📚 A-Level OCR Psychology: Formula and Theorem Quick Reference | A-Level OCR 心理学：公式定理速查手册
This concise handbook gathers the essential statistical formulas, decision rules, and research-method principles required for the OCR A-Level Psychology specification. It is designed as a last-minute revision aid, a pocket reference, and a structured guide to the calculations that underpin psychological investigations. From descriptive statistics to non-parametric tests, every key formula is presented with a clear explanation of its use, assumptions, and real-world application in psychological research.
这本速查手册汇集了 OCR A-Level 心理学课程所需的必备统计公式、决策规则和研究方法原理。它既适合考前冲刺复习，也可作为随时查阅的口袋指南，系统梳理支撑心理学研究的所有计算要点。从描述统计到非参数检验，每一关键公式都配有明确的用途说明、前提假设以及在心理学真实研究中的应用解读。
1. Measures of Central Tendency | 集中量数
The mean (x̄) is the arithmetic average of a set of scores, calculated by summing all values and dividing by the number of observations: x̄ = Σx / n. It is sensitive to extreme scores and is most appropriate for interval or ratio data that are normally distributed.
均值（x̄）是一组分数的算术平均值，由所有数值之和除以观测个数得出：x̄ = Σx / n。它对极端分数非常敏感，最适合用于呈正态分布的等距或等比数据。
The median is the middle score when data are arranged in rank order. If n is even, it is the midpoint of the two central values. The median is unaffected by outliers and is preferred for skewed distributions or ordinal data.
中位数是将数据按大小排序后位于正中间的数值。当 n 为偶数时，取中间两个数的平均值。中位数不受极端值影响，适用于偏态分布或顺序数据。
The mode is the most frequently occurring value in a data set. A data set can have one mode (unimodal), two modes (bimodal), or more. The mode is the only measure of central tendency suitable for nominal data.
众数是数据集中出现频率最高的数值。一个数据集可以有一个众数（单峰）、两个众数（双峰）或更多。众数是唯一适用于称名数据的集中量数。
2. Measures of Dispersion | 离势量数
The range is the simplest measure of spread: range = highest score − lowest score. It gives a quick sense of variability but is heavily influenced by a single extreme value.
全距是最简单的离散程度度量：全距 = 最高分 − 最低分。它能快速反映变异程度，但极易受个别极端值的影响。
The variance (s²) quantifies the average squared deviation of each score from the mean. For a sample, s² = Σ(x − x̄)² / (n − 1). The subtraction of 1 (Bessel’s correction) provides an unbiased estimate of the population variance.
方差（s²）衡量每个分数与均值偏差的平方的平均水平。对于样本，s² = Σ(x − x̄)² / (n − 1)。分母减去 1（贝塞尔校正）是为了得到总体方差的无偏估计。
The standard deviation (s) is the square root of the variance: s = √[Σ(x − x̄)² / (n − 1)]. It returns the measure of spread to the original units of measurement and is widely reported alongside the mean in psychological research.
标准差（s）是方差的平方根：s = √[Σ(x − x̄)² / (n − 1)]。它将离散程度还原至数据的原始测量单位，通常在心理学研究报告中与均值一并呈现。
3. Correlation: Pearson’s r | 相关：皮尔逊积差相关
Pearson’s product-moment correlation coefficient (r) measures the strength and direction of a linear relationship between two interval-level variables. It is calculated as the covariance of x and y divided by the product of their standard deviations: r = Σ(x − x̄)(y − ȳ) / √[Σ(x − x̄)² Σ(y − ȳ)²].
皮尔逊积差相关系数（r）衡量两个等距变量之间线性关系的强度与方向。其计算方式为 x 与 y 的协方差除以两者标准差的乘积：r = Σ(x − x̄)(y − ȳ) / √[Σ(x − x̄)² Σ(y − ȳ)²]。
The value of r ranges from −1 to +1. A coefficient close to +1 indicates a strong positive correlation, while a value near −1 indicates a strong negative correlation. A value around 0 suggests no linear relationship. Pearson’s r requires normally distributed variables with homoscedasticity and no significant outliers.
r 的取值范围在 −1 到 +1 之间。系数接近 +1 表示强正相关，接近 −1 表示强负相关，接近 0 则表明不存在线性关系。皮尔逊 r 要求变量服从正态分布、具有方差齐性且无明显异常值。
4. Correlation: Spearman’s Rho | 相关：斯皮尔曼等级相关
Spearman’s rank-order correlation coefficient (ρ or rs) is a non-parametric alternative to Pearson’s r. It evaluates the monotonic relationship between two variables based on ranked data. When there are no tied ranks, it is computed as: rs = 1 − [6 Σd² / n(n² − 1)], where d is the difference between the ranks of each pair and n is the number of paired scores.
