📚 IGCSE CCEA Psychology: Formula & Theorem Quick Reference Handbook | IGCSE CCEA 心理学：公式定理速查手册
This handbook collects the most important formulas, theorems and statistical tools you will encounter in the IGCSE CCEA Psychology course. Having these formulas at your fingertips helps you confidently analyse data, interpret research findings and answer exam questions on research methods.
本手册收录了你在 IGCSE CCEA 心理学课程中将遇到的最重要的公式、定理和统计工具。将这些公式随时放在手边，有助于你自信地分析数据、解读研究结果并回答研究方法相关的考试题目。
1. Measures of Central Tendency | 集中趋势的度量
The mean is calculated by summing all scores and dividing by the number of scores. Formula: Mean = Σx / N, where Σx is the sum of all values and N is the number of observations.
平均值通过将所有分数相加再除以分数个数来计算。公式：平均值 = Σx / N，其中 Σx 是所有数值的总和，N 是观测值的个数。
The median is the middle value in an ordered list. For an even number of scores, it is the mean of the two middle numbers. No single formula is used, but the position can be found by (N+1)/2.
中位数是有序列表中的中间值。当分数个数为偶数时，中位数是中间两个数的平均值。不依赖单一公式，但位置可通过 (N+1)/2 找到。
The mode is the most frequently occurring score in a set of data. A data set may have no mode, one mode or multiple modes.
众数是一组数据中出现频率最高的分数。一组数据可能没有众数、有一个众数或有多个众数。
2. Measures of Dispersion | 离散程度的度量
The range is the difference between the highest and lowest scores. Range = Maximum – Minimum. It gives a quick idea of spread but is affected by outliers.
全距是最高分与最低分之间的差值。全距 = 最大值 – 最小值。它能快速反映数据的分散程度，但容易受异常值影响。
The standard deviation measures how much scores deviate from the mean. For a sample, use: s = √[ Σ(x – x̄)² / (n – 1) ], where x̄ is the mean and n is sample size. The square of this value is called the variance.
标准差衡量分数偏离平均值的程度。对于样本，使用： s = √[ Σ(x – x̄)² / (n – 1) ]，其中 x̄ 为平均值，n 为样本量。该值的平方称为方差。
Interquartile range (IQR) is the range of the middle 50% of data: IQR = Q3 – Q1, where Q1 is the lower quartile and Q3 is the upper quartile. It is more resistant to outliers.
四分位距（IQR）是中间 50% 数据的范围：IQR = Q3 – Q1，其中 Q1 为下四分位数，Q3 为上四分位数。它对异常值具有更强的抵抗力。
3. Frequency Distributions & Graphical Displays | 频数分布与图形展示
Frequency tables summarise how often each score occurs. From these, relative frequency can be computed: Relative frequency = Frequency of a score / Total number of scores.
频数表汇总了每个分数出现的次数。由此可以计算相对频数：相对频数 = 某个分数的频数 / 总分数个数。
Histograms display grouped frequency data. The area of each bar represents frequency; when bin widths are unequal, frequency density = Frequency / Class width must be used.
直方图展示分组频数数据。每个条形的面积代表频数；当组距不相等时，必须使用频率密度 = 频数 / 组距。
A normal distribution is symmetrical and bell‑shaped. In a perfect normal curve, mean = median = mode, and about 68% of data fall within ±1 standard deviation from the mean.
正态分布是对称的钟形曲线。在完美的正态曲线上，平均值 = 中位数 = 众数，且约 68% 的数据落在平均值 ±1 个标准差的范围内。
4. Correlation Coefficients | 相关系数
Pearson’s r measures the strength and direction of a linear relationship between two continuous variables. Formula: r = Σ[(x – x̄)(y – ȳ)] / √[Σ(x – x̄)² Σ(y – ȳ)²]. r ranges from –1 to +1.
皮尔逊相关系数 r 衡量两个连续变量之间线性关系的强度和方向。公式： r = Σ[(x – x̄)(y – ȳ)] / √[Σ(x – x̄)² Σ(y – ȳ)²]。 r 的取值范围为 –1 到 +1。
Spearman’s rho (ρ) is used for ordinal data. It follows the same principle but is calculated on ranked scores. Formula: ρ = 1 – [6 Σd² / n(n² – 1)], where d is the difference between ranks for each pair.
斯皮尔曼等级相关系数 ρ 用于顺序数据。它遵循同样的原理，但基于排名分数计算。公式： ρ = 1 – [6 Σd² / n(n² – 1)]，其中 d 是每一对数据的排名差值。
5. Chi-Square Test (χ²) | 卡方检验
The chi-square test for independence compares observed frequencies with expected frequencies in a contingency table. Formula: χ² = Σ [(O – E)² / E], where O = observed frequency, E = expected frequency.
独立性卡方检验比较列联表中的观测频数与期望频数。公式： χ² = Σ [(O – E)² / E]，其中 O = 观测频数，E = 期望频数。
Expected frequencies are calculated under the null hypothesis: E = (Row total × Column total) / Grand total. Degrees of freedom (df) = (number of rows – 1) × (number of columns – 1).
期望频数在零假设下计算： E = （行合计 × 列合计）/ 总计。自由度（df）= （行数 – 1） × （列数 – 1）。
If the computed χ² value exceeds the critical value from the chi‑square table at a chosen significance level (commonly p<0.05), the null hypothesis is rejected.
如果计算得出的 χ² 值大于所选显著性水平（常用 p<0.05）下的卡方临界值，则拒绝零假设。
6. t-Test for Independent Means | 独立均值 t 检验
The independent t‑test compares the means of two unrelated groups. Formula: t = (x̄₁ – x̄₂) / √[sₚ²(1/n₁ + 1/n₂)], where sₚ² is the pooled variance.
