📚 IGCSE WJEC Psychology: Formula & Theorem Quick Reference Handbook | IGCSE WJEC 心理学：公式定理速查手册
This concise handbook gathers essential statistical formulas, calculation steps, and quantitative principles from the WJEC IGCSE Psychology specification. Use it as a quick revision tool to master the numeracy skills required for research methods, data handling, and key psychophysical laws.
本手册汇集了 WJEC IGCSE 心理学课程中必不可少的统计公式、计算步骤和定量原理。可作为快速复习工具，帮助你掌握研究方法、数据处理以及关键心理物理学定律涉及的数学技能。
1. Measures of Central Tendency | 集中趋势量数
Central tendency provides a single value that represents the centre of a data set. The three main measures are the mean, median and mode.
集中趋势量数提供一个能够代表数据集中趋势的单一数值。三种主要量数是平均值、中位数和众数。
The mean is calculated by summing all scores and dividing by the number of scores: Mean = Σx / n, where Σx is the sum of observations and n is the number of observations. It is sensitive to extreme values.
平均值通过将所有数据值相加后除以数据个数得到：平均值 = Σx / n，其中 Σx 代表各观测值的总和，n 代表数据的个数。平均值容易受极端值的影响。
The median is the middle value when data are ordered. For an odd number of scores, it is the centre score. For an even number, it is the average of the two middle scores. The median is not distorted by outliers.
中位数是将数据按大小顺序排列后位于中间的数。当数据个数为奇数时，中位数就是正中间的数值；若为偶数，则为中间两个数值的平均数。中位数不受极端值的干扰。
The mode is the most frequently occurring value in a data set. A set can have one mode (unimodal), more than one (multimodal) or no mode at all. The mode is the only measure suitable for nominal data.
众数是数据集中出现次数最多的数值。一个数据集可以有一个众数（单峰）、多个众数（多峰）或者没有众数。众数是唯一适用于命名数据的集中量数。
In psychological research the mean is the most commonly reported average because it takes all data into account and is used in further calculations such as the standard deviation.
在心理学研究中，平均值是最常报告的平均数，因为它考虑了所有数据，并可用于进一步的计算，例如标准差。
2. Measures of Dispersion | 离散量数
Dispersion tells us how spread out the scores are around the central value. The main measures are the range, variance and standard deviation.
离散量数反映了数据围绕中心值的分散程度。主要量数包括全距、方差和标准差。
The range is the simplest measure: Range = Maximum value – affected by a single extreme score and does not show how data are distributed internally.
全距是最简单的离散量数：全距 = 最大值 – 最小值。它极易受到一个极端值的影响，无法反映数据内部的分布情况。
The standard deviation indicates the average distance of each score from the mean. A low standard deviation means data are tightly clustered around the mean; a high standard deviation signals greater variability. The sample standard deviation is given by:
标准差反映了每个数据值与平均值之间的平均距离。标准差小意味着数据紧密聚集在平均值周围；标准差大则表示数据差异性更大。样本标准差公式如下：
s = √[ Σ(x – x̄)² / (n – 1) ]
where x is each individual score, x̄ is the sample mean, and n is the sample size. For populations the denominator becomes N.
其中 x 代表每个单独的数值，x̄ 是样本平均值，n 是样本容量。对于总体，分母变为 N。
The variance is simply the square of the standard deviation: s² = Σ(x – x̄)² / (n – advanced analyses but is measured in squared units, making it less intuitive to interpret.
方差就是标准差的平方：s² = Σ(x – x̄)² / (n – 1)。它用于更高级的分析，但由于单位是平方单位，解释起来不够直观。
When comparing two sets of data, researchers check both the mean and the standard deviation to understand group differences.
当比较两组数据时，研究者会同时考察平均值和标准差以理解组间差异。
3. Percentiles and Quartiles | 百分位数与四分位数
Percentiles divide a data set into 100 equal parts, showing the percentage of scores that fall below a particular value. A score at the 80th percentile means that 80% of scores are lower.
