📚 Year 10 Cambridge Psychology: Quick Reference Formulas & Theorems Handbook | Year 10 剑桥心理学：公式定理速查手册
Mastering the essential formulas and principles in Year 10 Cambridge Psychology gives you a solid foundation for analysing data, interpreting research, and understanding the quantitative laws behind human behaviour. This handbook compiles the key statistical tools, psychophysical laws, and theoretical models you will encounter across topics such as research methods, memory, perception, and social influence. Each entry presents the core concept in a concise English explanation followed immediately by its Chinese counterpart, ensuring you grasp both the reasoning and the terminology needed for your exams and internal assessments.
掌握 Year 10 剑桥心理学中的核心公式与定理，能为你分析数据、解读研究以及理解人类行为背后的量化规律奠定坚实基础。本手册汇集了你在研究方法、记忆、知觉和社会影响等主题中将遇到的关键统计工具、心理物理学定律和理论模型。每个条目先用英文简明解释，紧接着给出对应的中文阐述，确保你既能掌握原理，又能熟悉考试所需的术语。
1. Measures of Central Tendency | 集中趋势的测量
The mean is the arithmetic average of a data set and provides a single representative score. It is calculated by summing all the values and dividing by the number of data points.
平均数是一组数据的算术平均值，能够给出一个代表性的分数。计算方法是将所有数值相加，再除以数据点的个数。
Sample Mean: x̄ = Σx ÷ n
The median is the middle value when data are ordered from smallest to largest; it is not affected by extreme scores.
中位数是将数据从小到大排序后处于正中间位置的值，它不受极端分数的影响。
The mode is the most frequently occurring value in a distribution and is useful for identifying the most common category in nominal data.
众数是一组数据中出现次数最多的数值，对于识别名义数据中最常见的类别尤为有用。
2. Measures of Dispersion | 离散度的测量
The range is the simplest measure of spread, calculated as the difference between the highest and lowest scores in a data set.
范围（全距）是最简单的离散度测量指标，由数据集中最高分与最低分的差值求得。
Range = Maximum value − Minimum value
Standard deviation indicates how much scores deviate from the mean on average. A small standard deviation means data are tightly clustered around the mean; a large one indicates a wide spread.
标准差反映的是各数据点与平均数之间的平均偏差。标准差小表示数据紧密围绕均值聚集，标准差大表示分布较分散。
Sample standard deviation: s = √[ Σ(x − x̄)² ÷ (n − 1) ]
3. The Normal Distribution and z-Scores | 正态分布与 Z 分数
Many psychological variables follow a normal distribution – a bell-shaped curve symmetrical around the mean. In a normal distribution, approximately 68% of scores fall within one standard deviation of the mean, 95% within two, and 99.7% within three standard deviations.
许多心理学变量呈正态分布——一条围绕均值对称的钟形曲线。在正态分布中，大约 68% 的分数落在均值 ±1 个标准差内，95% 落在 ±2 个标准差内，99.7% 落在 ±3 个标准差内。
A z-score expresses how many standard deviations a raw score lies from the mean, enabling comparison across different scales.
Z 分数表示一个原始分数距离均值有多少个标准差，可用于不同尺度间的比较。
z = (X − μ) ÷ σ
Years 10 students use z-scores when exploring if a participant's score is typical or extreme relative to the reference population.
Year 10 学生在判断参与者的分数相对于参照总体是典型还是极端时，会用到 Z 分数。
4. Correlation | 相关
Correlation measures the strength and direction of a relationship between two variables. The correlation coefficient (r) ranges from -1 to +1. A positive value means as one variable increases, the other tends to increase; a negative value indicates an inverse relationship.
相关衡量两个变量之间关系的强度与方向。相关系数 (r) 的取值范围为 -1 到 +1。正值表示一个变量增加时，另一个变量往往也增加；负值表示反向关系。
For ranked data, Spearman's rank correlation coefficient is often used in IGCSE Psychology to determine how far two sets of ranks agree.
