📚 Year 8 CCEA Psychology: Formula & Theorem Quick Reference Handbook | CCEA 8年级心理学公式定理速查手册
Welcome to the Year 8 CCEA Psychology Formula & Theorem Quick Reference Handbook. This guide brings together the most essential laws, equations and formulas you will meet in your Year 8 psychology course. Each entry explains the theory behind the formula, shows the key expression in clear mathematical notation, and gives a practical example you can quote in a topic test. Use this handbook to revise for topic tests, to support your homework, and to make your own revision flashcards.
欢迎使用 CCEA 8年级心理学公式定理速查手册。本指南汇总了8年级心理学课程中最核心的定律、方程和公式。每个条目都阐释了公式背后的理论，用清晰的数学符号给出关键表达式，并提供一个可用于单元测验的实际示例。你可以用这本手册来准备单元测验、辅助完成课后作业，并制作自己的复习闪卡。
1. Miller's Law (The Magical Number 7 ± 2) | 米勒定律（神奇数字 7 ± 2）
George Miller's classic 1956 paper "The Magical Number Seven, Plus or Minus Two" revealed that the capacity of short-term memory (STM) is severely limited. Most people can hold about seven meaningful pieces of information at once, but the exact number varies from person to person. This is why a typical phone number is seven digits long – it fits comfortably into our STM span. If you try to remember a string of nine or ten random digits, you will probably forget the middle ones unless you use a memory strategy.
乔治·米勒1956年的经典论文《神奇的数字七，加减二》揭示，短时记忆的容量非常有限。大多数人一次能记住大约七个有意义的信息片段，但具体数字因人而异。这就是为什么典型的电话号码是七位数——它恰好能舒适地放在我们的短时记忆跨度中。如果你试着去记一串九个或十个随机数字，你很可能会忘掉中间的几个，除非你使用记忆策略。
STM capacity ≈ 7 ± 2 chunks
短时记忆容量 ≈ 7 ± 2 个组块
The term "chunk" letters F, B and I count as three chunks unless they are grouped into the familiar acronym "FBI" into each slot. Expert chess players use chunking to remember complex board positions by grouping pieces into familiar patterns.
"组块"指的是有意义的信息组合。例如，字母 F、B、I 算作三个组块，但如果组合成熟悉的缩略词 "FBI"，就成了一个组块。组块化使我们能够突破 7±2 的限制，将更多材料打包进每一个槽位。国际象棋高手运用组块化，将棋子归类为熟悉的模式，从而记住复杂的棋局。
In an exam answer, always link the law to a concrete situation: revise in groups of about seven items, then pause or change topic before adding more.
在考试作答时，一定要把这条定律联系到具体情境：每次复习大约七个条目，然后暂停或换一个主题，再继续增加新内容。
2. Ebbinghaus's Forgetting Curve and Savings Formula | 艾宾浩斯遗忘曲线与节省量公式
Hermann Ebbinghaus was one of the first psychologists to study memory experimentally. He memorised lists of nonsense syllables (like "WID" or "ZOF") so that previous meaning could not help him, and then tested himself after delays of 20 minutes, 1 hour, 1 day and longer. He found that forgetting is extremely rapid at first and then levels off – a pattern known as the forgetting curve.
赫尔曼·艾宾浩斯是最早用实验方法研究记忆的心理学家之一。他背诵 WID、ZOF 这类无意义音节，使已有意义无法提供帮助，然后在 20 分钟、1 小时、1 天以及更长的时间后自测。他发现遗忘最初极为迅速，随后逐渐趋缓——这一模式被称为遗忘曲线。
The curve is an exponential decay function, where R is the percentage of material retained, t is the delay since learning, and S is the strength of the memory trace. A larger S produces a flatter curve and slower forgetting.
该曲线是一个指数衰减函数，其中 R 是保持的材料百分比，t 是学习后的延迟时间，S 是记忆痕迹的强度。S 越大，曲线越平缓，遗忘越慢。
R = e−t/S × 100%
Ebbinghaus also invented the savings formula. Savings measure how much faster you relearn material compared with the first time you learned it, which proves that traces of "forgotten" information still remain in memory.
