📚 Year 12 CIE Psychology: Quick Reference for Formulas & Theorems | Year 12 CIE 心理学：公式定理速查手册
Psychology may not appear to be a formula-heavy subject, but the AS Level research methods component demands confident handling of statistical tools. This quick reference handbook distills the essential formulas, inferential tests, and fundamental theorems you need for CIE Year 12 Psychology. From descriptive statistics to choosing between parametric and non-parametric tests, every entry is presented with clear bilingual explanations to reinforce your understanding and exam readiness.
心理学或许看起来不像一门充满公式的学科，但 AS 阶段的研究方法要求你熟练掌握统计工具。本速查手册提炼了 CIE Year 12 心理学所需的核心公式、推论检验和基本定理。从描述统计到参数与非参数检验的选择，每个条目都配有清晰的中英双语解释，帮助你巩固理解、备战考试。
1. Descriptive Statistics and Central Tendency | 描述统计与集中趋势
Descriptive statistics summarise data sets using measures of central tendency and dispersion. The mean (M) is calculated by dividing the sum of all scores by the number of observations: Mean = Σx / n. The median is the middle value in an ordered list; the mode is the most frequent score. The range, Range = x_max – x_min, gives a basic spread of values.
描述统计通过集中趋势和离散程度的度量来汇总数据。平均值（M）为所有得分之和除以观测数：平均值 = Σx / n。中位数是排序列表中的中间值；众数是出现次数最多的分数。极差公式为 极差 = 最大值 – 最小值，它提供了最基本的离散信息。
These measures help psychologists understand the typical value and variability in their data before running inferential tests.
这些度量能帮助心理学工作者在进行推论检验之前了解数据的典型值和变异性。
2. Normal Distribution and Skewness | 正态分布与偏态
A normal distribution is a symmetrical, bell-shaped curve where the mean, median, and mode coincide. The empirical rule states that approximately 68% of data falls within ±1 standard deviation, 95% within ±2 SD, and 99.7% within ±3 SD. This theorem underpins many parametric tests.
正态分布是一种对称的钟形曲线，其均值、中位数和众数重合。经验法则指出，约 68% 的数据落在 ±1 个标准差内，95% 落在 ±2 SD 内，99.7% 落在 ±3 SD 内。这一特性是许多参数检验的基础。
Skewness describes asymmetry: a positively skewed distribution has a long right tail (mean > median), while a negatively skewed distribution has a long left tail (mean < median). Skewness affects the choice of statistical test.
偏度描述不对称性：正偏态分布的右尾较长（均值 > 中位数），负偏态分布左尾较长（均值 < 中位数）。偏度会影响统计检验的选择。
3. Standard Deviation and Variance Formula | 标准差与方差公式
Standard deviation quantifies the average distance of scores from the mean. The sample standard deviation is given by:
标准差衡量各数据点与平均值之间的平均距离。样本标准差的计算公式为：
s = √[ Σ(x – x̄)² / (n – 1) ]
Where x is each raw score, x̄ is the sample mean, and n is the sample size. The denominator (n – 1) corrects bias in estimating the population parameter. Variance is simply s².
其中 x 为各原始分数，x̄ 为样本均值，n 为样本量。分母 (n – 1) 用于修正估计总体参数时的偏差。方差即为 s²。
4. Inferential Statistics, Null and Alternative Hypotheses | 推论统计、零假设与备择假设
Inferential statistics allow researchers to draw conclusions about populations from sample data. The null hypothesis (H₀) posits no effect or no difference, e.g., ‘There is no difference in recall between conditions.’ The alternative hypothesis (H₁) states the predicted effect, which can be directional (one-tailed) or non-directional (two-tailed).
推论统计使研究者能够根据样本数据推断总体特征。零假设（H₀）假定没有效应或差异，如“不同条件下的记忆成绩无差异”。备择假设（H₁）陈述预测的效应，可分为定向（单尾）或非定向（双尾）。
The P-value tells you the probability of obtaining the observed result if H₀ were true. If p is less than the chosen significance level, H₀ is rejected.
P 值表示在 H₀ 为真时获得当前结果的概率。若 p 小于选定的显著性水平，则拒绝 H₀。
5. Significance Levels, p-Values and Errors | 显著性水平、p 值与两类错误
The significance level (α) is the threshold for rejecting H₀, typically set at 0.05 (5%). A result is statistically significant when p < 0.05. A Type I error (false positive) occurs when H₀ is wrongly rejected; the risk equals α. A Type II error (false negative) occurs when a real effect is missed (β).
显著性水平（α）是拒绝 H₀ 的门槛，通常设为 0.05（5%）。当 p < 0.05 时，结果具有统计显著性。第一类错误（假阳性）发生在错误地拒绝了真实的 H₀；其风险等于 α。第二类错误（假阴性）发生在漏掉了真实效应（β）。
To minimise errors, psychologists use appropriate sample sizes and carefully select tests. Power (1 – β) is the ability to detect a genuine effect.