对于等级数据，IGCSE 心理学中常使用斯皮尔曼等级相关系数，来判断两组排名之间的一致性程度。
Spearman's rₛ = 1 − [ 6Σd² ÷ n(n² − 1) ]
where d is the difference between ranks for each pair and n is the number of pairs. Remember, correlation does not imply causation; an observed relationship might be due to a third confounding variable.
公式中 d 是每对数据的等级差，n 为配对数量。切记：相关关系并不意味着因果关系；观察到的关系可能源于第三个混淆变量。
5. Probability and Statistical Significance | 概率与统计显著性
Probability (P) in experiments describes the likelihood that an observed result occurred by chance. It is computed by dividing the number of successful outcomes by the total number of possible outcomes.
实验中的概率 (P) 描述的是观察到的结果由偶然因素造成的可能性大小。计算方法是：成功结果的数量除以所有可能结果的总数。
P(Event) = Number of favourable outcomes ÷ Total number of outcomes
In psychological research, a result is considered statistically significant if the probability of it occurring by chance is very low – typically less than 5% (p < 0.05). This threshold helps researchers decide whether to reject the null hypothesis.
在心理学研究中，如果某个结果出于偶然的概率非常低（通常低于 5%，即 p < 0.05），则该结果被认为具有统计显著性。这一阈值帮助研究者决定是否拒绝零假设。
Understanding significance levels allows you to evaluate whether the independent variable genuinely had an effect on the dependent variable or whether the difference could be due to random variation.
理解显著性水平能让你评估自变量是否真的对因变量产生了影响，还是差异可能仅仅来自随机波动。
6. Weber's Law | 韦伯定律
Weber's Law is a key principle in psychophysics stating that the just noticeable difference (JND) between two stimuli is proportional to the magnitude of the original stimulus. For a change to be detected, a constant proportion rather than a constant absolute amount must be added.
韦伯定律是心理物理学中的关键原理，指两个刺激之间的最小可觉差 (JND) 与原始刺激的强度成正比。要被察觉到变化，增加的不是固定的绝对量，而是固定的比例。
ΔI / I = k
Where ΔI is the increment needed for detection, I is the initial stimulus intensity, and k is the Weber fraction (constant for a given sensory modality). For example, the Weber fraction for lifted weights is about 0.02, meaning you can detect a 2% increase more easily than a fixed 2-gram increase across different base weights.
其中 ΔI 是引发察觉所需增加的强度，I 是初始刺激强度，k 是韦伯分数（对于特定感觉通道为一常数）。例如，提起重量的韦伯分数约为 0.02，这意味着在不同基础重量下，你更容易察觉出 2% 的增加，而不是固定 2 克的增加。
7. Fechner's Law | 费希纳定律
Fechner built on Weber's work and proposed a logarithmic relationship between physical stimulus intensity and the perceived psychological magnitude. As stimulus intensity grows geometrically, sensation increases arithmetically.
费希纳在韦伯研究的基础上，提出了物理刺激强度与心理感受量之间的对数关系。当物理刺激强度呈几何级数增加时，感觉量只呈算术级数增加。
S = k log I
Here S is the sensation magnitude, I is the physical intensity, and k is a constant that varies with the sensory system. This law explains why doubling the brightness of a light does not make it seem twice as bright to an observer.
式中 S 为感觉量，I 为物理强度，k 为随感觉系统变化的常数。该定律解释了为什么将灯的物理亮度加倍，观察者感受到的主观亮度并不会翻倍。
8. Signal Detection Theory | 信号检测论
Signal detection theory separates the decision-making process into a sensory sensitivity component (d') and a response bias (criterion). It explains how people detect faint signals against a background of noise, accounting for hits, misses, false alarms, and correct rejections.
信号检测论将决策过程区分为感官敏感性成分 (d') 和反应偏向（判断标准）。它解释了人们如何在噪音背景下检测微弱信号，并量化了击中、漏报、虚报和正确拒绝的情况。
d' = Z(Hit rate) − Z(False alarm rate)
A high d' indicates a good ability to discriminate signal from noise. The decision criterion (β or c) reflects how conservative or liberal the observer is in reporting a signal. In Year 10 psychology, this theory helps students understand why the same physical stimulus may sometimes be reported and sometimes missed, depending on expectation and motivation.
d' 值越高，表示分辨信号与噪音的能力越强。判断标准 (β 或 c) 反映观察者在报告信号时是保守还是宽松。在 Year 10 心理学中，这一理论有助于学生理解：同样的物理刺激，为何有时被报告、有时被漏报，这可能取决于期望和动机。
9. Ebbinghaus Savings Formula | 艾宾浩斯节省公式
Hermann Ebbinghaus pioneered the experimental study of memory by measuring how much effort is saved when relearning previously memorised material. His savings method quantifies retention as a percentage reduction in the time (or trials) needed to relearn.