独立样本 t 检验比较两个不相关组的平均值。公式： t = (x̄₁ – x̄₂) / √[sₚ²(1/n₁ + 1/n₂)]，其中 sₚ² 为合并方差。
Pooled variance is calculated as sₚ² = [(n₁–1)s₁² + (n₂–1)s₂²] / (n₁ + n₂ – 2). This assumes equal variances in both populations.
合并方差计算公式为 sₚ² = [(n₁–1)s₁² + (n₂–1)s₂²] / (n₁ + n₂ – 2)。这假定两个总体的方差相等。
Degrees of freedom for this test: df = n₁ + n₂ – 2. Compare the t value to critical values from the t‑distribution table to decide significance.
该检验的自由度为： df = n₁ + n₂ – 2。将 t 值与 t 分布表中的临界值比较，从而判断显著性。
7. Effect Size (Cohen’s d) | 效应量（科恩 d）
Cohen’s d quantifies the size of the difference between two means in standard deviation units. Formula: d = (x̄₁ – x̄₂) / sₚ, where sₚ is the pooled standard deviation (the square root of the pooled variance).
科恩 d 以标准差为单位量化两个平均值之间差异的大小。公式： d = (x̄₁ – x̄₂) / sₚ，其中 sₚ 为合并标准差（合并方差的平方根）。
Guidelines for interpretation: small effect d ≈ 0.2, medium effect d ≈ 0.5, large effect d ≈ 0.8. Reporting effect size is now essential in psychological research because it gives meaning beyond statistical significance.
解读指南：小效应 d ≈ 0.2，中等效应 d ≈ 0.5，大效应 d ≈ 0.8。汇报效应量在心理学研究中如今必不可少，因为它能提供统计显著性之外的实际意义。
8. Percentages, Ratios & Proportions | 百分比、比率与比例
Percentage change is commonly used to describe differences over time or between conditions: Percentage change = [(New value – Original value) / Original value] × 100.
百分比变化常用于描述随时间或在不同条件间发生的差异：百分比变化 = [（新值 – 原始值）/ 原始值] × 100。
A ratio expresses the relative size of two quantities, for example the ratio of males to females. A proportion is a part of a whole, expressed as a fraction or decimal of 1.
比率表示两个数量的相对大小，例如男性与女性的比率。比例是整体的一部分，用分数或 1 的小数表示。
9. Probability & Significance Levels | 概率与显著性水平
In inferential statistics, probability (p) expresses how likely a result is to have occurred by chance. p = Number of favourable outcomes / Total number of possible outcomes.
在推断统计中，概率（p）表示某个结果由偶然因素产生的可能性。 p = 有利结果的数量 / 所有可能结果的总数。
The most common significance level is α = 0.05 (5%). If p ≤ 0.05, the result is statistically significant, meaning the null hypothesis can be rejected. Type I error occurs when a true null hypothesis is rejected; Type II error occurs when a false null hypothesis is not rejected.
最常见的显著性水平是 α = 0.05（5%）。若 p ≤ 0.05，则结果具有统计显著性，意味着可以拒绝零假设。第一类错误是拒绝了正确的零假设；第二类错误是未能拒绝错误的零假设。
10. Z‑Scores & the Normal Curve | Z 分数与正态曲线
A z‑score tells you how many standard deviations a raw score is from the mean. Formula: z = (x – μ) / σ for a population, or z = (x – x̄) / s for a sample.
Z 分数表示一个原始分数距离平均值多少个标准差。公式：总体 z = (x – μ) / σ，样本 z = (x – x̄) / s。
Using z‑tables, you can find the proportion of scores that fall above or below a given value, or between two values. For a normal distribution, z = ±1.96 cuts off the extreme 5% (2.5% in each tail), which is central to hypothesis testing.
利用 Z 表，你可以找出落在给定值之上、之下或两个值之间的分数比例。在正态分布中，z = ±1.96 切掉了极端的 5%（每尾 2.5%），这是假设检验的核心。
11. Reliability & Validity Coefficients | 信度与效度系数
Split‑half reliability correlates the scores from two halves of a test. The Spearman‑Brown prophecy formula corrects this to estimate the reliability of the full test: rₜₒₜₐₗ = 2rₕₐₗf / (1 + rₕₐₗf), where rₕₐₗf is the correlation between halves.
分半信度将测验两半的分数进行相关。斯皮尔曼–布朗预测公式将之校正以估计完整测验的信度： rₜₒₜₐₗ = 2rₕₐₗf / (1 + rₕₐₗf)，其中 rₕₐₗf 为两半之间的相关系数。
Cronbach’s alpha is a more general measure of internal consistency. Although its full formula is complex, for exam purposes you should know it ranges from 0 to 1, with values above 0.7 generally indicating acceptable reliability.
克隆巴赫 α 系数是内部一致性的更通用度量。虽然其完整公式较为复杂，但考试中你只需知道它的取值范围在 0 到 1 之间，高于 0.7 通常表示可接受的信度。
12. Study Design: Sample Size & Power | 研究设计：样本量与统计功效
In experiments, the number of participants per condition can affect the statistical power. While there is no single formula at IGCSE level, a larger sample reduces the standard error of the mean: Standard error (SE) = s / √n.
在实验中，每个条件的被试数量会影响统计功效。尽管在 IGCSE 阶段没有单一公式，但更大的样本会减小均值的标准误：标准误 (SE) = s / √n。
Power is the probability of correctly rejecting a false null hypothesis. It is influenced by sample size, effect size and significance level. Understanding these relationships helps you evaluate the quality of psychological studies.
统计功效是正确拒绝错误零假设的概率。它受样本量、效应量和显著性水平的影响。理解这些关系有助于你评估心理学研究的质量。
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