为减少错误，心理学家会采用适当的样本量并谨慎选择检验方法。检验功效（1 – β）指检测出真实效应的能力。
6. Choosing the Right Statistical Test | 选择合适的统计检验
Three questions guide test selection: (1) Is the research looking for a difference or a correlation? (2) What is the level of measurement – nominal, ordinal, or interval/ratio? (3) Is the design independent groups, repeated measures, or matched pairs? Parametric tests (e.g., t-test) require interval data and normally distributed scores; non-parametric tests (e.g., Mann-Whitney) are used when assumptions are violated or data are ordinal.
选择检验方法需依据三个问题：（1）研究是寻找差异还是相关？（2）测量水平是称名、顺序还是等距/等比？（3）实验设计是独立组、重复测量还是配对组？参数检验（如 t 检验）要求等距数据和正态分布；当假设违反或数据为顺序变量时，则采用非参数检验（如曼-惠特尼 U 检验）。
7. Chi-Squared Test (χ²) | 卡方检验
The chi-squared test analyses nominal (categorical) data in the form of frequencies. It compares observed frequencies (O) with expected frequencies (E) under the null hypothesis:
卡方检验用于分析称名（分类）数据的频数。它比较观察频数（O）与零假设下的期望频数（E）：
χ² = Σ[ (O – E)² / E ]
Degrees of freedom (df) for a contingency table: df = (rows – 1) × (columns – 1). The calculated χ² is compared with a critical value from the chi-squared table at a chosen α and df.
列联表的自由度：df = (行数 – 1) × (列数 – 1)。将算得的 χ² 值与选定 α 及自由度下的临界值比较。
A significant χ² indicates an association between variables, e.g., gender and voting preference. CIE requires you to state whether the result is significant using critical value tables.
显著的 χ² 表明变量间存在关联，如性别与投票偏好。CIE 要求你依据临界值表判断结果是否显著。
8. Independent and Related t-Tests | 独立样本与相关样本 t 检验
The independent samples t-test compares two separate groups when the dependent variable is interval/ratio and data are normally distributed. The formula is:
独立样本 t 检验比较两个独立组别，因变量为等距/等比且数据正态分布。其公式为：
t = (x̄₁ – x̄₂) / √[ s₁²/n₁ + s₂²/n₂ ]
For repeated measures or matched pairs designs, the related t-test uses difference scores (d = each pair’s difference) and tests whether the mean difference (d̄) deviates from zero:
对于重复测量或配对组设计，相关样本 t 检验采用差值分数（d = 每对数据的差），并检验平均差（d̄）是否偏离零：
t = d̄ / (sd / √n)
where sd is the standard deviation of the difference scores. Compare the obtained t with the critical t at the given df and α.
其中 sd 为差值分数的标准差。将算得的 t 值与给定自由度和 α 下的临界 t 值比较。
9. Spearman’s Rank Correlation Coefficient | 斯皮尔曼等级相关
Spearman’s rho (rₛ or ρ) measures the strength and direction of association between two ordinal variables or when data violate parametric assumptions. The formula is:
斯皮尔曼等级相关系数（rₛ 或 ρ）测量两个顺序变量之间的关联强度与方向，适用于不满足参数假设的情形。其公式为：
rₛ = 1 – (6 Σ d²) / (n(n² – 1))
where d is the difference between the ranks of each pair, and n is the number of pairs. rₛ ranges from -1 to +1. A value close to +1 indicates a strong positive correlation, whereas -1 indicates a strong negative correlation. A zero value suggests no monotonic relationship.
其中 d 为每对数据的等级差，n 为配对数目。rₛ 的取值范围为 -1 到 +1。接近 +1 表示强正相关，接近 -1 表示强负相关，零则提示无单调关系。
Compare rₛ with the critical value in the Spearman table; if rₛ exceeds the critical value, the correlation is significant.
将 rₛ 与斯皮尔曼临界值表比较；若 rₛ 大于临界值，则相关显著。
10. Mann-Whitney U and Wilcoxon Signed-Rank Tests | 曼-惠特尼 U 检验与威尔科克森符号秩检验
Mann-Whitney U is the non-parametric counterpart of the independent t-test for ordinal data or when normality is violated. It tests whether two independent groups come from the same population. The test statistic is:
曼-惠特尼 U 检验是独立样本 t 检验的非参数替代，适用于顺序数据或违反正态性的情形。它检验两个独立组是否来自同一总体。检验统计量为：
U = n₁n₂ + n₁(n₁+1)/2 – R₁
where R₁ is the sum of ranks for group 1. The smaller of U and U’ (U’ = n₁n₂ – U) is compared with the critical U.
其中 R₁ 为第 1 组的秩和。将 U 和 U’（U’ = n₁n₂ – U）中的较小值与
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