📚 A-Level CAIE Psychology: Quick Reference to Formulas and Theorems | A-Level CAIE 心理学：公式定理速查手册
Psychological research relies heavily on quantitative analysis. Knowing the key formulas and statistical theorems is essential for success in the A-Level CAIE Psychology exam. This handbook provides a quick reference to the most important mathematical and inferential tools you will need, from descriptive statistics to hypothesis testing.
心理学研究高度依赖定量分析。掌握关键公式和统计定理对于在A-Level CAIE心理学考试中取得成功至关重要。本手册为你提供最重要的数学与推断工具速查，涵盖描述统计到假设检验的全部核心内容。
1. Measures of Central Tendency | 集中趋势测量
The mean (x̄ is the arithmetic average: sum all scores and divide by the number of scores N. Formula: x̄ e median is the middle value when scores are ordered; the mode is the most frequent score.
平均数（x̄）是算术平均值：将所有分数求和后除以分数个数 N。公式：x̄ = Σx / N。中位数是按顺序排列后位于中间的数值；众数是出现次数最多的分数。
Use the mean for interval/ratio data without extreme outliers. The median is preferred for skewed distributions or ordinal data. The mode is the only measure suitable for nominal data.
在无极端异常值的等距/等比数据中使用平均数。对于偏态分布或顺序数据，优先使用中位数。众数是唯一适用于名义数据的集中量数。
Mean (x̄) = Σx / N
2. Measures of Dispersion | 离散趋势测量
deviation from the mean. For a population: σ² = Σ(x - μ)² / N. For a sample: s² = Σ(x - x̄)² / (n - 1).
全距是最大值与最小值之差。方差测量的是各数据与平均数之差的平方的平均值。对于总体：σ² = Σ(x - μ)² / N。对于样本：s² = Σ(x - x̄)² / (n - 1)。
returning to original units. Population SD: σ = √[Σ(x - μ)² / N]. Sample SD: s = √[Σ(x - x̄)² / (n - 1)]. A larger SD indicates greater spread.
标准差是方差的平方根，回到原始单位。总体标准差：σ = √[Σ(x - μ)² / N]。样本标准差：s = √[Σ(x - x̄)² / (n - 1)]。标准差越大，数据分布越分散。
Sample SD: s = √[Σ(x - x̄)² / (n - 1)]
3. Normal Distribution and Skewness | 正态分布与偏态
ibution is a symmetrical bell-shaped curve where mean, median, and mode coincide. About 68% of data falls within ±1 SD of the mean, 95% within ±2 SD, and 99.7% within ±3 SD.
正态分布是一条对称的钟形曲线，平均数、中位数和众数重合。约68%的数据落在平均数±1个标准差内，95%在±2个标准差内，99.7%在±3个标准差内。
Skewness describes asymmetry. Positive skew: mean > median > mode; tail extends to the right. Negative skew: mean < median < mode; tail extends to the left. Skew affects choice of central tendency and inferential tests.
偏态描述不对称性。正偏态：平均数 > 中位数 > 众数，尾部向右延伸。负偏态：平均数 < 中位数 < 众数，尾部向左延伸。偏态影响集中趋势测量和推断检验的选择。
Positive skew: mean > median > mode
4. Correlation: Pearson's r and Spearman's ρ | 相关：皮尔逊r与斯皮尔曼ρ
Pearson' ous variables. It ranges from -1 (perfect negative) to +1 (perfect positive). r = Σ[(x - x̄)(y - ȳ)] / √[Σ(x - x̄)² Σ(y - ȳ)²].
皮尔逊r测量两个连续变量之间线性关系的强度和方向。取值范围从-1（完全负相关）到+1（完全正相关）。r = Σ[(x - x̄)(y - ȳ)] / √[Σ(x - x̄)² Σ(y - ȳ)²]。
Spearman' each pair, then ρ = 1 - (6ΣD² / n(n² - 1)). It is a non-parametric alternative to Pearson's r.
斯皮尔曼ρ适用于顺序数据或非正态分布数据。将两组数据分别排序，计算每对秩次之差D，然后ρ = 1 - (6ΣD² / n(n² - 1))。这是皮尔逊r的非参数替代方法。
Spearman's ρ = 1 - 6ΣD² / [n(n² - 1)]
5. Chi-Square Test (χ²) | 卡方检验
served frequency and E = expected frequency. E = (row total × column total) / grand total.
独立性卡方检验评估两个类别变量是否相关。公式：χ² = Σ[(O - E)² / E]，其中O为观察频数，E为期望频数。E = (行合计 × 列合计) / 总计。
er of columns - 1). Compare the calculated χ² to a critical value from the chi-square distribution table. If calculated > critical, reject the null hypothesis.
列联表的自由度：df = (行数 - 1) × (列数 - 1)。将计算出的χ²值与卡方分布表中的临界值比较。若计算值 > 临界值，拒绝零假设。
χ² = Σ[(O - E)² / E]
6. t-Test: Independent and Related Designs | t检验：独立组与相关设计
t-test compares the means of two unrelated groups. Formula for equal sample sizes: t = (x̄₁ - x̄₂) / √[(s₁²/n₁) + (s₂²/n₂)]. It assumes normal distribution and homogeneity of variance.
