📚 Formulas and Theorems Quick Reference Handbook for Year 13 OCR Psychology | Year 13 OCR 心理学：公式定理速查手册
This streamlined handbook brings together all the key statistical formulas, decision rules, and conceptual 'theorems' you need for the research methods component of OCR A-Level Psychology. From measures of central tendency to effect sizes, each entry is presented with its practical application and calculator-friendly notation, helping you move confidently from data to conclusion.
这本精简手册汇集了 OCR A-Level 心理学研究方法模块中所有关键统计公式、决策规则和概念性“定理”。从集中趋势测量到效应量，每个条目都配有实际应用说明和便于使用计算器的符号，帮助你自信地从数据走向结论。
1. Descriptive Statistics: Central Tendency | 描述统计：集中趋势
Measures of central tendency summarise the typical score in a dataset. The mean (x̄) is the arithmetic average, calculated as Σx/n. The median is the middle value when scores are ordered. The mode is the most frequent score. In psychology, the mean is used for interval/ratio data without extreme outliers, while the median is preferred for ordinal data or skewed distributions.
集中趋势测量概括了数据集中的典型分数。平均数 (x̄) 是算术平均值，计算公式为 Σx/n。中位数 是排序后中间的数值。众数 是出现频率最高的分数。在心理学中，平均数用于无极端异常值的等距/比率数据，而中位数则适用于顺序数据或偏态分布。
2. Measures of Dispersion & Standard Deviation | 离散测量与标准差
Dispersion tells us how spread out the scores are. The range is simply the highest score minus the lowest. The standard deviation (s) quantifies the average distance of each score from the mean. The formula for a sample is:
离散测量告诉我们分数的分散程度。全距 就是最高分减最低分。标准差 (s) 量化了每个分数与平均值的平均距离。样本标准差的公式为：
s = √[ Σ (x - x̄)² / (n - 1) ]
A larger standard deviation indicates greater variability in the data, which can affect the sensitivity of inferential tests.
标准差越大，说明数据的变异性越大，这会影响推论检验的灵敏度。
3. Normal Distribution & the 68–95–99.7 Theorem | 正态分布与 68–95–99.7 定理
Many psychological variables (e.g. IQ scores) are normally distributed. In a normal curve, the mean, median and mode are equal. The empirical rule states that approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2 SDs, and 99.7% within 3 SDs. This property underpins probability calculations for parametric tests.
许多心理学变量（如智商分数）呈正态分布。在正态曲线中，平均数、中位数和众数相等。经验法则表明，约 68% 的数据落在平均值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。这一性质是参数检验概率计算的基础。
4. Inferential Statistics: Key Concepts | 推论统计：关键概念
Inferential statistics allow psychologists to determine whether a result is likely due to a real effect or chance. The null hypothesis (H₀) states there is no effect or difference. The significance level (α) is typically set at 0.05. If the calculated p-value is less than α, we reject H₀. A Type I error is a false positive, while a Type II error is a false negative.
推论统计使心理学家能够判断结果是否可能由真实效应引起，还是出于偶然。零假设 (H₀) 声明没有效应或差异。显著性水平 (α) 通常设为 0.05。如果计算出的 p 值小于 α，我们就拒绝 H₀。I 型错误 是假阳性，II 型错误 是假阴性。
5. Choosing the Right Test: Decision Theorem | 选择正确的检验：决策定理
The choice of statistical test follows a logical chain. Ask three questions: (1) Am I looking for a difference or a correlation? (2) What is the experimental design – independent groups, repeated measures, or correlational? (3) What is the level of measurement – nominal, ordinal, or interval? The decision table below summarises the rules for OCR Psychology.
统计检验的选择遵循一条逻辑链。问三个问题：(1) 我要寻找的是差异还是相关？ (2) 实验设计是什么——独立组、重复测量还是相关设计？ (3) 数据的测量水平是什么——类别、顺序还是等距？下面的决策表总结了 OCR 心理学中的规则。
| Test of Difference / 差异检验 | Design / 设计 | Data Type / 数据类型 |
|---|---|---|
| Chi-square (χ²) | Independent groups | Nominal (frequencies) |
| Binomial Sign Test | Repeated measures | Nominal |
| Mann–Whitney U | Independent groups | Ordinal (or interval if not normal) |
| Wilcoxon Signed Ranks | Repeated measures | Ordinal / Interval |
| Spearman's Rho / Pearson's r | Correlation | Ordinal / Interval |
6. Chi-Square Test (χ²) | 卡方检验
Chi-square tests for an association between two categorical variables, or a difference in frequency distribution. The observed frequencies (O) are compared with expected frequencies (E). The formula is:
卡方检验用于检验两个分类变量之间的关联，或频数分布的差异。将观察频数 (O) 与期望频数 (E) 进行比较。公式如下：
χ² = Σ (O - E)² / E
he obtained χ² value is compared against a critical value from the chi-square table at the chosen α level.