百分位数将数据划分为 100 等份，反映了低于某一特定值的百分比。一个处于第 80 百分位的分数意味着有 80% 的数据低于它。
The quartiles split the ordered data into four equal groups: Q1 (25th percentile) is the first quartile, Q2 (50th percentile) is the median, and Q3 (75th percentile) is the third quartile. The interquartile range (IQR = Q3 – Q1) describes the middle 50% of scores and is resistant to outliers.
四分位数将有序数据分成四个相等的组：Q1（第 25 百分位）是第一四分位数，Q2（第 50 百分位）即为中位数，Q3（第 75 百分位）是第三四分位数。四分位距（IQR = Q3 – Q1）描述了中间 50% 的数据，它不受异常值的干扰。
To determine a percentile position, a common method uses Position = (p/100) × (n + 1). If the position is a decimal, interpolation is employed, but at IGCSE level a simpler nearest-rank approach is often accepted.
要确定百分位数的位置，常用的方法是 位置 = (p/100) × (n + 1)。如果位置为小数，需进行插值；不过在 IGCSE 阶段，通常可采用较简单的最近等级法。
Percentiles and quartiles are widely used when reporting norm-referenced test results, such as IQ scores or school percentile ranks.
百分位数和四分位数广泛用于报告常模参照测验的结果，例如 IQ 分数或学校百分位排名。
4. Correlation and Spearman's Rank Correlation Coefficient | 相关与斯皮尔曼等级相关系数
Correlation measures the strength and direction of a relationship between two variables. The Spearman's rank correlation coefficient (rₛ) is a non-parametric test suitable for ordinal data or when assumptions of parametric tests are not met.
相关衡量两个变量之间关系的强度和方向。斯皮尔曼等级相关系数 (rₛ) 是一种非参数检验，适用于顺序数据或参数检验假设不满足的情况。
The formula for Spearman's rₛ is:
斯皮尔曼等级相关系数 rₛ 的公式如下：
rₛ = 1 – (6 Σ d²) / (n(n² – 1))
where d is the difference between the ranks of each pair of observations, and n is the number of pairs. The value of rₛ ranges from –1 (perfect negative correlation) to +1 (perfect positive correlation). A value near 0 indicates no relationship.
其中 d 为每对观测数据的秩次之差，n 为数据对的数量。rₛ 的取值介于 –1（完全负相关）到 +1（完全正相关）之间，接近 0 则表示没有关系。
Steps for calculating rₛ: (1) Rank each variable separately, giving the highest score rank 1. (2) Find the difference d between the two ranks for each participant. (3) Square each d to obtain d². (4) Sum all d² and insert into the formula together with n. (5) Compare the obtained rₛ with a critical value table to assess significance.
计算 rₛ 的步骤：(1) 分别对两个变量进行排序，最高分给予秩次 1。(2) 找出每位参与者两个秩次的差值 d。(3) 将每个 d 平方，得到 d²。(4) 将所有 d² 求和，并同 n 一起代入公式。(5) 将求得的 rₛ 与临界值表比较，以评估其显著性。
Spearman's rₛ is often used in IGCSE psychology when students analyse scatter graphs and test for a relationship, for example between age and memory score.
在 IGCSE 心理学中，学生在分析散点图并检验变量间关系时（例如年龄与记忆分数）常使用斯皮尔曼等级相关系数。
5. Percentage Change | 百分比变化
Psychologists frequently report changes in behaviour using percentage change, especially when comparing pre-test and post-test results in an experiment.
心理学家在报告行为变化时经常使用百分比变化，尤其是在比较实验中的前测与后测结果时。
The basic formula is:
基本公式如下：
Percentage Change = ((New Value – Original Value) / Original Value) × 100
indicates an increase; a negative result shows a decrease. For instance, if a participant's reaction time decreased from 250 ms to 200 ms, the percentage change is ((200 – 250) / 250) × 100 = –20%.
计算结果为正表示增加，为负表示减少。例如，若某参与者的反应时间从 250 ms 降至 200 ms，百分比变化为 ((200 – 250) / 250) × 100 = –20%。
This formula also helps to calculate percentage increase or percentage decrease in studies of cognitive improvement or decline. Always be clear about which value serves as the baseline.