独立样本t检验比较两个不相关组别的平均数。等样本量公式：t = (x̄₁ - x̄₂) / √[(s₁²/n₁) + (s₂²/n₂)]。假设数据正态分布且方差齐性。
ated measures or matched pairs. Calculate the difference score D for each pair, then t = D̄ / (sD / √n), where D̄ is the mean difference and sD is the standard deviation of differences.
相关（配对）t检验用于重复测量或配对设计。计算每对数据的差值D，然后t = D̄ / (sD / √n)，其中D̄是差值的平均数，sD是差值的标准差。
Related t-test: t = D̄ / (sD / √n)
7. Mann-Whitney U and Wilcoxon Signed-Ranks Tests | 曼-惠特尼U与威尔科克森符号秩检验
ted. Rank all data together, sum ranks for each group (R₁ and R₂). U₁ = n₁n₂ + n₁(n₁+1)/2 - R₁. The smaller U is compared to critical values.
曼-惠特尼U检验是独立t检验的非参数等效方法，用于顺序数据或违背正态假设时。将所有数据混合排序，计算每组的秩和（R₁和R₂）。U₁ = n₁n₂ + n₁(n₁+1)/2 - R₁。取较小的U与临界值比较。
The Wilcoxon signed-ranks test is used for related designs when data is not interval/ratio or not normally distributed. Rank the absolute differences, assign signs, and sum positive and negative ranks. The test statistic T is the smaller sum of signed ranks.
威尔科克森符号秩检验用于相关设计，当数据不是等距/等比或不符合正态分布时。对差值的绝对值排序，赋予符号，分别计算正秩和与负秩和。检验统计量T是较小的符号秩和。
Mann-Whitney U: U = n₁n₂ + n₁(n₁+1)/2 - R₁
8. Effect Size: Cohen's d | 效应量：科恩d
Cohen' difference between two means independent of sample size. For an independent t-test: d = (x̄₁ - x̄₂) / spooled, where spooled = √[((n₁-1)s₁² + (n₂-1)s₂²) / (n₁ + n₂ - 2)].
科恩d表示两个平均数之差的效应大小，不受样本量影响。对于独立t检验：d = (x̄₁ - x̄₂) / spooled，其中spooled = √[((n₁-1)s₁² + (n₂-1)s₂²) / (n₁ + n₂ - 2)]。
ect, 0.5 medium, and 0.8 large. Reporting effect size is required in the exam alongside significance tests to show practical importance.
解释：d = 0.2为小效应，0.5为中效应，0.8为大效应。考试中要求在报告显著性检验的同时报告效应量，以体现实际意义。
Cohen's d = (x̄₁ - x̄₂) / spooled
9. Probability and Significance Levels (p-values) | 概率与显著性水平
ignificance level α is 0.05. If p ≤ 0.05, we reject H₀ and conclude a statistically significant effect.
p值是在零假设为真的前提下，获得当前观测结果（或更极端结果）的概率。心理学中，常规显著性水平α为0.05。若p ≤ 0.05，拒绝H₀，认为存在统计上显著的效应。
A one-tailed test predicts direction; a two-tailed test does not. Critical values are more extreme for one-tailed tests. Always justify the choice. Probability can also be expressed in decimal, percentage, or fraction form in exam questions.
单尾检验预测方向；双尾检验不预测方向。单尾检验的临界值更极端。选择需说明理由。考试题目中概率可以小数、百分数或分数形式表示。
10. Inferential Statistics Decision Tree | 推断统计决策树
Choosing the correct statistical test is vital. Start with the research design: (1) Is it a test of difference or correlation? (2) What is the level of measurement? (3) Is the design independent groups or repeated measures/matched pairs?
选择正确的统计检验至关重要。从研究设计开始：(1) 是差异检验还是相关？(2) 测量水平是什么？(3) 设计是独立组还是重复测量/配对？
r assumptions are violated: independent groups → Mann-Whitney U; repeated measures → Wilcoxon signed-ranks. For correlation: interval/ratio normal → Pearson's r; ordinal/non-normal → Spearman's ρ. For nominal data association → Chi-square.
等距/等比数据的差异检验：独立组 → 独立t检验；重复测量 → 相关t检验。若顺序数据或假设违背：独立组 → 曼-惠特尼U；重复测量 → 威尔科克森符号秩检验。相关：等距/等比且正态 → 皮尔逊r；顺序/非正态 → 斯皮尔曼ρ。名义数据的关联 → 卡方检验。
- Difference, interval, independent → independent t-test
- Difference, interval, related → related t-test
- Difference, ordinal, independent → Mann-Whitney U
- Difference, ordinal, related → Wilcoxon
- Correlation, interval → Pearson's r
- Correlation, ordinal → Spearman's ρ
- Association, nominal → Chi-square
差异，等距，独立 → 独立t检验；差异，等距，相关 → 相关t检验；差异，顺序，独立 → 曼-惠特尼U；差异，顺序，相关 → 威尔科克森符号秩检验；相关，等距 → 皮尔逊r；相关，顺序 → 斯皮尔曼ρ；关联，名义 → 卡方。
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