Year 13 Edexcel Psychology: Formula & Theorem Quick Reference Handbook | Year 13 Edexcel 心理学：公式定理速查手册

📚 Year 13 Edexcel Psychology: Formula & Theorem Quick Reference Handbook | Year 13 Edexcel 心理学：公式定理速查手册

This quick reference handbook brings together every statistical formula and key theorem you will need for the research methods and data analysis components of Year 13 Edexcel Psychology. Bookmark this page so you can quickly check a formula, refresh a concept or verify the steps for an inferential test before your exam.

本速查手册汇集了 Year 13 Edexcel 心理学研究方法与数据分析部分所需的每一项统计公式和关键定理。收藏此页面，你便可在考试前快速查阅公式、重温概念或核对推断检验的步骤。

1. Descriptive Statistics: Central Tendency and Dispersion | 描述统计：集中趋势与离差

Descriptive statistics summarise large data sets. The three measures of central tendency describe a typical score, while measures of dispersion tell us how spread out the scores are.

描述统计用于概括大型数据集。三种集中趋势量数描述典型分数，而离差量数则告诉我们分数的分散程度。

Mean (x̄): sum of all scores divided by the number of scores. It uses every value but is sensitive to outliers.
平均值 (x̄)：所有分数之和除以分数的个数。它使用了每一个数值，但对异常值敏感。

Median: the middle score when data are ordered. It is unaffected by extreme scores and best for skewed distributions.
中位数：数据排序后位于中间位置的分数。它不受极端分数影响，最适合偏态分布。

Mode: the most frequently occurring score. Useful for nominal data and can be used with any level of measurement.
众数：出现频率最高的分数。适用于名义数据，可用于任何测量水平。

Range: the difference between the highest and lowest score. A crude measure of dispersion; heavily influenced by outliers.
全距：最高分与最低分之间的差值。一种粗略的离差量数，极易受异常值影响。

2. Standard Deviation and Variance Formulas | 标准差与方差公式

Variance and standard deviation are more precise measures of dispersion. They indicate how much scores typically deviate from the mean. In Edexcel Psychology, you may be asked to calculate standard deviation using the sample formula.

方差与标准差是更精确的离差量数。它们表示分数通常偏离平均值的程度。在 Edexcel 心理学中，你可能需要使用样本公式计算标准差。

Sample Variance: s² = Σ(x – x̄)² / (n – 1)

样本方差：s² = Σ(x – x̄)² / (n – 1)

Sample Standard Deviation: s = √[ Σ(x – x̄)² / (n – 1) ]

样本标准差：s = √[ Σ(x – x̄)² / (n – 1) ]

Where Σ means ‘sum of’, x represents each individual score, x̄ is the sample mean, and n is the number of scores. Subtracting 1 from n (Bessel’s correction) gives an unbiased estimate of the population parameter.

其中 Σ 表示“求和”，x 代表每个个体的分数，x̄ 是样本平均值，n 是分数的个数。用 n 减去 1（贝塞尔校正）可得到对总体参数的无偏估计。

3. Normal Distribution and Probability | 正态分布与概率

The normal distribution is a bell-shaped, symmetrical curve where the mean, median and mode all coincide. Many psychological variables are assumed to be normally distributed in the population.

正态分布是一种钟形对称曲线，其平均值、中位数和众数均重合。许多心理学变量被假设在总体中呈正态分布。

The 68-95-99.7 rule is a theorem describing the proportion of data within standard deviations of the mean: approximately 68% of scores fall within ±1 SD, 95% within ±2 SD, and 99.7% within ±3 SD. This rule underpins the logic of significance testing.

68-95-99.7 规则是一条描述数据落在平均值几个标准差内比例的原理：约 68% 的分数落在 ±1 SD 内，95% 落在 ±2 SD 内，99.7% 落在 ±3 SD 内。这条规则是显著性检验逻辑的基础。

Probability (p) expresses the likelihood that a result is due to chance. In psychology, the conventional significance level is p < 0.05, meaning a less than 5% probability that the observed effect occurred by chance alone.

