📚 Year 8 CAIE Psychology: Formula & Theorem Quick Reference Handbook | Year 8 CAIE 心理学：公式定理速查手册
This handbook brings together the most important formulas, laws, and theorems you will encounter in the Year 8 CAIE Psychology course. Mastering these will help you explain how we sense the world, how memory works, and how psychologists analyse data from experiments and observations.
本手册汇集了 Year 8 CAIE 心理学课程中最重要的公式、定律和定理。掌握这些内容将帮助你解释我们如何感知世界、记忆如何运作，以及心理学家如何分析来自实验和观察的数据。
1. Weber's Law | 韦伯定律
Weber's Law describes the relationship between a physical stimulus and the smallest change needed for a person to notice the difference. The 'just noticeable difference' (JND) is a constant proportion of the original stimulus intensity.
ΔI / I = k
韦伯定律描述了物理刺激与一个人能察觉到的最小变化之间的关系。“最小可觉差”（JND）是原始刺激强度的恒定比例。公式中，ΔI 是差别阈限，I 是初始刺激强度，k 是韦伯常数。
| Sense 感官 | Weber constant (k) 韦伯常数 |
|---|---|
| Vision (brightness) 视觉（亮度） | 0.016 |
| Hearing (pitch) 听觉（音高） | 0.003 |
| Lifted weight 举重 | 0.02 |
, you would need 4 g extra. The bigger the original stimulus, the larger the change must be.
例如，如果你正拿着一个 100 克的砝码（I = 100），你需要大约额外增加 2 克（ΔI）才能感觉到差异，因为 k ≈ 0.02。如果砝码是 200 克，你就需要增加 4 克。原始刺激越大，所需的变化就越大。
2. Fechner's Law | 费希纳定律
Fechner's Law builds on Weber's findings and proposes that the subjective sensation (S) increases as the logarithm of the physical stimulus intensity (I). This means that when a stimulus is already strong, you need a much bigger physical increase to notice the same psychological change.
S = k log I
费希纳定律在韦伯发现的基础上提出，主观感觉（S）随物理刺激强度（I）的对数增加而增加。这意味着当刺激已经很强时，你需要大得多的物理增量才能感受到相同的心理变化。k 是一个常数，取决于感官类型。比如，黑暗房间中增加1支蜡烛你会立刻察觉，但在明亮房间里增加1支蜡烛几乎注意不到。
This logarithmic relationship explains why our sensory systems can deal with an enormous range of energies without being overwhelmed. It is why we perceive brightness and loudness on a compressed scale.
这种对数关系解释了为什么我们的感觉系统能够处理巨大的能量范围而不被压垮。这就是为什么我们对亮度和响度的感知是经过压缩的。
3. Yerkes-Dodson Law | 耶克斯-多德森定律
The Yerkes-Dodson Law describes the relationship between arousal (alertness, stress) and performance. It is often shown as an inverted-U curve: performance improves as arousal increases up to an optimal point, but if arousal becomes too high, performance declines.
Performance = f(Arousal)
耶克斯-多德森定律描述了唤醒水平（警觉性、压力）与表现之间的关系。通常表现为一条倒U形曲线：表现随着唤醒水平升高而提升，直至最佳点，但如果唤醒水平过高，表现就会下降。因此，适度的紧张能帮你考试发挥好，但过度焦虑会让你大脑空白。
For simple or well-practised tasks, the optimal level of arousal is higher; for complex or new tasks, a lower level of arousal leads to best performance. This is important for designing study routines and managing exam stress.
对于简单或熟练的任务，最佳唤醒水平较高；对于复杂或新的任务，较低的唤醒水平能带来最佳表现。这对安排学习计划和应对考试压力非常重要。
4. Ebbinghaus Savings Formula | 艾宾浩斯节省公式
Hermann Ebbinghaus studied memory by learning lists of nonsense syllables and measuring how much time he saved when relearning them after a delay. The savings score shows how much information remains in memory.
Savings (%) = (Original Learning Time − Relearning Time) / Original Learning Time × 100%
赫尔曼·艾宾浩斯通过学习无意义音节表，并测量延迟后重新学习所节省的时间来研究记忆。节省分数显示了多少信息仍然保留在记忆中。
nd a week later you can relearn it in 4 minutes, your savings score is (10 − 4) / 10 × 100% = 60% ting happens within the first hour, and the forgetting curve gradually flattens over time.
例如，如果你最初花了10分钟记住一份清单，一星期后重新学习只用了4分钟，那么节省分数是 (10 − 4) / 10 × 100% = 60%。节省百分比越高，记忆保持得越好。艾宾浩斯发现，大部分遗忘发生在第一个小时内，遗忘曲线随时间逐渐变平缓。
5. Miller's Law (7 ± 2) | 米勒定律 (7 ± 2)
George Miller found that the capacity of short-term memory is about seven items (plus or minus two). This is why phone numbers are typically grouped into chunks of 3 or 4 digits.