该公式还可用于计算认知改善或衰退研究中的增长百分比或减少百分比。务必清楚哪个值作为基准。
6. Weber's Law and Difference Threshold | 韦伯定律与差别阈限
In the study of perception, Weber's Law explains how we detect changes in stimuli. The difference threshold (just noticeable difference, JND) is the minimum change in a stimulus that can be detected 50% of the time.
在感知觉研究中，韦伯定律解释了人们如何察觉刺激的变化。差别阈限（恰可察觉差异，JND）是指在 50% 的情况下可被察觉到的最小刺激变化。
Weber's Law is expressed as:
韦伯定律的表达式为：
ΔI / I = k
where ΔI is the just noticeable difference in intensity, I is the initial stimulus intensity, and k is the Weber fraction (a constant that varies across sensory modalities). For example, if the Weber fraction for lifted weights is 0.02, a weight of 100 g would need a 2 g increase before the difference is noticed.
其中 ΔI 是刚能察觉的强度差异，I 是初始刺激强度，k 是韦伯分数（一个随感觉通道不同而变化的常数）。例如，若提举重量的韦伯分数为 0.02，则 100 g 的重量需增加 2 g 才能被察觉出变化。
Consequently, the JND can be predicted as JND = k × I. Weber's Law helps psychologists design experiments where stimulus changes need to be detectable.
因此，恰可察觉差异可预测为 JND = k × I。韦伯定律有助于心理学家在设计实验时确保刺激变化可被察觉。
7. The Normal Distribution and Z-Scores | 正态分布与Z分数
Many psychological variables, such as IQ, are normally distributed. The normal curve is bell-shaped, symmetric, and the mean, median and mode coincide at the centre.
许多心理学变量（如智商）呈正态分布。正态曲线呈钟形、对称，且平均值、中位数与众数在中心重合。
A z-score standardises a raw score by expressing how many standard deviations it lies from the mean:
Z分数通过将原始分数转化为距离平均值若干个标准差来标准化：
z = (x – μ) / σ
n, and σ is the population standard deviation. For sample data, the formula becomes z = (x – x̄) / s.
其中 x 是原始分数，μ 是总体平均值，σ 是总体标准差。对于样本数据，公式变为 z = (x – x̄) / s。
Z-scores allow researchers to compare scores from different distributions. They are particularly useful when discussing probability under the normal curve. Approximately 68% of values fall within ±1 standard deviation, 95% within ±2, and 99.7% within ±3.
Z分数使研究者能够比较来自不同分布的分数。它在讨论正态曲线下的概率时尤为有用。大约 68% 的数值落在 ±1 个标准差内，95% 落在 ±2 个标准差内，99.7% 落在 ±3 个标准差内。
| ±1 SD | 68.26% of scores |
| ±2 SD | 95.44% of scores |
| ±3 SD | 99.74% of scores |
This is sometimes called the empirical rule and is referenced when psychologists judge how unusual a given observation is.
这有时被称为经验法则，在心理学家判断某个观察值有多不寻常时会被提及。
8. Significance Levels and p-Values | 显著性水平与p值
Inferential statistics help psychologists decide whether their results reflect a real effect or chance. The p-value represents the probability of obtaining the observed result (or one more extreme) if the null hypothesis were true.
推断统计帮助心理学家确定研究结果反映的是真实效应还是偶然误差。p值表示在虚无假设为真的情况下，获得当前结果或更极端结果的概率。
A conventional significance level is p ≤ 0.05. If p is less than or equal to 0.05, the result is deemed statistically significant, and the null hypothesis is rejected. This means there is less than a 5% probability the result occurred by chance.
常见的显著性水平是 p ≤ 0.05。如果 p 小于或等于 0.05，则结果被认为具有统计显著性，从而拒绝虚无假设。这意味着结果由偶然因素所致的概率小于 5%。
ger evidence against the null hypothesis. The choice of significance level must be stated before the study begins to avoid bias.
其他常见的阈值还有 p ≤ 0.01 或 p ≤ 0.001，它们代表着更强有力的反对虚无假设的证据。显著性水平必须在研究开始前予以说明，以避免偏差。
When using Spearman's rₛ, the calculated coefficient is compared with a critical value that corresponds to the chosen significance level and the degrees of freedom (
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