艾宾浩斯还发明了节省量公式。节省量衡量的是，与第一次学习相比，你重新学习同一材料时快了多少，这证明"被遗忘"的信息痕迹仍然留在记忆中。
Savings (%) = (first learning trials − relearning trials) ÷ first learning trials × 100%
节省量 (%) = (第一次学习次数 − 重学次数) ÷ 第一次学习次数 × 100%
st and only 6 trials to relearn it a week later, your savings score is (10 − 6) ÷ 10 × 100% = 40%. This is why spaced review works better than one long cramming session.
如果你第一次用 10 次尝试记住一个列表，一周后只用 6 次就能重新记住，那么节省量 = (10 − 6) ÷ 10 × 100% = 40%。这正是分散复习比一次性长时间突击更有效的原因。
3. Weber's Law and the Just Noticeable Difference | 韦伯定律与最小可觉差
Ernst Weber showed that the smallest difference we can detect between two stimuli is not a fixed amount; it is a constant fraction of the original stimulus. That fraction is called the Weber fraction, k, and the difference we can just detect is called the just noticeable difference (JND).
恩斯特·韦伯表明，我们能够觉察到的两个刺激之间的最小差异并不是一个固定数量，而是原始刺激的一个恒定比例。这个比例称为韦伯分数 k，而我们刚刚能觉察到的差异称为最小可觉差（JND）。
ΔI ÷ I = k
l 100 g from 102 g but not 100 g from 101 g, then k = 2 ÷ 100 = 0.02. For a 1000 g weight you would need an extra 20 g before noticing the change.
其中 I 是原始刺激强度，ΔI 是刚好能被觉察所需要的增加量，k 是韦伯分数。在重量实验中，如果你能分辨 100 g 与 102 g，却分辨不出 100 g 与 101 g，那么 k = 2 ÷ 100 = 0.02。对于 1000 g 的重物，你需要额外增加 20 g 才能察觉到变化。
Weber's Law applies to brightness, loudness, taste and many other senses, which makes it a very efficient way to summarise sensory sensitivity in one small number.
韦伯定律适用于亮度、响度、味觉以及许多其他感觉，这使它成为用一个小数字概括感觉敏感度的非常高效的方式。
4. Fechner's Law | 费希纳定律
Gustav Fechner extended Weber's work by asking how the strength of a sensation grows as the physical stimulus grows. His answer was that sensation increases with the logarithm of stimulus intensity, so each equal step in sensation requires a larger and larger physical increase.
古斯塔夫·费希纳扩展了韦伯的研究，追问感觉强度如何随物理刺激的增强而增长。他的答案是：感觉随刺激强度的对数而增加，因此感觉上每一个相等的提升，都需要越来越大的物理增量。
S = k × log I
In this equation S is the subjective sensation, I is the physical intensity of the stimulus, and k is a constant that depends on the sense being measured. A practical consequence is that turning a lamp from 10 to 20 units feels like a much bigger change than turning it from 100 to 110 units, even though the physical increase is the same.
在这个方程中，S 是主观感觉，I 是刺激的物理强度，k 是取决于所测感觉的常数。一个实际推论是：把灯从 10 个单位调到 20 个单位，感觉上的变化远大于从 100 调到 110，尽管物理增量相同。
Fechner's Law is a good model for middle-range intensities, but it breaks down at very high intensities, which is why Stevens later proposed a power function instead.
费希纳定律在中段强度上是很好的模型，但在极高强度下会失效，这正是史蒂文斯后来提出幂函数替代它的原因。
5. Stevens's Power Law | 史蒂文斯幂定律
S. S. Stevens argued that the relationship between stimulus and sensation is better described by a power function than by a logarithm. His law states that sensation is proportional to stimulus intensity raised to an exponent, a.
S. S. 史蒂文斯认为，刺激与感觉之间的关系用幂函数描述比用对数更合适。他的定律指出，感觉与刺激强度的 a 次幂成正比。
S = k × Iᵃ
Here S is the perceived magnitude, I is the stimulus intensity, k is a scaling constant, and a is the exponent that decides the shape of the curve. When a is greater than 1, as with electric shock, sensation grows faster than the physical stimulus. When a is less than 1, as with brightness, sensation grows more slowly. When a equals 1, the two grow together.
其中 S 是知觉到的量值，I 是刺激强度，k 是比例常数，a 是决定曲线形状的指数。当 a 大于 1 时（例如电击），感觉增长快于物理刺激。当 a 小于 1 时（例如亮度），感觉增长慢于物理刺激。当 a 等于 1 时，两者同步增长。
compressing, a = 1 linear. Examiners often ask you to match a sense to the correct shape.
