📚 Year 8 Cambridge Psychology: Formula & Theorem Quick Reference Handbook | 剑桥八年级心理学：公式定理速查手册
Welcome to your go-to formula and theorem quick reference for Year 8 Cambridge Psychology. This handbook covers the key numerical relationships, statistical tools, and psychological laws you will encounter when studying human behaviour scientifically. Keep this guide handy for revision, homework, and experiments.
欢迎使用这本剑桥八年级心理学公式定理速查手册。本手册涵盖了你用科学方法研究人类行为时会遇到的关键数量关系、统计工具和心理学定律。在复习、做作业和进行实验时，请将这份指南放在手边。
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. It helps explain how we perceive changes in brightness, loudness, and weight.
韦伯定律指出，两个刺激之间的最小可觉差（JND）与原始刺激强度成恒定比例。它有助于解释我们如何感知亮度、响度和重量的变化。
ΔI / I = k
In the formula, I is the initial stimulus intensity, ΔI is the smallest change in intensity a person can detect, and k is the Weber fraction – a constant that depends on the sensory system. For example, for lifted weights, k is about 0.02, meaning a 100 g weight must change by about 2 g before a difference is noticed.
在公式中，I 是初始刺激强度，ΔI 是一个人能察觉到的最小强度变化，而 k 是韦伯分数——一个取决于感觉系统的常数。例如，对于举起的重量，k 约为 0.02，这意味着一个 100 克的重量必须变化约 2 克才能被察觉到差异。
This law applies to many senses but breaks down at very low or very high stimulus levels. It forms the foundation for measuring sensory thresholds and is often used in psychophysics experiments.
该定律适用于许多感官，但在刺激水平极低或极高时会失效。它构成了测量感觉阈值的基础，并常用于心理物理学实验。
2. Fechner’s Law | 费希纳定律
Fechner’s Law builds on Weber’s findings and describes the mathematical relationship between the physical intensity of a stimulus and its perceived magnitude. It suggests that sensation increases as the logarithm of the stimulus intensity.
费希纳定律建立在韦伯发现的基础上，描述了刺激的物理强度与其感知大小之间的数学关系。它表明感觉随着刺激强度的对数而增加。
S = k log I
Here S stands for the perceived sensation magnitude, I is the physical stimulus intensity, and k is a constant that varies across sensory modalities. This means that to make a light seem twice as bright, you need to increase its physical intensity by a factor of about 10.
这里 S 代表感知到的感觉大小，I 是物理刺激强度，k 是一个因感觉模态而异的常数。这意味着要使光线看起来亮两倍，你需要将其物理强度增加约 10 倍。
Fechner’s Law is especially useful in discussing visual and auditory perception. It explains why sound volume is measured in logarithmic decibels rather than linear units.
费希纳定律在讨论视觉和听觉感知时特别有用。它解释了为什么音量用对数分贝来衡量，而不是线性单位。
3. Miller’s Law – The Magic Number 7 ± 2 | 米勒定律——神奇数字 7±2
Miller’s Law states that the average person can hold about seven (plus or minus two) items in their short-term memory at once. This is a cornerstone of cognitive psychology when studying memory span and chunking.
米勒定律指出，普通人一次能在短时记忆中保持大约 7 个（加减 2 个）项目。这是认知心理学研究记忆广度和组块时的基石。
Short-term memory capacity ≈ 7 ± 2 chunks
The formula reminds us that memory span can range from 5 to 9 meaningful units or ‘chunks’. A chunk can be a digit, a letter, a word, or a group of associated information. This is why phone numbers are often grouped into chunks of 3 or 4 digits.
该公式提醒我们，记忆广度可以从 5 到 9 个有意义的单元或“组块”。一个组块可以是一个数字、一个字母、一个词或一组相关信息。这就是为什么电话号码通常被分成 3 或 4 位数字的组块。
Understanding Miller’s Law helps in designing effective revision strategies, such as breaking long lists into smaller, meaningful groups.
理解米勒定律有助于设计有效的复习策略，例如将长列表分解为更小、更有意义的组。
4. Ebbinghaus Forgetting Curve Equation | 艾宾浩斯遗忘曲线方程
Hermann Ebbinghaus discovered that memory retention declines rapidly just after learning and then levels off. The forgetting curve can be modelled mathematically to predict how much information is retained over time.
