📚 Year 9 Cambridge Psychology: Formula & Theorem Quick Reference Handbook | Year 9剑桥心理学：公式定理速查手册
This handbook brings together the essential formulas, mathematical relationships and key laws you will encounter in Year 9 Cambridge Psychology. From calculating IQ scores and summarising data to understanding how we perceive the world, remember information and perform under pressure, these tools will help you describe, explain and evaluate psychological findings with precision.
本手册汇集了你在Year 9剑桥心理学中将会遇到的核心公式、数学关系和关键定律。从计算智商分数、汇总数据，到理解我们如何感知世界、记忆信息以及在压力下表现，这些工具将帮助你精确地描述、解释和评估心理学发现。
1. Intelligence Quotient (IQ) Formula | 智力商数 (IQ) 公式
The original Stanford‑Binet formula defines IQ as the ratio of mental age to chronological age, multiplied by 100. Although modern tests use deviation scores, this formula remains a cornerstone for understanding how intelligence was first quantified.
最初的斯坦福‑比奈公式将智商定义为心理年龄与实际年龄的比值，再乘以100。尽管现代测试使用离差分数，但这一公式仍然是理解智力最早如何被量化的基石。
IQ = (MA / CA) × 100
where MA = mental age and CA = chronological age.
其中 MA 为心理年龄，CA 为实际年龄。
When a child's mental age equals their chronological age, IQ = 100, which is always the average score.
当一个孩子的心理年龄等于其实际年龄时，IQ = 100，这始终是平均分数。
2. Measures of Central Tendency and Range | 集中趋势与范围的度量
Psychologists summarise large sets of data using central tendency and spread. These three measures describe where the centre of the data lies, while the range captures how spread out the scores are.
心理学家使用集中趋势和离散程度来总结大量数据。以下三个度量描述数据的中心位置，而范围则反映分数的分散程度。
Mean (arithmetic average):
平均数（算术平均值）：
Mean = (Σx) / n
where Σx is the sum of all scores and n is the number of scores.
其中 Σx 是所有分数的总和，n 是分数的个数。
Median ed in : the middle value when scores are arranged in order. If n is even, median = (value at position n/2 + value at position n/2+1) / 2.
中位数：将分数按顺序排列后的中间值。若 n 为偶数，中位数 =（第 n/2 位值 + 第 n/2+1 位值）/ 2。
Mode: the most frequently occurring score.
众数：出现频率最高的分数。
Range:
范围：
Range = Highest score − Lowest score
These descriptors are the foundation of any descriptive statistics section in a psychological report.
这些描述量是任何心理学报告中描述统计部分的基础。
3. Percentage Change and Ratio Calculations | 百分比变化与比率计算
Many psychological experiments report changes in behaviour as percentages. Whether it is the proportion of participants who conformed in Asch’s line study or the increase in recall after a memory strategy, a simple percentage formula is used repeatedly.
许多心理学实验以百分数报告行为的变化。无论是阿希线段研究中从众参与者的比例，还是使用记忆策略后回忆率的提高，都会反复使用一个简单的百分比公式。
Percentage (%) = (Part / Whole) × 100
For two conditions, percentage change is:
对于两种条件，百分比变化为：
% change = [(New value − Old value) / Old value] × 100
ct, conforming answer / total critical trials) × 100. Across all critical trials, about 37% of responses were conforming errors.
在阿希从众范式中，从众率计算为（参与者给出错误从众答案的试验次数 / 总关键试验次数）× 100。在所有关键试验中，约 37% 的回答是从众错误。
4. Weber’s Law | 韦伯定律
Weber’s law describes the just noticeable difference (JND) —— the smallest change in a stimulus that can be detected. The JND is a constant proportion of the original stimulus intensity.
韦伯定律描述了最小可觉差（JND）—— 能够被察觉到的刺激的最小变化。最小可觉差是原始刺激强度的恒定比例。
ΔI / I = k
where ΔI is the JND, I is the initial stimulus intensity, and k is the Weber fraction (a constant that differs across sensory modalities).
其中 ΔI 是最小可觉差，I 是初始刺激强度，k 是韦伯分数（一个因感觉通道不同而异的常数）。
For example, for lifted weights k ≈ 0.02, meaning you can detect about a 2% increase in weight. For brightness of light, k is approximately 0.08.
例如，对于提举重物，k ≈ 0.02，意味着你可以察觉到重量约 2% 的增加。对于光的亮度，k 大约为 0.08。
Weber’s law laid the groundwork for psychophysics and shows that perception is relative, not absolute.