自由度 (df) = (行数 - 1) × (列数 - 1)。将计算得到的 χ² 值与卡方分布表中选定 α 水平下的临界值进行比较。
7. Binomial Sign Test | 二项符号检验
The binomial sign test is a non-parametric test for repeated measures / matched pairs design with nominal data. It uses the number of pluses and minuses to assess the probability of a pattern occurring by chance. The observed value (S) is the count of the less frequent sign. No complex formula is required; you compare S to the critical number from binomial tables using N (number of pairs ignoring ties) and α.
二项符号检验是一种适用于重复测量/配对组设计、名义数据的非参数检验。它利用“+”和“-”的数量来评估特定模式偶然出现的概率。观察值 (S) 是出现频率较低的那个符号的次数。无需复杂公式；你只需根据 N（忽略相同评分的配对数量）和 α，将 S 与二项分布表中的临界值进行比较。
8. Wilcoxon Signed Ranks Test | 威尔科克森符号秩检验
This test is used for a repeated measures design with at least ordinal data. It considers both the direction and magnitude of differences between paired scores. After calculating the difference for each pair, rank the absolute differences ignoring zeroes, then sum the ranks of positive (T₊) and negative (T₋) differences. The test statistic T is the smaller sum of ranks. Compare T to the critical value table; the result is significant if T is less than or equal to the critical value.
该检验用于至少为顺序数据的重复测量设计。它同时考虑配对分数之间差异的方向和大小。计算每对分数的差值后，忽略零值并对差值的绝对值进行排秩，然后将正差值秩和 (T₊) 与负差值秩和 (T₋) 相加。检验统计量 T 是较小的秩和。将 T 与临界值表比较；若 T 小于或等于临界值，则结果显著。
9. Mann–Whitney U Test | 曼-惠特尼 U 检验
Mann–Whitney U is the independent groups counterpart to Wilcoxon, suited to ordinal data or interval data that violate parametric assumptions. First, rank all the scores from both groups together. Second, sum the ranks for group 1 (R₁) and group 2 (R₂). Calculate U for each group:
曼-惠特尼 U 检验是独立组设计对应于威尔科克森的检验，适用于顺序数据或违反参数假设的等距数据。首先，将两组所有分数混合排秩。其次，分别计算组1的秩和 (R₁) 和组2的秩和 (R₂)。计算两组的 U 值：
U₁ = n₁n₂ + n₁(n₁+1)/2 - R₁
U₂ = n₁n₂ + n₂(n₂+1)/2 - R₂
The observed U is the smaller of U₁ and U₂. Compare this with the Mann–Whitney critical value table; significance is found if U is less than or equal to the critical value.
观察值 U 取 U₁ 和 U₂ 中较小者。将其与曼-惠特尼临界值表进行比较；当 U 小于或等于临界值时，结果显著。
10. Spearman’s Rank Correlation (ρ) | 斯皮尔曼等级相关
Spearman's rho assesses the strength and direction of a monotonic association between two ordinal variables. Rank each variable separately, then find the difference (d) between the ranks for each participant. The formula is:
斯皮尔曼等级相关系数评估两个顺序变量之间单调关联的强度和方向。分别对每个变量排秩，然后求出每位参与者秩次的差值 (d)。公式为：
ρ = 1 - (6 Σ d²) / (n(n² - 1))
The obtained ρ value ranges from -1 to +1. Compare it to the Spearman critical value table. The correlation is significant if the absolute ρ is greater than or equal to the critical value.
计算出的 ρ 值介于 -1 到 +1 之间。将其与斯皮尔曼临界值表比较。若 ρ 的绝对值大于或等于临界值，则相关显著。
11. Pearson’s Product–Moment Correlation (r) | 皮尔逊积差相关
Pearson' used, but the conceptual formula highlights covariance over the product of standard deviations: r = Σ[(x - x̄)(y - ȳ)] / √[Σ(x - x̄)² Σ(y - ȳ)²]. The degrees of freedom are n - 2. Values near +1 or -1 indicate a strong correlation.
皮尔逊积差相关是用于两个正态分布的等距/比率变量之间线性相关的参数检验。几乎总是使用计算器来完成计算，但其概念公式强调了协方差除以标准差乘积：r = Σ[(x - x̄)(y - ȳ)] / √[Σ(x - x̄)² Σ(y - ȳ)²]。自由度为 n - 2。接近 +1 或 -1 的值表示强相关。
12. Effect Size & Statistical Power | 效应量与统计功效
While OCR does not require calculation of effect size formulas, you must understand the concept. Cohen's d estimates the size of a difference in standard deviation units: d = (x̄₁ - x̄₂) / spooled. Effect size adds meaning to significance: a p-value tells us whether an effect exists, while effect size tells us how large that effect is. Power is the probability of correctly rejecting a false null hypothesis and is influenced by sample size, α level, and effect size.
虽然 OCR 不要求计算效应量公式，但你必须理解这个概念。科恩 d 值 以标准差为单位估计差异大小：d = (x̄₁ - x̄₂) / spooled。效应量为显著性增添意义：p 值告诉我们效应是否存在，而效应量则告诉我们这个效应有多大。统计功效 是正确拒绝错误零假设的概率，受样本量、α 水平和效应量的影响。
Published by TutorHao | Psychology Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图