赫尔曼·艾宾浩斯发现，记忆保持在刚学习后会迅速下降，然后趋于平稳。遗忘曲线可以用数学方法建模，以预测随时间推移保留了多少信息。
R = e–t / S
In this equation, R is the proportion of material retained, t is time elapsed since learning, and S is the relative strength of memory. The exponential decay shows that forgetting is fastest in the first hours and days, and later slows down.
在这个方程中，R 是保留材料的比例，t 是学习后经过的时间，S 是记忆的相对强度。指数衰减表明遗忘在最初几小时和几天内最快，随后减慢。
This formula reminds students that spaced repetition and timely revision can greatly improve long-term retention by resetting the curve.
该公式提醒学生，间隔重复和及时复习可以通过重置曲线来大大提高长期记忆保持。
5. Serial Position Effect: Probability of Recall | 序列位置效应：回忆概率
When a list of words is presented, people tend to remember the first few (primacy effect) and the last few (recency effect) best. The recall probability follows a U-shaped curve when plotted against the position in the list.
当呈现一系列词语时，人们往往最容易记住开头几个（首因效应）和最后几个（近因效应）。回忆概率与列表位置绘成图形时呈 U 形曲线。
P(recall) = f(primacy) + f(recency) – forgetting middle items
The formula is conceptual, showing that the probability of recalling a word depends on both its position near the beginning (primacy) and near the end (recency), minus interference for middle items. In experiments, this can be expressed numerically: early items might have a recall rate of 0.70, late items 0.80, and middle items only 0.40.
该公式是概念性的，表明回忆一个词的概率取决于它靠近开头（首因）和靠近结尾（近因）的位置，减去对中间项目的干扰。在实验中，这可以用数字表达：开头项目的回忆率可能为 0.70，末尾项目为 0.80，中间项目仅为 0.40。
Understanding this effect helps in structuring study sessions – place the most important information at the start and end of a learning block.
理解这一效应有助于安排学习时间——将最重要的信息放在学习模块的开头和结尾。
6. Measures of Central Tendency: Mean | 集中趋势测量：平均数
The arithmetic mean is the most common measure of central tendency in psychology. It summarises a set of scores by finding the balance point of the data.
算术平均数是心理学中最常用的集中趋势测量。它通过找到数据的平衡点来总结一组分数。
M = Σx / n
M is the mean, Σx (sigma x) is the sum of all individual scores, and n action times are 250, 280, 310, 260, 300 ms, the mean is (250+280+310+260+300)/5 = 280 ms.
M 是平均数，Σx（西格玛 x）是所有单个分数的总和，n 是分数的总数。例如，五个反应时间分别为 250、280、310、260、300 毫秒，则平均数为 (250+280+310+260+300)/5 = 280 毫秒。
The mean is sensitive to extreme scores (outliers), so it may not always represent the typical response when data are skewed.
平均数对极端分数（离群值）敏感，因此当数据偏斜时，它可能并不总能代表典型反应。
7. Measures of Central Tendency: Median | 集中趋势测量：中位数
The median is the middle value when all scores are arranged in order. It is often used when the data set contains outliers, as it is not affected by extreme values.
中位数是将所有分数按顺序排列后的中间值。当数据集中包含离群值时，它经常被使用，因为它不受极端值的影响。
Median position = (n + 1) / 2
For a set of scores, first order them from smallest to largest. If n is the number of scores, the position of the median is given by (n+1)/2. For example, with 7 scores, the 4th score is the median. With an even number of scores, take the average of the two middle scores.
对于一组分数，首先将它们从小到大排序。如果 n 是分数的个数，中位数的位置由 (n+1)/2 给出。例如，有 7 个分数时，第 4 个分数就是中位数。如果分数个数为偶数，则取中间两个分数的平均值。
In psychology studies, the median is frequently reported for reaction times or when a few participants have extremely slow or fast responses.
在心理学研究中，中位数常用于报告反应时间，或当少数参与者的反应极慢或极快时。
8. Measures of Central Tendency: Mode | 集中趋势测量：众数
The mode is the most frequently occurring value in a data set. It is the only measure of central tendency that can be used with nominal (categorical) data.
众数是数据集中出现频率最高的值。它是唯一可用于名义（分类）数据的集中趋势测量。
Mode = Value with highest frequency
No complex formula is needed: simply count how many times each value appears; the one with the highest count is the mode. A data set can have one mode (unimodal), two modes (bimodal), or more.
无需复杂公式：只需计数每个值出现的次数；计数最高的那个就是众数。一个数据集可以有一个众数（单峰）、两个众数（双峰）或更多。
In questionnaires where participants select a favourite colour or a yes/no answer, the mode tells you the most popular choice.