赫尔曼·艾宾浩斯通过测量重学先前记忆材料时省去的努力，开创了记忆的实验研究。他的节省法将保持量化为重学所需时间（或次数）减少的百分比。
Savings % = (Original learning time − Relearning time) ÷ Original learning time × 100%
This formula gave rise to the famous forgetting curve, which shows a steep drop in retention shortly after learning, followed by a more gradual decline. Revisiting material before the retention drops dramatically is a practical application of this principle in revision planning.
这一公式催生了著名的遗忘曲线，显示学习后短时间内保持量急剧下降，随后下降趋于平缓。在遗忘急剧发生前温习材料，正是该原理在复习规划中的实际应用。
10. 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 limited to about seven items (plus or minus two). This "magic number" refers to meaningful chunks of information rather than individual bits.
乔治·米勒 (1956) 提出，短期记忆 (STM) 的容量大约限于 7 个项目，可增减 2 个。这个“神奇数字”指的是有意义的组块，而非单个信息单元。
STM capacity ≈ 7 ± 2 chunks
Chunking is the process of grouping separate pieces of information into larger, meaningful units, which effectively expands the capacity of working memory. For example, the digit string 1-9-1-2-0-1-4 is easier to recall when chunked as 1912 and 014, representing familiar dates or patterns.
组块化是将分离的信息片段组合成更大、更有意义单元的过程，能有效扩展工作记忆的容量。例如，数字串 1-9-1-2-0-1-4 若组块为 1912 和 014（熟悉的年份与模式），则更容易回忆。
11. Social Impact Theory | 社会影响理论
Latane's Social Impact Theory describes how individuals are influenced by others as a multiplicative function of three factors: Strength (the importance of the influencing group or person), Immediacy (closeness in time and space), and Number (how many sources are present).
拉塔内的社会影响理论指出，个体受他人影响的力度，是三个因素乘积函数：强度（影响源的重要性）、接近性（在时间和空间上的接近程度）和数量（有多少个影响源同时在场）。
Social Impact = f(Strength × Immediacy × Number)
The theory predicts that impact increases with each factor, but the effect of number follows a power function: each additional source adds progressively less influence. This helps explain phenomena like conformity, obedience, and the diffusion of responsibility in bystander situations.
该理论预测，影响力随各因素增强而增大，但数量的效应遵循幂函数规律：每增加一个影响源，所增加的额外影响力逐渐减小。这有助于解释从众、服从以及旁观者情境中的责任分散等现象。
12. Research Variables and Hypothesis Formulation | 研究变量与假设构建
Experiments in psychology manipulate an independent variable (IV) to observe its effect on a dependent variable (DV), while controlling confounding variables (CVs). A well-written hypothesis states the predicted relationship between the IV and the DV in operationalised terms.
心理学实验通过操纵自变量 (IV) 来观察其对因变量 (DV) 的影响，同时控制混淆变量 (CV)。一个表述恰当的假设，会用可操作化的术语阐明白变量与因变量之间的预测关系。
The general formula for a directional (one-tailed) hypothesis can be written as:
方向性（单侧）假设的通用表达公式为：
If IV changes (in a specific direction), then DV will change (in a predicted direction).
Vs are held constant. Being able to formalise hypotheses in this way helps you design robust investigations and clearly identify the independent, dependent, and control variables in scenario-based exam questions.
从概念上，这种效应可表达为函数：DV = f(IV)，同时保持混淆变量不变。能够以这种方式将假设形式化，有助于你设计可靠的研究，并在基于情境的考试题目中清晰地识别自变量、因变量和控制变量。
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