复习时可以把三种情况记成一张快速清单：a 大于 1 为扩张型，a 小于 1 为压缩型，a 等于 1 为线性。考官常要求你把某种感觉与正确的曲线形状配对。
6. The Spearman-Brown Formula and Test Reliability | 斯皮尔曼-布朗公式与测验信度
Psychologists need to know whether a test measures something consistently. One common method is the split-half method: divide a test into two halves, correlate the two scores, and then correct the result because each half is only half as long as the full test. The Spearman-Brown formula makes that correction.
心理学家需要知道一个测验是否稳定地测量某种东西。常用方法之一是分半法：把测验分成两半，求两半得分的相关，然后因为每一半只有完整测验的一半长，需要对结果进行校正。斯皮尔曼-布朗公式就是用来做这一步校正的。
rxx = 2r ÷ (1 + r)
In this formula r is the correlation between the two half-tests and rxx estimated reliability of the full test. If the split-half correlation is r = 0.60, then rxx .75. A reliability above about 0.70 is usually considered acceptable for a classroom test.
该公式中，r 是两个半测验之间的相关，rxx 是完整测验的估计信度。如果分半相关 r = 0.60，那么 rxx = (2 × 0.60) ÷ (1 + 0.60) = 1.20 ÷ 1.60 = 0.75。对课堂测验来说，信度高于约 0.70 通常被认为可以接受。
The general message is that longer tests tend to be more reliable, but the gain gets smaller each time you add items, so simply making a test longer is not always the best fix.
总体结论是：更长的测验往往更可靠，但每增加题目带来的收益会逐渐变小，所以单纯把测验加长并不总是最好的解决办法。
7. Pearson's r and Cohen's d | 皮尔逊相关系数与科恩 d 值
Pearson's r measures the strength and direction of a linear relationship between two variables. It ranges from −1 through 0 to +1. A value near +1 means that as one variable rises the other also rises; a value near −1 means that as one rises the other falls; a value near 0 means there is little linear relationship.
皮尔逊 r 衡量两个变量之间线性关系的强度和方向。它的取值范围从 −1 经过 0 到 +1。接近 +1 的值意味着一个变量上升时另一个也上升；接近 −1 的值意味着一个上升时另一个下降；接近 0 的值意味着几乎没有线性关系。
r = Σ[(X − MX)(Y − MY)] ÷ (N × SDX × SDY)
When you compare two group means, Cohen's d expresses the size of the difference in standard deviation units, which makes it comparable across different experiments.
当你比较两个组的平均数时，科恩 d 值用标准差单位表示差异大小，使不同实验之间的结果可以直接比较。
d = (M₁ − M₂) ÷ SDpooled
t. Always report the effect size alongside the means, because a difference can be statistically real but practically tiny.
常用的判断标准是：d = 0.2 为小效应，d = 0.5 为中等效应，d = 0.8 及以上为大效应。报告平均数时一定要同时报告效应量，因为差异在统计上可能真实存在，但在实践上却微不足道。
8. Descriptive Statistics: Mean, Standard Deviation and z Scores | 描述统计：平均数、标准差与 z 分数
Descriptive statistics summarise a data set with a few numbers. The mean is the arithmetic average, found by adding all scores and dividing by the number of scores. The standard deviation measures how far scores typically spread away from the mean.
描述统计用少数几个数字概括一组数据。平均数是算术平均值，由所有分数相加再除以分数个数得到。标准差衡量分数通常偏离平均数多远。
Mean M = ΣX ÷ N
SD = √[Σ(X − M)² ÷ N]
A z score converts a raw score into the number of standard deviations it lies above or below the mean. This lets you compare scores from tests with different maximum marks.
z 分数把一个原始分数转换成它位于平均数之上或之下多少个标准差。这让你能够比较满分不同的测验分数。
z = (X − M) ÷ SD
÷ 8 = +1.5, meaning you scored one and a half standard deviations above the class average.
如果你的测验分数是 72，班级平均数是 60，标准差是 8，那么 z = (72 − 60) ÷ 8 = +1.5，意味着你比班级平均分高出一个半标准差。
9. Signal Detection Theory: d' and β | 信号检测论：d' 与 β
Signal detection theory separates a person's actual sensitivity from their decision bias. In a detection task
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