概率 (p) 表示结果由偶然因素导致的可能性。在心理学中，常规显著性水平为 p < 0.05，即观察到的效应单纯由随机因素造成的概率小于 5%。

4. Inferential Testing: The Logic of Hypothesis Testing | 推断检验：假设检验的逻辑

Inferential tests allow psychologists to decide whether to accept or reject a null hypothesis. The null hypothesis (H₀) states there is no effect or no difference; the alternative hypothesis (H₁) states there is an effect or difference.

推断检验使心理学家能够决定接受还是拒绝零假设。零假设 (H₀) 指没有效应或没有差异；备择假设 (H₁) 则指有效应或有差异。

A test statistic (e.g., U, T, rₛ, χ²) is calculated and compared with a critical value from a statistical table. If the observed value is equal to or more extreme than the critical value, the result is significant and H₀ is rejected.

计算出一个检验统计量（例如 U, T, rₛ, χ²），并将其与统计表中的临界值进行比较。若观测值等于或比临界值更极端，则结果显著，拒绝 H₀。

Two types of error can occur: Type I error (false positive) – rejecting H₀ when it is true; Type II error (false negative) – failing to reject H₀ when it is false. Choosing p < 0.05 balances these risks.

可能发生两类错误：第 I 类错误（假阳性）—— H₀ 为真却被拒绝；第 II 类错误（假阴性）—— H₀ 为假却未被拒绝。选择 p < 0.05 可平衡这两种风险。

5. Sign Test: Formula and Steps | 符号检验：公式与步骤

The Sign Test is a non-parametric test for a repeated measures or matched pairs design, used when data are at least nominal (difference in direction). It simply counts the number of positive and negative signs.

符号检验是一种适用于重复测量或配对设计的非参数检验，当数据至少为名义水平（方向差异）时使用。它只计算正号和负号的个数。

Step 1: Calculate the sign of the difference for each pair (e.g., condition B minus condition A). Ignore ties (no difference). Count the number of ‘+’ signs and ‘−’ signs. The test statistic S is the smaller of these two counts.

步骤 1：计算每对数据的差异符号（例如条件 B 减条件 A）。忽略无差异的对。统计 ‘+’ 号和 ‘−’ 号的数量。检验统计量 S 取这两者中较小的那个。

S = smaller of (number of pluses, number of minuses)

S = (正号个数，负号个数) 中较小的值

Step 2: Compare S with the critical value from a Sign Test table, using the number of paired scores (excluding ties, N) and your chosen significance level (one-tailed or two-tailed). If S ≤ critical value, the result is significant.

步骤 2：将 S 与临界值比较，使用符号检验表，根据配对分数数量（排除无差异项，N）和所选的显著性水平（单尾或双尾）。若 S ≤ 临界值，结果显著。

6. Wilcoxon Signed-Rank Test | 威尔科克森符号秩检验

The Wilcoxon signed-rank test is a non-parametric alternative for a repeated measures or matched pairs design when the data are at least ordinal (difference scores can be ranked). It uses both the direction and magnitude of differences.

威尔科克森符号秩检验是用于重复测量或配对设计的非参数替代方法，适用于至少为顺序水平的数据（差值可以排序）。它同时利用差异的方向和大小。

Steps: Calculate the difference for each pair. Rank the absolute differences from 1 (smallest) to N (largest), ignoring zero differences. Attach the original sign (+ or −) to each rank to get signed ranks. Sum the positive signed ranks (T⁺) and the negative signed ranks (T⁻). The test statistic T is the smaller of T⁺ and T⁻.

步骤：计算每对数据的差值。对差值的绝对值按从 1（最小）到 N（最大）排秩，忽略差值为零的对。给每个秩附上原符号（+ 或 −），得到符号秩。分别计算正符号秩之和 (T⁺) 与负符号秩之和 (T⁻)。检验统计量 T 取 T⁺ 与 T⁻ 中较小者。

T = smaller of (Σ positive signed ranks, Σ negative signed ranks)

T = (正符号秩之和，负符号秩之和) 中的较小值

Compare T with the critical Wilcoxon T value using N (number of non-zero differences). For significance, the observed T must be equal to or less than the critical value.