斯皮尔曼等级相关系数（ρ 或 rs）是皮尔逊 r 的非参数替代方法。它基于排序数据评估两个变量之间的单调关系。在没有相同等级的情况下，计算公式为：rs = 1 − [6 Σd² / n(n² − 1)]，其中 d 是每一对数据的等级差，n 是配对数。
Spearman’s rho is used when data are ordinal, or when the assumptions of Pearson’s r (normality, linearity) are violated. A correction factor is applied when tied ranks occur, but the logic of interpreting the coefficient remains the same: −1 to +1, with zero signifying no monotonic association.
当数据为顺序数据，或皮尔逊 r 的假设（正态性、线性）不满足时，使用斯皮尔曼 rho。出现相同等级时需要引入校正因子，但系数的解读逻辑保持不变：取值范围 −1 到 +1，0 表示无单调关联。
5. Inferential Statistics: Null Hypothesis and Significance | 推断统计：零假设与显著性
Inferential statistics allow researchers to determine whether observed differences or relationships are likely to be genuine or due to chance. The null hypothesis (H₀) states that there is no effect or no difference in the population, while the alternative hypothesis (H₁ or Ha) suggests a real effect exists. A directional hypothesis predicts the direction of the effect (one-tailed test), whereas a non-directional hypothesis does not (two-tailed test).
推断统计帮助研究者判断所观察到的差异或关系更可能是真实的还是由偶然因素所致。零假设（H₀）声称总体中不存在效应或差异，而备择假设（H₁ 或 Ha）则认为存在真实的效应。方向性假设会预测效应的方向（单尾检验），非方向性假设则不预测方向（双尾检验）。
The significance level (α) is the probability threshold for rejecting the null hypothesis, typically set at 0.05 (5%). The p-value is the probability of obtaining the observed result, or a more extreme one, if H₀ were true. If p ≤ α, the result is statistically significant. Also crucial are the degrees of freedom (df), which influence the critical value against which the computed statistic is compared.
显著性水平（α）是拒绝零假设的概率阈值，通常设为 0.05（5%）。p 值是指当 H₀ 为真时，获得当前结果或更极端结果的概率。若 p ≤ α，则结果具有统计学显著性。此外，自由度（df）也至关重要，它影响用于与计算值进行比较的临界值。
6. The Sign Test | 符号检验
The sign test is the simplest non-parametric test for a difference in related data. It uses nominal-level data (positive and negative signs) to decide whether there is a significant difference between two conditions. The test statistic (S) is the smaller of the number of plus signs or minus signs. The sign test requires pairs of related observations, and any pairs showing no change (ties) are excluded from the analysis.
符号检验是最简单的非参数检验，用于检验相关数据的差异。它利用称名层次的数据（正号与负号）来判断两种条件之间是否存在显著差异。检验统计量（S）是正号个数与负号个数中的较小者。符号检验要求配对的关联观测值，任何未表现出变化的对（相同结果）需从分析中剔除。
To determine significance, compare the observed S to a critical value from a binomial distribution table for the given n (number of non-tied pairs) and chosen α. If S is less than or equal to the critical value, the null hypothesis is rejected. The sign test is most appropriate when the data are ordinal but lack the precision required for the Wilcoxon test.
为判决显著性，需将观测到的 S 值与二项分布表中对应 n（非相同对的数量）及所选 α 水平的临界值进行比较。若 S 小于或等于临界值，则拒绝零假设。当数据为顺序数据但缺乏威尔科克森检验所需的精度时，符号检验最为适用。
7. Wilcoxon Signed-Ranks Test | 威尔科克森符号秩检验
The Wilcoxon signed-ranks test is a non-parametric test for a difference between two related conditions, using ranked differences. For each pair, calculate the difference score, rank the absolute differences, and then sum the ranks of the positive differences (T+) and negative differences (T−). The test statistic T is the smaller of these two sums. Ties and differences of zero receive special handling.
