📚 Psychology Formulas and Principles Quick Reference Handbook | 心理学公式定理速查手册
This handbook brings together the essential formulas, laws, and quantitative principles you encounter in Year 8 CIE Psychology. From calculating the mean of a data set to understanding Fechner's logarithmic law, each entry is presented with a clear formula or statement, followed by a paired English–Chinese explanation. Use this as a quick refresher when revising research methods, cognitive experiments, or the famous laws that link physical stimuli to our psychological experience.
本手册汇集了 Year 8 CIE 心理学课程中涉及的重要公式、定律和量化原理。从计算一组数据的平均数到理解费希纳对数定律，每一项都给出了清晰的公式或命题，并配有中英文对照说明。在复习研究方法、认知实验或那些把物理刺激与心理体验联系起来的经典定律时，可以用它快速查阅。
1. Measures of Central Tendency: Mean, Median, Mode | 集中趋势的度量：平均数、中位数、众数
The mean is the arithmetic average of a set of scores. It is obtained by adding all the values together and then dividing by the total number of values.
平均数是一组分数的算术平均值，计算方法是把所有数值相加，再除以数值的总个数。
Mean (X̄) = Σx ÷ N
Here Σx (sigma x) stands for the sum of all observations, and N is the number of observations.
其中 Σx（西格玛 x）表示所有观察值的总和，N 表示观察值的个数。
The median is the middle score when all the scores are arranged in order from smallest to largest. If there is an even number of scores, the median is the mean of the two middle scores.
中位数是把所有分数从小到大排列后位于中间位置的数值。如果分数的个数是偶数，中位数就是中间两个分数的平均数。
The mode is simply the most frequently occurring score in the data set. A data set can have one mode, more than one mode, or no mode at all.
众数就是数据集中出现次数最多的分数。一组数据可以有一个众数、多个众数，或者没有众数。
2. Measure of Spread: Range | 离散程度的度量：极差
The range tells us how spread out the scores are. It is calculated by subtracting the smallest score from the largest score.
极差表示分数的分散程度，计算方法是用最大值减去最小值。
Range = Highest score – Lowest score
Although the range is quick to calculate, it is affected by extreme values (outliers). A single unusually high or low score can make the range very large, giving a misleading picture of the spread of most of the data.
尽管极差计算起来很快，但它容易受极端值（离群值）的影响。只要有一个异常高或低的分数，极差就会变得非常大，从而扭曲了大多数数据的分散情况。
3. Percentages and Percentage Change | 百分比与百分比变化
Psychologists often convert raw frequencies into percentages so that groups of different sizes can be compared fairly. A percentage answers the question 'out of 100'.
心理学家常常把原始频数转换成百分比，以便能公平地比较不同规模的组。百分比回答的是“在 100 份中占多少”的问题。
Percentage = (Part ÷ Whole) × 100%
8 out of 30 participants recall a word correctly, the percentage recall is (18 ÷ 30) × 100% = 60%.
例如，如果 30 名参与者中有 18 人正确回忆了一个单词，那么回忆的百分比就是 (18 ÷ 30) × 100% = 60%。
Percentage change shows how much a value has increased or decreased compared to the original value.
百分比变化表示某个值相对于原始值增加或减少了多少。
Percentage change = [(New value – Original value) ÷ Original value] × 100%
f reaction time drops from 250 ms to 200 ms after practice, the percentage change is [(200 – 250) ÷ 250] × 100% = –20%, a 20% decrease.
如果练习后反应时间从 250 毫秒降到 200 毫秒，百分比变化就是 [(200 – 250) ÷ 250] × 100% = –20%，表示减少了 20%。
4. Weber's Law: Just Noticeable Difference | 韦伯定律：最小可觉差
Weber's Law describes the relationship between the physical intensity of a stimulus and the smallest change in intensity that we can detect. This smallest detectable change is called the just noticeable difference (JND).
韦伯定律描述的是刺激的物理强度与我们能察觉到的最小强度变化之间的关系。这个能被察觉到的最小变化量称为最小可觉差（JND）。
ΔI ÷ I = k
In this formula, ΔI is the size of the just noticeable difference, I is the original stimulus intensity, and k is a constant that depends on the type of sensory judgement (Weber fraction).
在这个公式中，ΔI 表示最小可觉差的量，I 表示原始刺激强度，k 是一个取决于感觉判断类型的常数（韦伯分数）。
then you would need to add about 2 g (ΔI = 2) before you notice the weight feels heavier.
举个例子，如果你手里拿着一个 100 g 的砝码（I = 100），而重量感觉的韦伯分数大约是 0.02，那么你需要再添加大约 2 g（ΔI = 2）才能察觉到重量变重了。
5. Fechner's Law: The Psychophysical Scaling | 费希纳定律：心理物理量表
Fechner extended Weber's work and proposed that the perceived intensity of a sensation grows as the logarithm of the physical stimulus intensity. This is often called Fechner's logarithmic law.
费希纳发展了韦伯的工作，提出感觉的知觉强度是物理刺激强度对数的函数，这常被称为费希纳对数定律。
S = k log I
Here S stands for the perceived sensation magnitude, I is the physical intensity of the stimulus, and k is a scaling constant that varies with the type of stimulus.
其中 S 代表知觉到的感觉量，I 是刺激的物理强度，k 是一个随刺激类型变化的度量常数。
This means that when a physical intensity is multiplied by a constant factor, the perceived intensity increases by approximately equal steps. For instance, to make a light appear twice as bright, the physical intensity must be increased by a much larger factor than two.