将 T 与 Wilcoxon T 临界值进行比较，使用 N（非零差值的数量）。要达到显著，观测的 T 必须等于或小于临界值。

7. Mann-Whitney U Test | 曼-惠特尼 U 检验

The Mann-Whitney U test is used for an independent groups design when data are at least ordinal. It tests whether two independent samples come from the same population by comparing their ranks.

曼-惠特尼 U 检验适用于独立组设计，数据至少为顺序水平。它通过比较秩来检验两个独立样本是否来自同一总体。

Calculating U: Pool all scores from both groups and rank them from 1 (smallest) to N (largest). Sum the ranks for each group (R₁ and R₂). Use the formulas:

计算 U：将两组所有分数合并后从 1（最小）到 N（最大）排秩。分别求出每组秩和（R₁ 和 R₂）。使用公式：

U₁ = n₁n₂ + ½ n₁(n₁ + 1) – R₁

U₂ = n₁n₂ – U₁ (or U₂ = n₁n₂ + ½ n₂(n₂ + 1) – R₂)

The test statistic U is the smaller of U₁ and U₂. Compare U with the critical Mann-Whitney U value using n₁ and n₂. For significance, the observed U must be equal to or less than the critical value.

检验统计量 U 取 U₁ 和 U₂ 中较小者。将 U 与曼-惠特尼 U 临界值比较，使用 n₁ 和 n₂。要达到显著，观测的 U 必须等于或小于临界值。

8. Spearman’s Rank Correlation Coefficient | 斯皮尔曼秩相关系数

Spearman’s rho (rₛ) is a non-parametric test of correlation between two co-variables, both measured on at least ordinal scales. It assesses the strength and direction of a monotonic relationship.

斯皮尔曼 rho (rₛ) 是两列共变量之间相关性的非参数检验，两者均至少为顺序水平。它评估单调关系的强度和方向。

Formula:

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. Rank each variable separately, compute d for each participant, square each d, sum them, then substitute into the formula.

其中 d 是每对数据的秩差值，n 是对子的数量。分别对每个变量排秩，计算每位参与者的 d，对每个 d 取平方，求和，然后代入公式。

rₛ ranges from +1 (perfect positive relationship) to -1 (perfect negative relationship). Compare the absolute value of rₛ with the critical Spearman rho value. If |rₛ| ≥ critical value, the correlation is significant.

rₛ 的取值范围从 +1（完全正相关）到 -1（完全负相关）。将 rₛ 的绝对值与斯皮尔曼 rho 临界值比较。若 |rₛ| ≥ 临界值，则相关显著。

9. Chi-Squared Test of Association | 卡方关联性检验

The chi-squared (χ²) test examines whether there is a significant association between two categorical variables (nominal data) in an independent groups design. It compares observed frequencies with expected frequencies.

卡方 (χ²) 检验考察两个类别变量（名义数据）在独立组设计中是否存在显著关联。它比较观测频数与期望频数。

χ² = Σ [ (O – E)² / E ]

O is the observed frequency in each cell; E is the expected frequency under the null hypothesis. Expected frequency for a cell is calculated as:

O 是每个单元格的观测频数；E 是在零假设下的期望频数。每个单元格的期望频数计算如下：

E = (row total × column total) / grand total

The calculated χ² is compared with a critical value from the chi-squared distribution table. Degrees of freedom (df) must be calculated: df = (number of rows – 1) × (number of columns – 1). To be significant, χ² must be equal to or greater than the critical value.

将计算所得的 χ² 与卡方分布表中的临界值比较。必须计算自由度 (df)：df = (行数 – 1) × (列数 – 1)。若要显著，χ² 必须等于或大于临界值。

10. Degrees of Freedom and Critical Values | 自由度与临界值

Degrees of freedom (df) represent the number of values in a calculation that are free to vary. Each inferential test uses df or sample size to determine the appropriate critical value.