在参与者选择最喜欢的颜色或是/否答案的问卷调查中，众数告诉你最受欢迎的选择。
9. Range and Spread of Data | 全距与数据分布
The range is the simplest measure of dispersion. It tells you how spread out the scores are by subtracting the lowest value from the highest value.
全距是最简单的离散程度测量。它通过用最大值减去最小值来告诉你分数的分布范围。
Range = Maximum value – Minimum value
For example, if memory test scores are 12, 15, 18, 14, 22, the range is 22 – nge indicates that scores are clustered together; a large range suggests high variability.
例如，如果记忆测试分数为 12、15、18、14、22，则全距为 22 – 12 = 10。全距小表明分数集中在一起；全距大表明变异性高。
While easy to calculate, the range only uses two values and can be distorted by a single outlier. Researchers often pair it with other measures like standard deviation in advanced study.
虽然全距容易计算，但它只使用了两个值，并且可能被单个离群值扭曲。研究人员在高级研究中通常将其与标准差等其他测量结合使用。
10. Percentage Change in Experiments | 实验中的百分比变化
Psychologists often need to compare how much a score or a behaviour has changed before and after an intervention. The percentage change formula gives a standardised way to express this difference.
心理学家经常需要比较干预前后一个分数或行为的变化程度。百分比变化公式提供了一种表达这种差异的标准化方法。
Percentage change = ((New value – Old value) / Old value) × 100%
ercentage. For example, if stress scores drop from 40 to 30, the percentage change is ((30–40)/40)×100% = –25%, meaning a 25% reduction in stress.
如果旧值是基线测量，任何增加都会产生正百分比，而减少则产生负百分比。例如，如果压力分数从 40 降到 30，则百分比变化为 ((30–40)/40)×100% = –25%，意味着压力降低了 25%。
This formula is widely used when evaluating the effectiveness of therapies, teaching methods, or memory improvement techniques in psychology projects.
在评估心理学项目中疗法、教学方法或记忆改善技术的有效性时，这一公式被广泛使用。
11. Simple Reaction Time Formula | 简单反应时间公式
Reaction time (RT) is a fundamental measure in cognitive psychology. It captures the time taken to process a stimulus and produce a motor response. A basic additive model helps break down the components.
反应时间（RT）是认知心理学中的一项基本测量。它捕捉处理刺激并做出运动反应所需的时间。一个基本的加法模型有助于分解其组成部分。
RT = Detection time + Decision time + Motor time
Here, detection time is the sensory processing stage, decision time involves cognitive interpretation and response selection, and motor time is the physical execution of the response. In a simple task where only one stimulus and one response are used, decision time is minimal, so RT mainly reflects detection and motor processes.
这里，察觉时间是感官处理阶段，决策时间涉及认知解释和反应选择，而运动时间是执行身体反应。在只使用一个刺激和一个反应的简单任务中，决策时间最小，因此 RT 主要反映察觉和运动过程。
Experiments measuring how RT changes under distraction or fatigue rely on this formula to identify which stage is affected.
测量 RT 在分心或疲劳下如何变化的实验，依靠该公式来确定哪一个阶段受到影响。
12. Correlation Coefficient: Sign and Strength | 相关系数：方向与强度
In psychology, researchers often explore relationships between two variables, such as stress and sleep quality. The correlation coefficient, symbolised as r, quantifies both the direction and strength of a linear relationship.
在心理学中，研究人员经常探索两个变量之间的关系，例如压力与睡眠质量。相关系数，符号为 r，量化线性关系的方向和强度。
–1 ≤ r ≤ +1
The value of r always lies between –1 and +1. A positive r (e.g., +0.8) means that as one variable increases, the other tends to increase. A negative r (e.g., –0.6) indicates an inverse relationship. Values close to 0 suggest a weak or no linear relationship. This single number summarises a scattergraph pattern.
r 的值始终在 –1 和 +1 之间。正 r（如 +0.8）意味着一个变量增加时，另一个也倾向于增加。负 r（如 –0.6）表示反向关系。接近 0 的值表明线性关系较弱或不存在。这个单一数字总结了散点图的模式。
While Year 8 students are not expected to calculate r by hand, understanding its sign and range helps in interpreting psychological findings critically.
虽然不要求八年级学生手动计算 r，但理解其符号和范围有助于批判性地解读心理学发现。
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