📚 Year 9 CAIE Psychology: Formula & Theorem Quick Reference | Year 9 剑桥国际心理学：公式定理速查手册
This quick reference handbook brings together the essential formulas, statistical calculations, and foundational theorems that appear in the Year 9 CAIE Psychology curriculum. It is designed to help you quickly recall how to compute descriptive statistics, interpret psychological models, and apply key principles in both research methods and core topic areas.
本速查手册汇集了 Year 9 剑桥国际心理学课程中必备的公式、统计计算方法和基础定理。它能帮助你快速回顾如何计算描述性统计量、解读心理学模型，以及在研究方法和核心主题领域中应用关键原理。
1. Mean, Median and Mode: The Core Trio | 平均值、中位数和众数：核心三要素
The mean is calculated by adding all the scores in a data set and then dividing by the total number of scores. It is the arithmetic average and is sensitive to extreme values.
平均值是将数据集中所有分数相加，再除以分数的总个数得到的数值。它是算术平均数，容易受极端值的影响。
Formula for the mean: Mean = (sum of all values) ÷ (number of values).
平均值公式：平均值 = 所有数值的总和 ÷ 数值的个数。
The median is the middle value when the data are arranged in ascending order. If there is an even number of scores, the median is the average of the two middle numbers.
中位数是将数据按升序排列后位于中间位置的数值。如果分数的个数是偶数，中位数则是中间两个数值的平均数。
The mode is the value that appears most frequently in a data set. A set may have one mode, more than one mode, or no mode at all.
众数是数据集中出现频率最高的数值。一个数据集可以有一个众数、多个众数，或者根本没有众数。
2. Range: Measuring Spread | 范围：衡量离散程度
The range gives a quick estimate of the spread of scores. It is found by subtracting the lowest value from the highest value in the set.
范围可以快速估算分数的离散程度，它由数据集中最高值减去最低值得到。
Formula: Range = Highest value – Lowest value.
公式：范围 = 最高值 – 最低值。
Because the range uses only two values, it does not tell us how the remaining scores are distributed, but it is useful for spotting extreme outliers.
由于范围只用到两个数值，它无法告诉我们其余分数是如何分布的，但它有助于发现极端异常值。
3. Percentage Calculations in Psychology | 心理学中的百分比计算
Percentages are widely used in psychology to summarise proportions, for example the percentage of participants who recalled a word correctly or the percentage of people who conformed in an experiment.
百分比在心理学中被广泛用来概括比例，例如正确回忆单词的参与者所占的百分比，或者在实验中表现出从众行为的人所占的百分比。
Formula: Percentage = (Part ÷ Whole) × 100%.
公式：百分比 = (部分 ÷ 整体) × 100%。
To find the percentage change, subtract the original value from the new value, divide by the original value, and multiply by 100. This is useful when comparing results before and after an intervention.
计算百分比变化时，用新值减去原值，除以原值，再乘以 100。这在比较干预前后的结果时非常有用。
4. Ratio and Proportional Reasoning | 比率与比例推理
Ratios express the relationship between two quantities. In a memory experiment, you might describe the ratio of words recalled from the beginning of a list to those recalled from the end as 2:1.
比率表示两个数量之间的关系。在一个记忆实验中，你可能把列表开头回忆出的单词与末尾回忆出的单词之比描述为 2:1。
Proportions tell us what fraction of the total a part represents. For example, if 15 out of 20 participants gave a correct answer, the proportion is 15/20 or 0.75.
比例告诉我们某一部分占整体的分数。例如，如果 20 名参与者中有 15 人给出了正确答案，比例就是 15/20 或 0.75。
Being comfortable with ratios and proportions helps in understanding sampling, effect sizes and comparing experimental conditions.
熟练掌握比率和比例有助于理解抽样、效应量以及比较不同的实验条件。
5. Miller's Law: The Magic Number 7 ± 2 | 米勒定律：神奇数字 7 ± 2
George Miller (1956) proposed that the capacity of short-term memory is about seven items, plus or minus two. This is often called Miller’s Law.
