IGCSE CAIE Psychology: Formula & Theorem Quick Reference | IGCSE CAIE 心理学：公式定理速查手册

📚 IGCSE CAIE Psychology: Formula & Theorem Quick Reference | IGCSE CAIE 心理学：公式定理速查手册

This quick-reference guide gathers the essential formulas, statistical calculations, and psychological theorems required for the IGCSE CAIE Psychology syllabus. It provides clear explanations and ready-to-use equations so you can tackle research-method questions with confidence.

这本速查手册汇集了 IGCSE CAIE 心理学大纲所要求的核心公式、统计算法和心理定理。每个要点都配有清晰的解释和直接可用的方程式，帮助你自信应对研究方法类试题。

1. Calculating the Mean | 计算平均数

The mean is the arithmetic average of a data set. You sum all the raw scores and then divide by the number of scores.

平均数是一组数据的算术平均值：将所有原始分数相加，然后除以分数的个数。

x̄ = Σx ÷ n

x̄ = mean score (平均数) | Σx = sum of all scores (所有分数之和) | n = number of scores (分数个数)

2. Finding the Median | 确定中位数

The median is the middle value when all scores are arranged in ascending order. If there is an even number of scores, the median is the mean of the two middle values.

中位数是将所有分数由小到大排列后处于中间位置的数值。若分数个数为偶数，中位数取中间两个数的平均数。

Position of median = (n + 1) ÷ 2

First rank the data smallest to largest, then locate the value at the calculated position. Always check for extreme scores that might distort the mean – the median is less affected by outliers.

先将数据从小到大排序，再找到计算所得位置对应的数值。始终留意可能拉偏平均数的极端值——中位数受异常值的影响较小。

3. Identifying the Mode | 确定众数

The mode is the score that appears most frequently in a data set. A set can have one mode, more than one mode (bimodal/multimodal), or no mode at all if all values are unique.

众数是一组数据中出现次数最多的数值。一个数据集可能有一个众数、多个众数（双众数/多众数），或者在所有数值都不重复时没有众数。

It is the simplest measure of central tendency and is the only one suitable for nominal (categorical) data, such as favourite colour or experimental condition.

它是最简单的集中趋势量数，也是唯一适用于称名数据（类别数据）的指标，比如最喜爱的颜色或实验条件。

4. Determining the Range | 计算全距

The range is the difference between the highest and lowest values. It gives a quick indication of spread but is heavily influenced by a single outlier.

全距是最大值与最小值之差，能快速反映离散程度，但极易受单一异常值的影响。

Range = Highest score − Lowest score

Alongside the range, always note the standard deviation for a fuller picture of dispersion.

在使用全距的同时，应留意标准差，以便更全面地了解离散情况。

5. Standard Deviation (Simplified) | 标准差 (简化版)

Standard deviation measures how spread out scores are around the mean. A small SD means scores cluster tightly around the mean; a large SD indicates wide dispersion.

标准差衡量分数围绕平均数的分散程度。标准差小，分数紧密围绕平均数；标准差大，则分数分布很散。

s = √[ Σ(x − x̄)² ÷ (n − 1) ]

s = sample standard deviation (样本标准差)
x = each individual score (单个分数)
x̄ = sample mean (样本平均数)
n − 1 = degrees of freedom for a sample (样本自由度)

In the exam you may be asked to interpret a given SD, not necessarily to compute it from raw data. Remember: approximately 68% of values fall within ±1 SD of the mean in a normal distribution.

考试中你可能需要解读给定的标准差，不一定要从原始数据计算。记住：在正态分布中，约68% 的数值落在平均数 ±1 个标准差之内。

6. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关系数

This statistic measures the strength and direction of a relationship between two sets of ranked data. It produces a coefficient between −1 and +1.

该统计量测量两组等级数据之间关系的强度和方向，得出的系数在 −1 到 +1 之间。

rₛ = 1 − [ 6 ΣD² ÷ n(n² − 1) ]

rₛ = Spearman’s rank coefficient (斯皮尔曼系数)
D = difference between each pair of ranks (每对等级之差)
n = number of paired scores (对数)

Interpret rₛ by comparing it to a critical values table. If the absolute value of rₛ exceeds the critical value for your n (at p<0.05), the correlation is statistically significant.

需要将 rₛ 与临界值表比较。若 rₛ 的绝对值大于你对应 n 的临界值（p<0.05），该相关就具有统计显著性。

7. Percentage Change | 百分比变化

Percentage change is often used to compare pre- and post-test scores in an experiment, or to express the effect of an independent variable.

百分比变化常用于比较实验中的前后测得分，或表达自变量的效应。

% change = [ (New value − Original value) ÷ Original value ] × 100

A positive result indicates an increase; a negative result indicates a decrease. Always state the direction clearly when writing a conclusion.

正值表示增加，负值表示减少。在撰写结论时务必清楚说明变化的方向。

8. Probability and Significance Levels | 概率与显著性水平

In inferential statistics, probability (p) tells us how likely the observed result is due to chance. The conventional significance level in psychology is p < 0.05.

在推论统计中，概率 (p) 告诉我们观察到的结果由偶然因素造成的可能性有多大。心理学中常用的显著性水平是 p < 0.05。

p < 0.05 means the probability of the result occurring by chance alone is less than 5%.

If p ≤ 0.05, the null hypothesis is rejected and the result is statistically significant. If p > 0.05, you must retain the null hypothesis. Never say ‘prove’ – use ‘suggests’ or ‘supports’.

若 p ≤ 0.05，则拒绝虚无假设，结果具有统计显著性。若 p > 0.05，则保留虚无假设。切记不说「证明」，而用「提示」或「支持」。

9. Weber’s Law | 韦伯定律

Weber’s Law states the just-noticeable difference (JND) is a constant proportion of the original stimulus intensity. It applies to sensory thresholds and psychophysics.

韦伯定律指出，最小可觉差 (JND) 是原始刺激强度的一个恒定比例，适用于感觉阈限和心理物理学。

ΔI ÷ I = k

ΔI = minimal change in intensity that can be detected (能被察觉的最小强度变化，即 JND)
I = original stimulus intensity (原始刺激强度)
k = Weber fraction (constant) | 韦伯分数 (常数)

For example, if k = 0.1 for weight, you would need a 10 g change to notice a difference when holding a 100 g weight, but a 20 g change for a 200 g weight.

例如，若重量的 k = 0.1，当你手拿 100 g 重物时，需要增减 10 g 才能察觉差别；而对于 200 g 则需要 20 g。

10. Yerkes-Dodson Law | 耶克斯–多德森定律

The Yerkes-Dodson law describes the relationship between arousal and performance as an inverted U-shaped curve. Optimal arousal level leads to peak performance.

耶克斯–多德森定律将唤醒水平与绩效的关系描述为一条倒 U 形曲线：适中的唤醒水平带来最佳表现。

Performance improves with arousal up to an optimal point; too much arousal causes performance to decline.

For simple or well-learned tasks, the optimal arousal level is higher; for complex or novel tasks, the optimal level is lower. This principle is used to explain phenomena such as social facilitation and evaluation apprehension.

对简单或熟练的任务，最佳唤醒水平偏高；对复杂或新任务，最佳唤醒水平偏低。该原理可解释社会助长效应、评价恐惧等现象。

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IGCSE CAIE Psychology: Formula & Theorem Quick Reference | IGCSE CAIE 心理学：公式定理速查手册