威尔科克森符号秩检验是一种非参数检验，用于检验两个相关条件之间的差异，基于差值的排序。对每一对数据计算差值，对差值的绝对值排序，再分别求出正差值的秩和（T+）与负差值的秩和（T−）。检验统计量 T 是这两个秩和中的较小值。相同差值和零差值需特殊处理。
The computed T is compared with a critical T value from the Wilcoxon table. Reject H₀ if T is less than or equal to the critical value. The test assumes data are at least ordinal and that the pairs are randomly drawn. When the sample size is large (n > 25), a normal approximation can be used.
将计算出的 T 值与威尔科克森表中的临界 T 值比较。若 T 小于或等于临界值，则拒绝 H₀。该检验要求数据至少为顺序数据，且配对样本随机选取。当样本较大（n > 25）时，可使用正态近似。
8. Mann-Whitney U Test | 曼-惠特尼 U 检验
The Mann-Whitney U test compares two independent groups when the dependent variable is at least ordinal. All scores from both groups are combined and ranked. The U statistic for each group is: U₁ = n₁n₂ + [n₁(n₁+1) / 2] − R₁, where R₁ is the sum of ranks for group 1. The smaller U is used as the test statistic. Tied ranks are handled by assigning the average rank.
曼-惠特尼 U 检验用于比较两个独立组，要求因变量至少是顺序数据。将两组所有分数合并并排序。每组 U 统计量为：U₁ = n₁n₂ + [n₁(n₁+1) / 2] − R₁，其中 R₁ 是第一组的秩和。取较小的 U 值作为检验统计量。相同等级按平均秩处理。
The observed U is compared with critical values from the Mann-Whitney table. If U is less than or equal to the critical value, the null hypothesis of no difference is rejected. When groups are large (n₁, n₂ > 20), a z-score approximation can be calculated.
将观测到的 U 值与曼-惠特尼表中的临界值比较。若 U 小于或等于临界值，则拒绝无差异的零假设。当两组样本量都较大（n₁, n₂ > 20）时，可计算 z 分数的近似值。
9. Chi-Square Test | 卡方检验
The chi-square test evaluates the association between two categorical variables (test of independence) or tests how well an observed distribution fits an expected distribution (goodness-of-fit). The test statistic is χ² = Σ (O − E)² / E, where O is the observed frequency and E is the expected frequency under the null hypothesis. Expected frequencies are calculated as (row total × column total) / grand total for the test of independence.
卡方检验用于评估两个分类变量之间的关联性（独立性检验），或检验观测分布与期望分布的拟合度（拟合优度检验）。检验统计量为 χ² = Σ (O − E)² / E，其中 O 是观测频数，E 是零假设下的期望频数。独立性检验中，期望频数按（行总计 × 列总计）/ 总计计算。
The degrees of freedom for the test of independence are (number of rows − 1) × (number of columns − 1). The computed χ² is compared with a critical value from the chi-square distribution. If χ² exceeds the critical value, H₀ is rejected. Chi-square requires nominal data, independent observations, and expected frequencies of at least 5 in each cell; otherwise, Fisher’s exact test or a correction may be necessary.
独立性检验的自由度为（行数 − 1）×（列数 − 1）。计算出的 χ² 值与卡方分布表中的临界值比较。若 χ² 大于临界值，则拒绝 H₀。卡方检验要求称名数据、独立的观测值且每个单元格的期望频数至少为 5；否则可能需要使用费希尔精确检验或进行校正。
10. Choosing a Statistical Test | 统计检验的选择
Selecting the appropriate inferential test depends on three questions: (i) Is the investigation testing for a difference or a relationship? (ii) What is the research design—independent groups, repeated measures, or matched pairs? (iii) What is the level of measurement—nominal, ordinal, or interval? The table below summarises the standard decision flow used in OCR Psychology.
选择合适的推断检验取决于三个问题：(i) 研究考察的是差异还是关系？(ii) 研究设计是独立组、重复测量还是配对组？(iii) 测量层次是称名、顺序还是等距？下表总结了 OCR 心理学中使用的标准决策流程。
| Research Aim | Design | Data Level | Test |
|---|---|---|---|
| Difference | Repeated measures / Matched pairs | Nominal | Sign test |
| Difference | Repeated measures / Matched pairs | Ordinal | Wilcoxon signed-ranks test |
| Difference | Independent groups | Ordinal | Mann-Whitney U test |
| Difference | Independent groups | Interval (normal) | Unrelated t-test* |
| Difference | Repeated measures | Interval (normal) | Related t-test* |
| Relationship | Correlation |
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