STM Capacity = 7 ± 2 chunks
乔治·米勒发现短时记忆的容量大约为 7 个项目（上下浮动 2 个）。这就是为什么电话号码通常被分成 3 或 4 个数字的组块。
An 'item' can be a letter, digit, word, or even a meaningful chunk of information. Chunking helps us expand our effective memory span. For example, the sequence 'FBICIAUK' has 8 letters, but if you chunk it into 'FBI', 'CIA', 'UK', you hold only 3 chunks, making it easy to remember.
一个“项目”可以是一个字母、一个数字、一个单词，甚至一个有意义的组块。组块化能帮助我们扩展有效记忆容量。例如，序列“FBICIAUK”有8个字母，但如果你把它组块成“FBI”、“CIA”、“UK”，你只需记住3个组块，就很容易记住了。
6. Measures of Central Tendency | 集中趋势量数
Psychologists summarise data using three measures of central tendency. These help identify a 'typical' score in a data set.
Mean = Σx / n
Median = middle value when data are ordered
Mode = most frequently occurring value
心理学家用三种集中趋势量数来概括数据，它们有助于确定数据集中“典型”的分数。平均数（Mean）是所有数值之和除以总数；中位数（Median）是将数据按顺序排列后的中间值；众数（Mode）是出现次数最多的值。
data are skewed by outliers, the median is often a better measure of central tendency than the mean.
一名研究者收集到测验分数：12, 15, 15, 18, 20。平均数 = (12+15+15+18+20)/5 = 16。中位数 = 15（第三个值）。众数 = 15（出现两次）。当数据受异常值影响而偏斜时，中位数通常比平均数更能反映集中趋势。
7. Range and Percentage Change | 全距与百分比变化
The range is the simplest measure of spread, showing how spread out the scores are. Percentage change is useful when comparing before-and-after data, such as the effect of an intervention.
Range = Maximum Value − Minimum Value
Percentage Change = (New Value − Old Value) / Old Value × 100%
全距是最简单的离散度量，显示分数的分散程度。百分比变化在比较干预前后的数据时很有用。如果某人的焦虑分数从 40 分降到 30 分，百分比变化 = (30 − 40)/40 × 100% = −25%，表示焦虑减少了25%。
es high variability among participants, while a small range suggests scores are clustered together.
在一个记忆实验中，最高分是22，最低分是8。全距是 22 − 8 = 14。全距大说明被试间的差异大，全距小则说明分数集中在一起。
8. Fixed Ratio (FR) Schedule of Reinforcement | 固定比率强化程式
In operant conditioning, a fixed ratio schedule delivers a reinforcer only after a set number of responses. The formula simply states the required number of responses per reward.
FR n: 1 reinforcer after every n responses
在操作性条件作用中，固定比率程式要求反应达到一定次数后才给予强化物。该公式仅表明每次奖励所需的反应次数。例如，FR5 表示每5次正确反应获得1次奖励。
This schedule typically produces a high, steady rate of responding because the faster an organism responds, the sooner it receives a reward. However, a brief pause may occur right after reinforcement, known as the post-reinforcement pause. Factory workers paid based on the number of items produced (piecework) follow an FR schedule.
这种程式通常会产生高且稳定的反应速率，因为有机体反应得越快，就能越早得到奖励。然而，在强化之后可能会出现短暂停顿，称为强化后停顿。按件计酬的工厂工人就遵循着固定比率程式。
9. Correlation Coefficient (r) | 相关系数 (r)
The correlation coefficient (r) measures the strength and direction of a linear relationship between two variables. It always falls between -1 and +1.
-1 ≤ r ≤ +1
相关系数 (r) 衡量两个变量之间线性关系的强度和方向。它的值始终在 -1 到 +1 之间。r = +1 表示完全正相关，r = -1 表示完全负相关，r = 0 表示没有线性关系。
gly. If r = -0.60 between screen time and sleep quality, more screen time is associated with poorer sleep. Crucially, correlation does not imply causation.
例如，如果复习时间与考试分数之间的 r = +0.85，这意味着复习时间增加时，考试分数也倾向于明显提高。如果屏幕时间与睡眠质量之间的 r = -0.60，则说明屏幕时间增加与较差的睡眠相关。关键是，相关并不代表因果关系。
10. Thorndike's Law of Effect | 桑代克效果律
Edward Thorndike's Law of Effect states that behaviours followed by satisfying consequences are more likely to be repeated, whereas behaviours followed by uncomfortable consequences are less likely to occur again. This principle forms the foundation of operant conditioning.
Satisfying consequence → Behaviour strengthened
Annoying consequence → Behaviour weakened
爱德华·桑代克的效果律指出，带来满意后果的行为更可能被重复，而带来不适后果的行为则较少再次发生。这一原理构成了操作性条件作用的基础。例如，学生在回答正确后得到表扬，就更愿意举手发言；若被嘲笑，则可能变得沉默。
Thorndike placed cats in puzzle boxes and observed that actions leading to escape (and food) were gradually learned, while unsuccessful movements were abandoned. The learning curve showed a gradual decline in time taken to escape, without sudden insight.
桑代克把猫放在迷笼中，观察到导致逃脱（并获得食物）的动作被逐步学会，而不成功的动作则被抛弃。学习曲线显示逃脱所需的时间逐渐下降，没有突然的顿悟。
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