢复和分化都可以用这种配对的规律变化来解释。
4. Operant Conditioning Consequences | 操作性条件作用结果
B.F. Skinner's operant conditioning centres on a simple rule: behaviour is shaped by its consequences. The probability of a behaviour occurring again is determined by the consequence that follows. We can present the four key equations: Positive Reinforcement: Behaviour + Pleasant stimulus → Behaviour increases. Negative Reinforcement: Behaviour – Unpleasant stimulus → Behaviour increases. Positive Punishment: Behaviour + Unpleasant stimulus → Behaviour decreases. Negative Punishment: Behaviour – Pleasant stimulus → Behaviour decreases. These four formulae are the bedrock of behaviour modification token economies and classroom management.
B.F. 斯金纳的操作性条件作用以一条简单规则为核心：行为由其后果所塑造。某一行为再次发生的概率取决于随之而来的后果。我们可以列出四个关键公式：正强化：行为 + 愉快刺激 → 行为增加。 负强化：行为 – 厌恶刺激 → 行为增加。 正惩罚：行为 + 厌恶刺激 → 行为减少。 负惩罚：行为 – 愉快刺激 → 行为减少。这四个公式是行为矫正代币制和课堂管理的基石。
5. Multi-Store Model of Memory Flow | 多储存记忆模型流程
Atkinson and Shiffrin (1968) proposed that memory consists of three linked stores. Information moves in a linear sequence: ensory Register → Attention → Short-Term Memory → Elaborative Rehearsal → Long-Term Memory. If attention is not paid, information in the sensory register decays within a fraction of a second. Without rehearsal, STM holds material for about 18–30 seconds before it is displaced or decayed. The maintenance rehearsal loop can keep information cycling in STM, but only elaborative rehearsal transfers it to LTM. This directional flow is the central theorem of the model.
阿特金森和希夫林（1968年）提出，记忆由三个相互连接的结构组成。信息按线性顺序移动：感觉登记 → 注意 → 短时记忆 → 精细复述 → 长时记忆。如果不加注意，感觉登记中的信息在几分之一秒内便衰退。如果没有复述，短时记忆保持材料约18–30秒后便会被替换或衰退。维持性复述环路可以使信息在短时记忆中循环，但只有精细复述才能将信息转入长时记忆。这种定向流动是该模型的核心定理。
6. Serial Position Effect Formula | 系列位置效应公式
When recalling a list of items in order, the likelihood of correct recall is not equal for all positions. The serial position curve shows a U-shaped pattern with two components. The primacy effect gives high recall for items at the beginning (early items transferred to LTM), and the recency effect gives high recall for items at the end (still active in STM). We can state the formula: Recall probability = f(Position), where f is bimodal, peaking at the first few and last few positions. Middle items suffer the poorest recall, which directly informs revision strategies: place the most important content at the beginning or end of a study session.
当按顺序回忆一系列项目时，正确回忆的概率并非对所有位置都相同。系列位置曲线呈现U型，包含两个成分。首因效应使位于开头项目回忆率高（早期项目转入了长时记忆），而近因效应使位于末尾项目回忆率高（仍活跃在短时记忆中）。我们可以提出如下公式：回忆概率 = f(位置)，其中f是双峰函数，在前几个和后几个位置达到峰值。中间项目回忆最差，这直接启示了复习策略：将最重要的内容放在学习阶段的开始或末尾。
7. Conformity and Group Size Theorem | 从众与团体规模定理
Solomon Asch's classic line-judgement experiments revealed a clear link between group size and conformity. When participants were faced with a unanimous but incorrect majority, the probability of conforming increased with group size. The theorem can be summarised: Conformity pressure ∝ Unanimous majority size (up to about 4 confederates). After the majority reaches 3 or 4 people, further increases in group size add little additional conformity. The presence of a single dissenter reduces conformity dramatically, a relationship captured by: Dissenter presence → Conformity drops by ~70%. These regularities highlight the power of social influence.
所罗门·阿希的经典线条判断实验揭示了团体大小与从众之间的明确联系。当参与者面对一个一致但错误的多数时，从众的概率随团体大小而增加。该定理可以概括为：从众压力 ∝ 一致多数的大小（最多约4名同谋）。当多数达到3或4人时，继续增加人数几乎不再提升从众率。若存在一个持异议者，会显著降低从众行为，这层关系可用公式表达：异议者出现 → 从众率下降约70%。这些规律凸显了社会影响的威力。
8. Piaget's Cognitive Development Stages | 皮亚杰认知发展阶段
Jean Piaget's theory specifies that children's thinking progresses through four major stages according to approximate age ranges. The theorem of age-bound cognitive shift is expressed as a sequence:
Sensorimotor: 0–2 years → Object permanence mastered.
Preoperational: 2–7 years → Symbolic thinking, egocentrism.
Concrete operational: 7–11 years → Logical thought about concrete events, conservation.
Formal operational: 11+ years → Abstract reasoning, hypothetical thinking.