这意味着当物理强度乘以某个常数倍时，知觉到的强度会以近似相等的步长增加。例如，为了使一盏灯看起来亮两倍，其物理强度需要增加的倍数远大于 2。
6. Ebbinghaus Forgetting Curve: Savings Method | 艾宾浩斯遗忘曲线：节省法
Hermann Ebbinghaus studied how quickly we forget newly learned material. He used himself as a participant, learning lists of nonsense syllables, and measured how much he retained after different time delays using the savings method.
赫尔曼·艾宾浩斯研究了我们对新学材料的遗忘速度。他以自己为被试，学习无意义音节表，并用节省法测量在不同时间间隔后的保持量。
Savings (%) = [(Original learning time – Relearning time) ÷ Original learning time] × 100%
The savings score tells us the percentage of information retained. A high savings percentage means less forgetting occurred over that period.
节省分数告诉我们保留信息的百分比。节省百分比高意味着在这段时间内发生的遗忘较少。
The famous forgetting curve shows a rapid drop in memory within the first hour, followed by a much slower decline. Ebbinghaus demonstrated this pattern can be modelled by a logarithmic relationship.
著名的遗忘曲线显示，记忆在第一个小时内迅速下降，随后减缓。艾宾浩斯证明这一模式可以用对数关系来建模。
7. Piaget's Conservation Principle | 皮亚杰守恒原理
Jean Piaget discovered that children develop the understanding that certain properties of objects remain the same even when their appearance changes. This is the principle of conservation.
让·皮亚杰发现，儿童逐渐理解物体的某些特性即使在外观改变时也保持不变，这就是守恒原理。
No single mathematical formula describes conservation, but the underlying rule can be stated as: 'quantity, mass, or volume remains invariant despite transformations in shape or arrangement.'
虽然没有单一的数学公式来描述守恒，但其核心规则可以表述为：“数量、质量或体积尽管形状或排列方式发生变化，但仍然保持不变。”
For instance, when water is poured from a short, wide beaker into a tall, thin cylinder, a child who has mastered conservation will say the amount of water is still the same, even though the water level looks different.
例如，当把水从一个矮宽的烧杯倒入一个高细的量筒时，已经掌握守恒的孩子会说水量仍然是一样的，尽管水位看起来不一样了。
8. Generalisation Gradient in Classical Conditioning | 经典条件作用中的泛化梯度
After conditioning a response to a specific stimulus, organisms often respond to similar stimuli as well. This is called stimulus generalisation. The strength of the response gradually decreases as the similarity between the test stimulus and the original conditioned stimulus decreases.
当对一个特定刺激建立了条件反应后，有机体常常也会对类似的刺激做出反应，这叫做刺激泛化。随着测试刺激与原始条件刺激之间相似性的降低，反应的强度会逐渐减弱。
This pattern is called the generalisation gradient. It can be represented as a curve where response strength is plotted on the vertical axis and stimulus similarity or physical distance along a dimension is plotted on the horizontal axis.
这种模式被称为泛化梯度。它可以绘制成一条曲线，纵轴表示反应强度，横轴表示刺激相似性或在一个维度上的物理距离。
For example, if a dog has been conditioned to salivate to a tone of 1000 Hz, it might also salivate to tones of 900 Hz or 1100 Hz, but less vigorously the further the tone is from the original frequency.
例如，如果一只狗已经对 1000 Hz 的声音形成了唾液分泌的条件反射，它可能也会对 900 Hz 或 1100 Hz 的声音分泌唾液，但离原始频率越远，分泌反应就越弱。
9. Multi-Store Model of Memory: Sequence and Capacities | 记忆的多储存模型：顺序与容量
The multi-store model (Atkinson and Shiffrin, 1968) 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（Atkinson 与 Shiffrin，1968）将记忆描述为信息在三个独立储存库中的流动：感觉记忆、短时记忆（STM）和长时记忆（LTM）。
The key 'formula' a limited capacity and duration except LTM, which is thought to have unlimited capacity and a potentially permanent duration.
这里的关键“公式”是信息流动的顺序：环境输入 → 感觉登记 → 注意 → 短时记忆 → 复述 → 长时记忆。除了长时记忆被认为具有无限容量和潜在的永久持续时间外，每个储存库的容量和持续时间都是有限的。
l figures often cited: sensory memory lasts up to 0.5 seconds for visual information and 2– auditory information. STM can hold about 7 ± 2 chunks of information for approximately 18–30 seconds without rehearsal.
通常引用的数据是：感觉记忆对视觉信息保持时间最多约 0.5 秒，对听觉信息约 2–3 秒。短时记忆在不进行复述的情况下，可以容纳大约 7 ± 2 个信息组块，持续约 18–30 秒。
10. Stroop Effect Interference Formula | 斯特鲁普效应干扰公式
The Stroop effect demonstrates that naming the ink colour of a colour word is harder when the word spells a different colour (e.g., the word 'RED' printed in blue ink). The interference can be quantified by comparing reaction times in congruent and incongruent trials.
斯特鲁普效应表明，当颜色词的语义与墨水颜色不一致时（如用蓝色墨水印“红”字），说出墨水颜色的名称会更困难。这种干扰可以通过比较一致试次和不一致试次的反应时间来量化。
Stroop interference = RTincongruent – RTcongruent
Here RTincongruent is the average reaction time for items where the word and ink colour do not match, and RTcongruent is the average reaction time for items where they match.
其中 RTincongruent 是词义与墨水颜色不匹配项目的平均反应时间，RTcongruent 是词义与墨水颜色匹配项目的平均反应时间。
A larger interference score indicates greater automaticity of reading words; the brain cannot ignore the word meaning even when trying to focus only on the ink colour.
干扰分数越大，表明阅读单词的自动化程度越高；即使只试图专注于墨水颜色，大脑也无法忽略单词的含义。
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