乔治·米勒（1956）提出，短时记忆的容量大约是 7 个项目，可以上下浮动 2 个，这常被称为米勒定律。
An item can be a digit, a letter, or a chunk of meaningful information. Chunking helps us remember more by grouping separate pieces into a larger meaningful unit.
一个项目可以是一个数字、一个字母或是一个有意义的信息组块。组块化可以将零散的信息组合成更大的有意义单元，从而帮助我们记住更多内容。
Capacity formula concept: STM span ≈ 7 ± 2 chunks.
容量公式概念：短时记忆广度 ≈ 7 ± 2 个组块。
6. The Multi-Store Model of Memory (Atkinson & Shiffrin) | 记忆的多储存模型
The Multi-Store Model describes memory as a flow of information through three separate stores: Sensory Memory, Short-Term Memory (STM) and Long-Term Memory (LTM).
多储存模型将记忆描述为信息依次经过三个独立的存储器：感觉记忆、短时记忆和长时记忆。
M (rehearsal) → LTM. Without rehearsal, information is lost through decay or displacement.
信息流程：感觉输入 → 感觉记忆（经注意）→ 短时记忆（经复述）→ 长时记忆。如果没有复述，信息就会因为消退或被替代而丢失。
| Store | 编码 Encoding | 容量 Capacity | 持续时间 Duration |
|---|---|---|---|
| Sensory Memory | Sense-specific (echoic, iconic) | Very large | 0.25–2 seconds |
| Short-Term Memory | Mainly acoustic | 7 ± 2 chunks | 18–30 seconds |
| Long-Term Memory | Mainly semantic | Potentially unlimited | Up to a lifetime |
These properties act as theorems explaining why we remember some material and forget others.
这些特征就像定理一样，解释了我们为什么会记住某些材料而忘记另一些。
7. Classical Conditioning Formulation | 经典条件反射公式
Classical conditioning, discovered by Ivan Pavlov, follows a predictable pattern that can be written as a chain of associations.
由伊万·巴甫洛夫发现的经典条件反射遵循一种可预测的模式，可以写成一系列联结链。
(UCS) → Unconditioned Response (UCR). Neutral Stimulus (NS) produces no specific response.
条件反射形成前：无条件刺激 (UCS) → 无条件反应 (UCR)。中性刺激 (NS) 不引发特定反应。
During acquisition: NS is paired with UCS repeatedly. NS + UCS → UCR.
习得阶段：中性刺激与无条件刺激反复配对。NS + UCS → UCR。
s become a Conditioned Stimulus (CS) and now elicits a Conditioned Response (CR). CS → CR.
条件反射形成后：原来的中性刺激变成了条件刺激 (CS)，现在它能引发条件反应 (CR)。CS → CR。
Extinction occurs when the CS is presented repeatedly without the UCS, and the CR gradually disappears.
当条件刺激多次单独出现而不伴随无条件刺激时，就会发生消退，条件反应逐渐消失。
8. Operant Conditioning Reinforcement Principles | 操作性条件反射强化原理
B.F. Skinner’s operant conditioning focuses on how consequences shape voluntary behaviour. The core law is that behaviour is controlled by its outcomes.
斯金纳的操作性条件反射关注结果如何塑造自愿行为。核心定律是，行为由其结果所控制。
Positive reinforcement: adding a pleasant stimulus after a desired behaviour, making the behaviour more likely to recur (e.g. praise).
正强化：在期待的行为之后增加一个令人愉快的刺激，使该行为更可能再次出现（例如表扬）。
Negative reinforcement: removing an unpleasant stimulus after a desired behaviour, which also increases the likelihood of that behaviour (e.g. turning off a loud alarm).
负强化：在期待的行为之后移除一个令人不快的刺激，这同样会增加该行为出现的可能性（例如关掉响亮的警报）。
Punishment: an unpleasant consequence following an undesired behaviour, which decreases the probability of that behaviour being repeated.
惩罚：在不良行为之后施加一个令人不快的后果，从而降低该行为被重复的概率。
e principles can be summarised as: Behaviour → Consequence → Change in future probability.