While ages are flexible, the invariant sequence of these stages is a core formula for understanding intellectual development. Each stage is characterised by distinct schemas and operations.
让·皮亚杰的理论指出，儿童思维按照大致的年龄范围经历四个主要阶段。年龄限定的认知转换定理用序列表达为：
感觉运动阶段：0–2岁 → 获得客体永久性。
前运算阶段：2–7岁 → 符号思维，自我中心。
具体运算阶段：7–11岁 → 对具体事件的逻辑思维，守恒。
形式运算阶段：11岁以上 → 抽象推理，假设性思维。
虽然年龄是灵活的，但这些阶段不变的顺序是理解智力发展的核心公式。每个阶段都以独特的图式和运算为特征。
9. Brain Area Functions Map | 脑区功能映射
Modern psychology relies on localisation of function — the theorem that specific brain regions perform specific tasks. The key mapping can be summarised as a functional equation:
Frontal lobe → Executive functions, decision-making, personality
Temporal lobe → Auditory processing, memory (hippocampus)
Parietal lobe → Sensory integration, spatial awareness
Occipital lobe → Visual processing
Cerebellum → Balance, motor coordination
Brainstem → Vital functions (breathing, heart rate)
This set of localisation relations is fundamental to understanding biological explanations in psychology, especially when linking damage to observed symptoms.
现代心理学依赖于功能定位学说——即特定脑区执行特定任务这一定理。关键映射可以概括为功能等式：
额叶 → 执行功能、决策、人格
颞叶 → 听觉加工、记忆（海马体）
顶叶 → 感觉整合、空间意识
枕叶 → 视觉加工
小脑 → 平衡、运动协调
脑干 → 维持生命的功能（呼吸、心率）
这一组定位关系是理解心理学中生物学解释的基础，尤其是在将脑损伤与观察到的症状联系起来时。
10. Descriptive Statistics: Mean, Median, Mode | 描述统计：平均数、中位数、众数
Data analysis in psychology relies on three central measures of central tendency, each with its own formula.
- Mean (X̄): the arithmetic average. Formula: X̄ = Σxᵢ / n, where Σxᵢ is the sum of all values and n is the number of scores.
- Median odd: median = the ((n+1)/2)th score. If n is even: median = mean of the two middle scores.
- Mode: the most frequently occurring value in the set. A distribution may be unimodal, bimodal, or have no mode.
心理学中的数据分析依赖于三种主要的集中趋势量数，每种都有自己的公式。
- 平均数（X̄）：算术平均值。公式：X̄ = Σxᵢ / n，其中Σxᵢ是所有数值的总和，n是分数个数。
- 中位数：有序列表中的中间值。若n为奇数：中位数 = 第((n+1)/2)个分数。若n为偶数：中位数 = 中间两个分数的平均数。
- 众数：数据集中出现频率最高的数值。分布可以是单峰的、双峰的或没有众数。
11. Independent Variable and Dependent Variable Formula | 自变量与因变量公式
Every experiment is built on the relationship between the independent variable (IV) and the dependent variable (DV). The standard experimental formula can be written as: DV = f(IV), meaning the dependent variable is a function of the independent variable. The IV is the variable the researcher manipulates or changes (the cause), and the DV is the variable measured for any effect (the outcome). A well-designed hypothesis takes the form: 'If the IV is changed, then the DV will change in a specific way.' Controlling extraneous variables ensures that the observed DV changes can truly be attributed to the IV.
每个实验都建立在自变量（IV）与因变量（DV）之间的关系之上。标准实验公式可以写作：DV = f(IV)，即因变量是自变量的函数。自变量是研究者操纵或改变的变量（原因），因变量是为观察效果而测量的变量（结果）。一个设计良好的假设采取如下形式：“如果自变量被改变，那么因变量将以特定方式改变。”控制额外变量能确保观察到的因变量变化真正归因于自变量。
12. The Yerkes-Dodson Law of Arousal | 耶克斯-多德森唤醒定律
Performance on tasks is linked to arousal level through an inverted U-shaped function. The Yerkes-Dodson Law states: Performance = f(Arousal), where performance increases with arousal up to an optimal point, then declines as arousal becomes too high. For complex or difficult tasks, the optimal arousal level is lower; for simple or well-learned tasks, it is higher. This can be expressed as a set of conditional statements: If task is difficult → optimal arousal low; if task is easy → optimal arousal high. This theorem explains why moderate stress can boost exam performance, but excessive anxiety impairs it.
任务表现与唤醒水平之间通过一条倒U型函数相关联。耶克斯-多德森定律指出：表现 = f(唤醒)，表现随唤醒增加而提高，直至达到最优点，然后随唤醒过高而下降。对于复杂或困难的任务，最佳唤醒水平较低；对于简单或熟练的任务，最佳唤醒水平较高。这可以表述为一组条件语句：若任务困难 → 最佳唤醒低；若任务容易 → 最佳唤醒高。这一定理解释了为什么适度压力能提升考试成绩，而过度焦虑则会削弱表现。
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