自由度 (df) 表示计算中可以自由变化的数值个数。每种推断检验都使用 df 或样本大小来确定对应的临界值。

Chi-squared test of association: df = (r – 1)(c – 1); r = rows, c = columns
Sign test, Wilcoxon T, Mann-Whitney U, Spearman’s rho: these tests do not use df directly; you look up critical value using N (number of observations after exclusions) or n₁ and n₂.

卡方关联性检验：df = (r – 1)(c – 1)；r = 行数，c = 列数
符号检验、Wilcoxon T、Mann-Whitney U、Spearman rho：这些检验不直接使用 df；你需使用 N（排除后的观察数）或 n₁、n₂ 查表。

Always use a one-tailed test when the alternative hypothesis predicts a direction; use two-tailed when the hypothesis is non-directional. Ensure the correct table is selected and the chosen significance level is reported (normally 0.05).

当备择假设预测方向时，始终使用单尾检验；当假设无方向时，使用双尾检验。确保选择正确的表格并报告所选显著性水平（通常为 0.05）。

11. Reporting Results: APA Format Essentials | 结果报告：APA 格式要点

In Edexcel Psychology, you are expected to report findings in a clear, standardised manner. A complete report includes the test statistic value, degrees of freedom (if applicable), N, p value, and whether the result is significant. Provide an effect size where relevant.

在 Edexcel 心理学中，你需要以清晰且标准化的方式报告结果。一份完整的报告包括检验统计量值、自由度（如适用）、N、p 值，以及结果是否显著。相关时需提供效应大小。

Example for a Wilcoxon test: ‘A Wilcoxon signed-rank test revealed that the number of words recalled was significantly higher in the quiet condition compared to the noisy condition, T = 12, N = 20, p < 0.05 (one-tailed).' For chi-squared: χ² (1, N = 80) = 6.42, p < 0.05.

Wilcoxon 检验示例：“Wilcoxon 符号秩检验显示，安静条件下回忆单词的数量显著高于嘈杂条件，T = 12，N = 20，p < 0.05 (单尾)。”卡方示例：χ² (1, N = 80) = 6.42，p < 0.05。

Never state that you ‘accept the null hypothesis’ – instead, state that the result was non-significant and the null hypothesis was retained.

切勿声明“接受零假设”——应说明结果不显著，零假设被保留。

12. Central Limit Theorem and Test Assumptions | 中央极限定理与检验假设

The Central Limit Theorem (CLT) states that as sample size increases, the sampling distribution of the mean approaches a normal distribution, regardless of the shape of the population distribution. This theorem justifies the use of parametric tests with large samples.

中央极限定理 (CLT) 指出，随着样本量增大，平均值的抽样分布会趋近正态分布，无论总体分布形状如何。该定理为在大样本中使用参数检验提供了依据。

However, non-parametric tests used in Edexcel make fewer assumptions. Key assumptions include:

然而，Edexcel 使用的非参数检验假设较少。关键假设包括：

Level of measurement: Sign test requires nominal data at minimum; Wilcoxon and Mann-Whitney require ordinal; Spearman uses ordinal; Chi-squared uses nominal. Parametric tests such as the related t-test (not assessed but conceptually linked) require interval/ratio data.
Design: Use repeated measures tests for related designs, independent groups tests for unrelated designs.
Independence: Observations must be independent; each participant contributes only once.

测量水平：符号检验至少需名义数据；Wilcoxon 和 Mann-Whitney 需顺序数据；Spearman 使用顺序数据；卡方使用名义数据。参数检验如相关 t 检验（不直接考查但概念相关）需间隔/比率数据。
设计：对相关设计使用重复测量检验，对无关设计使用独立组检验。
独立性：观测值必须独立；每个参与者只参与一次。

Adhering to these assumptions ensures the test’s accuracy. The non-parametric nature of the Edexcel test battery makes them robust, but the correct choice of test remains critical.

遵守这些假设能确保检验的准确性。Edexcel 考试中这套非参数检验工具稳健性较强，但正确选择检验仍然至关重要。

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Year 13 Edexcel Psychology: Formula & Theorem Quick Reference Handbook | Year 13 Edexcel 心理学：公式定理速查手册