这些原理可以概括为：行为 → 后果 → 未来发生概率的改变。
9. Piaget's Stage Theory of Cognitive Development | 皮亚杰认知发展阶段理论
Jean Piaget proposed that children pass through four fixed stages of intellectual growth. Each stage marks a qualitative shift in thinking.
让·皮亚杰提出，儿童会依次经历四个固定的智力发展阶段。每个阶段都标志着思维方式的一次质变。
| Stage | Approximate Age | Key Cognitive Feature |
|---|---|---|
| Sensorimotor | 0–2 years | Object permanence develops |
| Preoperational | 2–7 years | Egocentrism and animism |
| Concrete operational | 7–11 years | Conservation and logical reasoning about concrete events |
| Formal operational | 11+ years | Abstract and hypothetical thinking |
The sequence of stages is invariant, meaning every child moves through them in the same order, although the pace may vary.
阶段的顺序是不变的，也就是说每个孩子都按相同的顺序经历这些阶段，尽管速度可能不同。
10. Maslow's Hierarchy of Needs | 马斯洛需求层次理论
Abraham Maslow organised human needs into a pyramid-shaped hierarchy, where lower-level needs must be satisfied before higher-level needs become motivating.
亚伯拉罕·马斯洛将人类需求组织成一个金字塔形的层级结构，必须先满足较低层次的需求，高层次的需求才会产生激励作用。
ysiological needs → Safety needs → Love and belonging → Esteem needs → Self-actualisation.
从塔基到塔顶的金字塔依次为：生理需求 → 安全需求 → 爱与归属 → 尊重需求 → 自我实现。
Self-actualisation represents the fulfilment of one's potential and is a driving force for personal growth.
自我实现代表着一个人潜能的充分发挥，是推动个人成长的动力。
The hierarchy is often used in health and social care psychology to understand motivation and well-being.
这个需求层次在健康和社会心理学中常被用来理解动机与幸福感。
11. Ethical Guidelines in Psychological Research | 心理学研究中的伦理准则
Ethics in psychology are a set of principles that protect the welfare of participants. These can be treated as fundamental rules, or 'theorems of good practice'.
心理学伦理学是一套保护参与者福祉的原则，可以看成是基本规则，即“良好实践的定理”。
Key guidelines: Informed consent – participants must freely agree to take part after being told the aim and procedures.
核心准则：知情同意——参与者在被告知研究目的和程序后，必须自愿同意参与。
Right to withdraw – participants can leave the study at any time without penalty.
退出权——参与者可以在任何时间退出研究而不受惩罚。
Confidentiality – personal data must not be shared without permission.
保密性——未经许可不得分享个人数据。
Protection from harm – researchers must avoid physical or psychological distress.
保护免受伤害——研究者必须避免引发身体或心理上的痛苦。
12. Interpreting Data: Charts and Graph Quick Reference | 数据解读：图表与公式速查
Psychologists present data in bar charts, histograms, scatter graphs and line graphs. Understanding how to read these helps in evaluating research.
心理学家用条形图、直方图、散点图和折线图来呈现数据。懂得如何解读这些图表有助于评价研究。
Bar charts represent discrete categories (e.g. number of participants who conformed). Each bar is separated and the height shows frequency or mean.
条形图表示离散的类别（例如表现出从众行为的参与者人数）。每个条柱分开，高度表示频次或平均数。
Histograms display continuous data (e.g. reaction times). The bars touch each other to reflect the continuous scale.
直方图展示连续数据（例如反应时间）。条柱之间相互接触，以体现连续的量表。
Scatter graphs show the relationship between two variables. When points form an upward pattern, it suggests a positive correlation.
散点图显示两个变量之间的关系。当点的分布呈上升形态时，意味着存在正相关关系。
Formula for percentage change when comparing two bars or points: Change (%) = [(New value – Old value) ÷ Old value] × 100%.
比较两个条柱或数据点时的百分比变化公式：变化百分比 = [(新值 – 旧值) ÷ 旧值] × 100%。
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