韦伯定律为心理物理学奠定了基础，并表明感知是相对的，而非绝对的。
5. Fechner’s Law | 费希纳定律
Building on Weber’s work, Fechner proposed a logarithmic relationship between the physical intensity of a stimulus and the perceived sensation magnitude.
在韦伯工作的基础上，费希纳提出刺激的物理强度与感知到的感觉大小之间存在对数关系。
S = k log I
where S is the magnitude of sensation, I is the physical intensity of the stimulus, and k is a constant that depends on the sensory modality.
其中 S 是感觉大小，I 是刺激的物理强度，k 是取决于感觉通道的常数。
If a light is doubled in physical intensity, the perceived brightness does not double; instead it increases by a much smaller, logarithmic amount. This helps explain why we can detect a candle in a dark room but not in bright sunlight.
如果光的物理强度加倍，感知到的亮度并不会加倍；相反，它以对数方式增加，幅度小很多。这有助于解释为什么我们在暗室中能察觉一支蜡烛，而在明亮阳光下却不行。
6. Ebbinghaus Forgetting Curve | 艾宾浩斯遗忘曲线
Hermann Ebbinghaus studied memory using nonsense syllables and plotted the decline of memory retention over time. The resulting curve shows rapid forgetting immediately after learning, which then levels off.
赫尔曼·艾宾浩斯使用无意义音节研究记忆，并绘制了记忆保持率随时间下降的曲线。结果显示，学习后遗忘立刻迅速发生，随后趋于平缓。
A simplified mathematical form often used to describe this curve is:
一个常用于描述该曲线的简化数学形式是：
R = e−t/S
where R is retention (proportion remembered), t is time, and S is the relative strength of the memory trace.
其中 R 是保持率（记住的比例），t 是时间，S 是记忆痕迹的相对强度。
In classroom terms, if you do not review newly learned material, you may forget up to 50% within an hour. Regular, spaced revision dramatically flattens the forgetting curve.
在课堂语境中，如果你不复习新学的材料，一小时内可能遗忘多达 50%。定期、间隔的复习会显著使遗忘曲线趋于平坦。
7. Miller’s Law (Memory Span) | 米勒定律 (记忆广度)
George Miller’s classic paper ‘The Magical Number Seven, Plus or Minus Two’ argued that the capacity of short‑term memory is limited to about 7 ± 2 chunks of information.
乔治·米勒的经典论文《神奇的数字7±2》提出，短时记忆的容量限制为大约 7 ± 2 个信息组块。
Short‑term memory span ≈ 7 ± 2 items
An ‘item’ is not a single digit or letter but a meaningful chunk. For example, the string ‘FBICIAUSA’ is 9 letters, but if you chunk it as ‘FBI’, ‘CIA’, ‘USA’ it becomes three chunks, well within the span.
一个“项目”并非单个数字或字母，而是一个有意义的组块。例如，字符串“FBICIAUSA”是9个字母，但如果你将其组块为“FBI”、“CIA”、“USA”，它就变成三个组块，完全在广度之内。
This limitation explains why phone numbers are often broken into chunks and why students struggle when information is presented in long, unbroken strings.
这一限制解释了为什么电话号码常被分成组块，以及为什么当信息以冗长连续的字符串呈现时学生会感到困难。
8. Yerkes–Dodson Law | 耶克斯–多德森定律
The Yerkes–Dodson law describes the relationship between arousal and performance. It states that performance increases with arousal up to an optimal point, after which further arousal leads to a decline in performance.
耶克斯–多德森定律描述了唤醒与表现之间的关系。它指出，表现随唤醒水平提高而提高，直至达到最佳点，此后进一步唤醒会导致表现下降。
While there is no single mathematical equation, the law is often illustrated by an inverted‑U function:
虽然没有单一的数学方程，这一定律通常用倒U形函数来说明：
Performance = f(Arousal), with an optimum at moderate arousal
Furthermore, the optimal level of arousal depends on task difficulty: for simple, well‑learned tasks, a higher arousal level is best; for complex, novel tasks, lower arousal is preferable.
此外，最佳唤醒水平取决于任务难度：对于简单、熟练的任务，较高的唤醒水平最好；对于复杂、新颖的任务，较低的唤醒水平更合适。
This law is used to explain phenomena such as test anxiety and sports performance, where being slightly stressed can improve focus but excessive stress impairs it.
这一定律被用来解释考试焦虑和运动表现等现象，即适度的压力可以提升专注力，而过度压力则会损害表现。
Published by TutorHao | Psychology Revision Series | aleveler.com
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